DIOPHANTINE GEOMETRY OVER GROUPS V:

QUANTIFIER ELIMINATION

Z. Sela^{1,2}

posted October 9, 2001

This paper is the fifth in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the fifth paper we use the iterative procedure that validates the correctness of an AE sentence defined over a free group, to show that the Boolean algebra of AE sets defined over a free group is invariant under projections, hence, implies that every elementary set defined over a free group is in the Boolean algebra of AE sets. The procedures presented in this paper enable us to answer affirmatively some of Tarski’s questions on the elementary theory of a free group in the next paper in this sequence.

In the first 4 papers in the sequence on Diophantine geometry over groups we studied sets of solutions to systems of equations in a free group, and developed basic techniques and objects required for the analysis of sentences and elementary sets defined over a free group. In the first paper in this sequence we studied sets of solutions to systems of equations defined over a free group and parametric families of such sets, and associated a canonical Makanin-Razborov diagram that encodes the entire set of solutions to the system. Later on we studied systems of equations with parameters, and with each such system we associated a (canonical) graded Makanin-Razborov diagram, that encodes the Makanin-Razborov diagrams of the systems of equations associated with each specialization of the defining parameters. In the second paper we generalized Merzlyakov’s theorem on the existence of a formal solution associated with a positive sentence. We first constructed a formal solution on a general AE sentence which is known to be true over some variety, and then presented formal limit groups and graded formal limit groups that enable us to collect and analyze the collection of all such formal solutions. In the third paper we studied the structure of exceptional solutions of a parametric system of equations. We proved the existence of a global bound (independent of the specialization of the defining parameters) on the number of rigid solutions of a rigid limit group, and a global bound on the number of strictly solid families of solutions of a solid limit group. Using these bounds we studied the stratification of the ”base” of the ”bundle” associated with the set of solutions of a parametric system of equations in a free group, and showed that the set of specializations of the defining parameters in each of the stratas is in the Boolean algebra of AE sets.

1Hebrew University, Jerusalem 91904, Israel.

2Partially supported by an Israel academy of sciences fellowship.

In the fourth paper we applied the structural results obtained in the first two papers in the sequence, to analyze AE sentences. Given a true sentence of the form:

∀y ∃x Σ(x, y, a) = 1 ∧ Ψ(x, y, a) ≠ 1

we presented an iterative procedure, that produces a sequence of varieties and formal solutions defined over them. Since in order to define the completions of a variety, and the closures of these completions, additional variables are required, the varieties produced along the iterative procedure are determined by larger and larger sets of variables, and so are the formal solutions defined over them. Still, by carefully analyze these varieties, and properly measure the complexity of Diophantine sets associated with them, we were able to show that certain complexity of the varieties produced along the procedure strictly decreases, which finally forces the iterative procedure to terminate after finitely many steps. The outcome of the terminating iterative procedure is a collection of varieties, together with a collection of formal solutions defined over them. The varieties are determined by the original universal variables y, and extra (auxiliary) variables. The collection of varieties gives a partition of the initial domain of the universal variables y, which is a power of the original free group of coefficients, into sets which are in the Boolean algebra of Diophantine sets, so that on each such set the sentence can be validated using a finite family of formal solutions. Hence, the outcome of the iterative procedure can be viewed as a stratification theorem that generalizes Merzlyakov’s theorem from positive sentences to general AE ones.

In this paper we apply the tools and techniques presented in the previous 4 papers in the sequence, to prove quantifier elimination in the elementary theory of a free group. In order to prove quantifier elimination we show that the Boolean algebra of AE sets is invariant under projections. The projection of a set that is in the Boolean algebra of AE sets, is naturally an EAE set, hence, to show that the Boolean algebra of AE sets is invariant under projections, we need to show that a general EAE set is in the Boolean algebra of AE sets. Let:

EAE(p) = ∃w ∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1)

be a general EAE set. The set EAE(p) is, by definition, the set of all specializations p_{0} of the defining parameters p, for which the corresponding sentence:

∃w ∀y ∃x (Σ_{1}(x, y, w, p_{0}, a) = 1 ∧ Ψ_{1}(x, y, w, p_{0}, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p_{0}, a) = 1 ∧ Ψ_{r}(x, y, w, p_{0}, a) ≠ 1)

is a truth sentence. Hence, we start the analysis of the set EAE(p), by analyzing the set of tuples (w_{0}, p_{0}), for which the corresponding sentence:

∀y ∃x (Σ_{1}(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ_{1}(x, y, w_{0}, p_{0}, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ_{r}(x, y, w_{0}, p_{0}, a) ≠ 1)

is a truth sentence.

By the procedure for validation of an AE sentence, presented in the fourth paper in this sequence, if (w_{0}, p_{0}) is a tuple for which the corresponding sentence is a truth sentence, then there exists a (finite) collection of varieties and formal solutions defined over those varieties, that prove the validity of the sentence. However, the finite collection of varieties and formal solutions may depend on the particular specialization (w_{0}, p_{0}). Therefore, our first goal in analyzing the collection of tuples (w_{0}, p_{0}) is obtaining a ”uniformization of proof systems”. To get such ”uniformization”, we present an iterative procedure, that is based on the iterative procedure for validation of an AE sentence, that starts with our given EAE predicate, and produces a finite tree for which in each vertex we place a variety, that is graded with respect to the variables w, p, and a family of formal solutions defined over this variety, so that if (w_{0}, p_{0}) is a specialization for which the corresponding sentence:

∀y ∃x (Σ_{1}(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ_{1}(x, y, w_{0}, p_{0}, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ_{r}(x, y, w_{0}, p_{0}, a) ≠ 1)

is a truth sentence, then there exists a proof of the sentence, that goes along a finite collection of paths in the finite tree associated with the set EAE(p). Hence, the finite tree associated with the set EAE(p) encodes proofs for all truth sentences associated with a tuple (w_{0}, p_{0}).

Constructing such a finite tree, to analyze the set EAE(p), we need to look for all specializations p_{0} of the defining parameters p, for which there exists a specialization w_{0} of the variables w, so that the sentence corresponding to the tuple (w_{0}, p_{0}), has a proof that goes along a collection of paths in the given finite tree. To carry out the analysis of this set of specializations p_{0} of the defining parameters p, we present an iterative procedure which we view as a ”sieve procedure”, that produces an increasing sequence of sets of specializations of the defining parameters p, that are all in the Boolean algebra of AE sets, and are all approximations of the set EAE(p), and in particular are all contained in the set EAE(p). The sieve procedure we present terminates after finitely many steps, and the set it produces in the step it terminates, is equal to the set EAE(p), and is in the Boolean algebra of AE sets, hence, the sieve procedure finally enables us to show that the set EAE(p) is in the Boolean algebra of AE sets.

Since the iterative procedures are rather involved, we prefered to present them first in a special case, which is conceptually and technically simpler, but it already demonstrates some of the principals used in the general case. Hence, in the first section we present the two iterative procedures used for the analysis of an EAE set in the minimal rank case, i.e., for predicates for which the limit groups involved in their analysis are all of minimal possible rank, i.e., limit groups that do not admit an epimorphism onto a free group so that the coefficient group is mapped onto a proper factor. In the second section we present the procedure for the construction of the tree of stratified sets, i.e., the tree associated with a general EAE set that encodes proofs for all tuples (w_{0}, p_{0}) for which the corresponding sentence is a truth sentence. In the third section we analyze few special cases, in which it is possible to slightly modify the procedure used for the construction of the tree of stratified sets to get a sieve procedure. In the fourth section we present core resolutions and some of their basic properties. Core resolutions seem to be a basic tool in analyzing projections of varieties defined over a free group, and play an essential role in the sieve procedure that is finally presented in the fifth section.

§1. The minimal (graded) rank case

To obtain quantifier elimination of elementary predicates over a free group, our goal is showing that the Boolean algebra of AE sets is invariant under projections. For presentation purposes, we will first present our approach to the analysis of the projection of the Boolean algebra of AE sets assuming the (graded) limit groups that appear in our procedure are of minimal (graded) rank, and then analyze the general case. We start with the following immediate fact.

Lemma 1.1. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let the EA set EA(w, p) be defined as:

EA(w, p) = ∃y ∀x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1).

Then the projection of EA(w, p), i.e., the set:

πEA(p) = ∃w (w, p) ∈ EA(w, p)

is an EA set.

Proof: πEA(p) is defined by the EA predicate:

πEA(p) = ∃w, y ∀x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …

… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1).
□

Unlike lemma 1.1 the analysis of projections of a general set in the Boolean algebra of AE sets requires the techniques and results obtained in our previous papers, graded limit groups and their Makanin-Razborov diagrams, the construction of formal solutions and graded formal limit groups and the analysis of iterated quotients of completed resolutions similar to the one presented in analyzing AE sentences presented in [Se4]. We start with the following reduction of our main goal from general sets in the Boolean algebra of AE sets to AE sets.

Lemma 1.2. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let V (w, p) be a set in the Boolean algebra of AE sets over F_{k}. If the projection of every AE set defined over F_{k} is in the Boolean algebra of AE sets over F_{k}, then the projection of V (w, p) is in the Boolean algebra of AE sets.

Proof: By ([Se3],3.6) the set V (w, p) is the (finite) union of some AE sets, EA sets and sets which are the intersection of an AE and an EA set. The projection of a union of sets is the union of the projections of the individual sets. By lemma 1.1 the projection of an EA set is in the Boolean algebra of EA sets and by our assumptions so is the projection of every AE set. Therefore, to prove the lemma we only need to show that under our assumption of the projections of AE sets, the projection of the intersection of an EA set and an AE set is in the Boolean algebra of AE sets.

Let V (w, p) be the intersection between an AE and an EA sets. V (w, p) is defined by the predicate:

V (w, p) = ∃w (∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1)) ∧
∧ (∃t ∀u (Σ′_{1}(u, t, w, p, a) = 1 ∧ Ψ′_{1}(u, t, w, p, a) ≠ 1) ∨ …
… ∨ (Σ′_{r′}(u, t, w, p, a) = 1 ∧ Ψ′_{r′}(u, t, w, p, a) ≠ 1)).

Equivalently:

V (w, p) = ∃w ∃t ∀y ∀u ∃x Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …

… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1) ∧
∧ (Σ′_{1}(u, t, w, p, a) = 1 ∧ Ψ′_{1}(u, t, w, p, a) ≠ 1) ∨ …
… ∨ (Σ′_{r′}(u, t, w, p, a) = 1 ∧ Ψ′_{r′}(u, t, w, p, a) ≠ 1).

So, V (w, p) is the projection of an AE set, which under our assumption implies that V (w, p) is in the Boolean algebra of AE sets.

□

Given lemmas 1.1 and 1.2, the analysis of the projection of the Boolean algebra of AE sets reduces to the analysis of the projection of AE sets.

Theorem 1.3. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let the AE set AE(w, p) be defined as:

AE(w, p) = ∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1).

Then the projection of the AE set AE(w, p), i.e., the set:

EAE(p) = ∃w ∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1)

is in the Boolean algebra of AE sets.

For presentation purposes we will first show that the projection of an AE set, EAE(p), is in the Boolean algebra of AE sets in the case r = 1. The generalization to arbitrary r is fairly straightforward, and is presented afterwards.

Theorem 1.4. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let EAE(p) be a set defined by the predicate:

EAE(p) = ∃w ∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1.

Then EAE(p) is in the Boolean algebra of AE sets.

Let F_{y} be the free group F_{y} =< y_{1}, …, y_{ℓ} >, and let:

ψ_{1}(x, y, w, p, a) = 1, …, ψ_{q}(x, y, w, p, a) = 1

be the defining equations of the system Ψ(x, y, w, p, a) = 1. By ([Se2],1.2) for every p_{0} ∈ EAE(p) there exists some w_{0} and a formal solution x = x_{w_{0},p_{0}}(y, a) so that the words corresponding to the defining equations of the system Σ(x_{w_{0},p_{0}}(y, a), y, w_{0}, p_{0}, a) are trivial in the free group F_{k} ∗ F_{y} =< a, y >, and the sentence:

∃y ψ_{1}(x_{w_{0},p_{0}}(y, a), y, w_{0}, p_{0}, a) ≠ 1 ∧ … ∧ ψ_{q}(x_{w_{0},p_{0}}(y, a), y, w_{0}, p_{0}, a) ≠ 1

is a truth sentence in F_{k}.

By the construction of graded formal limit groups presented in section 3 of [Se2], viewing W P =< w, p > as the parameter subgroup, one can associate with the free group F_{y} and the system of equations Σ(x, y, w, p, a) = 1 a (canonical) finite collection of graded formal limit groups GF L_{1}(x, y, w, p, a), …, GF L_{r}(x, y, w, p, a) so that every formal solution x = x_{w_{0},p_{0}}(y, a) of the system Σ(x, y, w, p, a) factors through one of the resolutions of the graded formal Makanin-Razborov diagram of one of the graded formal limit groups GF L_{1}(x, y, w, p, a), …, GF L_{r}(x, y, w, p, a).

Viewing W P =< w, p > as the parameters subgroup, each graded formal resolution in the graded formal Makanin-Razborov diagrams of the graded formal limit groups GF L_{1}(x, y, w, p, a), …, GF L_{r}(x, y, w, p, a) terminates in either a rigid formal limit group of the form W P Rgd(h_{R}, w, p, a) ∗ F_{y}, where W P Rgd(h_{R}, w, p, a) is a rigid (not formal!) graded limit group (with respect to W P), or in a solid formal limit group of the form W P Sld(h_{S}, w, p, a) ∗ F_{y}, where W P Sld(h_{S}, w, p, a) is a solid (not formal!) graded limit group. Note that by ([Se3],2.3), for each specialization (w_{0}, p_{0}) there exists a global bound on the number of rigid solutions of the form (h_{R}, w_{0}, p_{0}, a) of any of the rigid graded limit groups W P Rgd(h_{R}, w, p, a), and by ([Se3],2.11), for each specialization (w_{0}, p_{0}) there exists a global bound on the number of solid families of solutions of the form (h_{S}, w_{0}, p_{0}, a) of any of the solid graded limit groups W P Sld(h_{S}, w, p, a).

Let W P Rgd(h_{R}, w, p, a) be one of the terminating rigid graded limit groups in the formal graded Makanin-Razborov diagram with respect to the defining parameters W P =< w, p >. The modular groups associated with each graded formal resolution that terminates in a rigid specialization of the rigid graded limit group W P Rgd(h_{R}, w, p, a) are compatible with the graded formal resolution that terminates in the rigid graded formal limit group W P Rgd(h_{R}, w, p, a) ∗ F_{y}, i.e., the ”tower” of modular groups in such formal graded resolution is independent of the particular rigid specialization of W P Rgd(h_{R}, w, p, a). Therefore, using the ”tower” of modular groups that lie ”above” each of the rigid formal graded limit groups W P Rgd(h_{R}, w, p, a) ∗ F_{y}, we can associate a (usually infinite) system of equations (in the variables (h_{R}, y, w, p) and coefficients in F_{k}) corresponding to each of the equations in the system Ψ(x, y, w, p, a) = 1. By Guba’s theorem [Gu], each such infinite system of equations is equivalent to a finite system of equations λ_{R}(h_{R}, y, w, p, a) = 1. Similarly, with each terminating solid formal graded limit group W P Sld(h_{S}, w, p, a) ∗ F_{y} we can associate a system of equations λ_{S}(h_{S}, y, w, p, a) = 1.

From now on we work with each of the graded formal resolutions W P GF Res_{i}(x, y, w, p, a), and its terminating rigid formal graded limit group W P Rgd_{i}(h_{R}, w, p, a) ∗ F_{y}, or solid graded formal limit group W P Sld_{i}(h_{S}, w, p, a) ∗ F_{y} in parallel, so we may restrict our intention to one of these graded formal resolutions, omit its index and denote it W P GF Res(x, y, w, p, a). Note that each of these formal resolutions are with respect to the parameter subgroups W P =< w, p >.

Suppose that the graded formal resolution W P GF Res(x, y, w, p, a) terminates in the rigid graded formal limit group W P Rgd(h_{R}, w, p, a) ∗ F_{y}. Let:

λW P RGL_{1}(h_{R}, y, w, p, a), …, λW P RGL_{d}(h_{R}, y, w, p, a)

be the canonical collection of maximal graded limit groups (with respect to the parameter subgroup W P =< w, p >) corresponding to the set of specializations (h_{R}, y, w, p, a) for which (h_{R}, w, p, a) is a rigid specialization of W P Rgd(h_{R}, w, p, a), (h_{R}, y, w, p, a) factors through the graded formal resolution W P GF Res(x, y, w, p, a), and through one of the systems of equations, λ_{R}(h_{R}, y, w, p, a) = 1, associated with the various equations in the system Ψ(x, y, w, p, a) = 1. Similarly, if the graded formal resolution W P GF Res(x, y, w, p, a) terminates in the solid formal graded limit group W P Sld(h_{S}, w, p, a)∗F_{y}, we associate with the solid graded limit group W P Sld(h_{S}, w, p, a) and the system of equations λ_{S}(h_{S}, y, w, p, a) the canonical collection of maximal graded limit groups:

λW P SGL_{1}(h_{S}, y, w, p, a), …, λW P SGL_{d}(h_{S}, y, w, p, a)

corresponding to the set of specializations (h_{S}, y, w, p, a), for which (h_{S}, w, p, a) is a solid specialization of W P Sld(h_{S}, w, p, a), (h_{S}, y, w, p, a) factors through the graded formal resolution W P GF Res(x, y, w, p, a), and λ_{S}(h_{S}, y, w, p, a) = 1.

At this point we need to collect the ”remaining” set of specializations of the variables y for each value of our parameters (w, p). Suppose that the terminating graded limit group of the formal graded resolution W P GF Res(x, y, w, p, a) is the rigid formal graded limit group Rgd(h_{R}, w, p, a)∗F_{y} or the solid graded limit group Sld(h_{S}, w, p, a). With each of the graded limit groups:

λW P RGL_{1}(h_{R}, y, w, p, a), …, λW P RGL_{d}(h_{R}, y, w, p, a)

or:

λW P RGL_{1}(h_{S}, y, w, p, a), …, λW P RGL_{d}(h_{S}, y, w, p, a)

(depending on whether the graded formal resolution W P GF Res(x, y, w, p, a) terminates in a rigid or solid limit group with respect to W P =< w, p >) we associate its taut graded Makanin-Razborov diagram with respect to the parameter subgroup < h_{R}, w, p > or < h_{S}, w, p > in correspondence. Each graded resolution λW P GRes(y, h_{R}, w, p, a) (or λW P GRes(y, h_{S}, p, a)) in one of the taut graded diagrams of the graded limit groups: λW P RGL_{1}(h_{R}, y, w, p, a), …, λW P RGL_{d}(h_{R}, y, w, p, a) (or:

λW P RGL_{1}(h_{S}, y, w, p, a), …, λW P RGL_{d}(h_{S}, y, w, p, a))

terminates in either a rigid graded limit group (with respect to < h_{R}, w, p > or < h_{S}, p >) which we denote λW P Rgd(g_{R}, h_{R}, w, p, a) (or λW P Rgd(g_{R}, h_{S}, w, p, a)) or a solid graded limit group which we denote λW P Sld(g_{S}, h_{R}, w, p, a) (or

λW P Sld(g_{S}, h_{S}, w, p, a)).

Before continuing with our iterative procedure for analyzing the structure of an EAE set we use the limit groups constructed in the first step of the procedure to give a first approximation of an EAE set, an approximation which is an EA set. To get the approximation we need to use the entire collection of limit groups: W P Rgd_{i}(h_{R}, w, p, a), W P Sld_{i}(h_{S}, w, p, a) and:

λW P Rgd_{j}(g_{R}, h_{R}, w, p, a), λW P Sld_{j}(g_{S}, h_{R}, w, p, a), λW P Rgd_{j}(g_{R}, h_{S}, w, p, a),

λW P Sld_{j}(g_{S}, h_{S}, w, p, a).

Lemma 1.5. Let the set U_{1}(p) be defined as follows.

(1) p_{0} ∈ U_{1}^{R}(p) if for some w_{0} , some index i and some hr_{0} (hr_{0}, w_{0}, p_{0}) is a specialization of the rigid limit group W P Rgd_{i}(h_{R}, w, p, a) and for every index j, there exist no specialization gr_{0} for which (gr_{0}, hr_{0}, w_{0}, p_{0}) is a specialization of λW P Rgd_{j}(g_{R}, h_{R}, w, p, a) and no specialization gs_{0} for which (gs_{0}, hr_{0}, w_{0}, p_{0}) is a specialization of W P Sld_{j}(g_{S}, h_{R}, w, p, a).

(2) p_{0} ∈ U_{1}^{S}(p) if for some w_{0} , some index i and some hs_{0} (hs_{0}, w_{0}, p_{0}) is a specialization of the solid limit group W P Sld_{i}(h_{S}, w, p, a) and for every index j, there exist no specialization gr_{0} for which (gr_{0}, hs_{0}, w_{0}, p_{0}) is a specialization of λW P Rgd_{j}(g_{R}, h_{S}, w, p, a) and no specialization gs_{0} for which (gs_{0}, hs_{0}, w_{0}, p_{0}) is a specialization of W P Sld_{j}(g_{S}, h_{S}, w, p, a).

(3) We define U_{1}(p) to be U_{1}(p) = U_{1}^{R}(p) ∪ U_{1}^{S}(p). Then:

(i) U_{1}(p) ⊂ EAE(p).

(ii) U_{1}(p) is an EA set.

Proof: Both conditions (1) and (2) can be stated by an EA predicate, so U_{1}(p) is indeed an EA set. If w_{0} satisfies either (1) or (2), then the sentence:

∀y∃x Σ(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ(x, y, w_{0}, p_{0}, a) ≠ 1

is a truth sentence by construction, so U_{1}(p) ⊂ EAE(p).

□

To construct the graded formal resolutions GF Res(x, y, w, p, a), we have collected all the formal solutions x_{(w,p)}(y, a), for which all the words corresponding to the equations in Σ(x_{(w,p)}(y, a), y, w, p, a) represent the trivial words in F_{a,y} =< a, y >. By theorem 1.18 of [Se2], if p_{0} ∈ EAE(p) then there must exist some ”witness” w_{0} and a formal solution x_{(w_{0},p_{0})}(y, a) so that the maximal limit groups corresponding to each of the equations in the system Ψ(x_{(w_{0},p_{0})}(y, a), y, w, p, a) = 1 are all proper quotients of the free group < a, y > = F_{k} ∗ F_{y}. Hence, for every p_{0} ∈ EAE(p) there must exist some witness w_{0}, and a rigid specialization (hr_{0}, w_{0}, p_{0}) of one of the rigid limit groups W P Rgd(h_{R}, w, p, a), or a solid specialization (hs_{0}, w_{0}, p_{0}) of one of the solid limit groups W P Sld(h_{S}, w, p, a), so that every ungraded resolution W P GRes(y, hr_{0}, w_{0}, p_{0}, a) (or W P GRes(y, hs_{0}, w_{0}, p_{0}, a)) does not correspond to the entire set of y’s but rather to a resolution of a limit group which is a proper quotient of the free group < a, y > = F_{k} ∗ F_{y}.

Therefore, the outcome of the first step of our ”trial and error” procedure is a decrease in the complexity (definitions 1.15 and 3.2 in [Se4]) of the ungraded resolutions of (the remaining) y’s associated with each p_{0} ∈ EAE(p), at least for one rigid or solid specialization (hr_{0}, w_{0}, p_{0}), or (hs_{0}, w_{0}, p_{0}). Each of the next steps of the procedure is meant to sequentially decrease the complexity of the ungraded resolutions of the remaining y’s. Once the iterative procedure terminates, we present a second iterative procedure, that uses the outcome of the first iterative procedure to sequentially approximate the set EAE(p) by sets which are all in the Boolean algebra of AE sets. Finally, we show that the approximations we construct in the second iterative procedure, become identical with the set EAE(p) after finitely many steps. Since the approximations are all in the Boolean algebra of AE sets, this will imply that an EAE set is indeed in the Boolean algebra of AE sets, which finally proves theorem 1.4.

In this section we present the iterative ”trial and error” procedure, and the proof of theorem 1.4 under the minimal (graded) rank assumption. i.e., from now on we will assume that if any of the graded limit groups λW P GL_{j}(y, h_{R}, w, p, a) or λW P GL_{j}(y, h_{S}, w, p, a) admit an epimorphism τ onto a free group F_{k} ∗ F where τ(< p >) < F_{k}, then F is the trivial group. Under the minimal graded rank assumption, the termination of the ”trial and error” procedure is based on the analysis of iterative quotients of completed resolutions in the minimal rank case, presented in section 1 of [Se4]. In the next sections we will use the analysis of iterative quotients of completed resolutions in the general case presented in section 2 of [Se4], to get a ”trial and error” procedure, hence, a proof of theorem 1.4, omitting the minimal graded rank assumption.

For the continuation of the iterative procedure we will denote (for brevity) each of the limit groups W P Rgd(h_{R}, w, p, a) or W P Sld(h_{S}, w, p, a) as W P H(h, w, p, a), and each of the limit groups:

λW P Rgd(g_{R}, h_{R}, w, p, a), λW P Rgd(g_{R}, h_{S}, w, p, a), λW P Sld(g_{S}, h_{R}, w, p, a),

λW P Sld(g_{S}, h_{S}, w, p, a)

as W P HG(g, h, w, p, a). Our treatment of these limit groups will be conducted in parallel, so we don’t keep the indices associated with each of these (finite collection of) limit groups. Also, the rest of our ”trail and error” procedure does not depend in an essential way on the type (rigid or solid) of the terminating graded limit groups in the first two steps, hence, we do not keep notation for the type of each of these terminating limit groups.

For each tuple (p_{0}, w_{0}, h_{0}, g_{0}), which is either a rigid or a solid specialization of the terminating limit group W P HG, there is an ungraded (well-structured) resolution associated. The associated ungraded resolution depends only on the solid family of the specialization in case the corresponding terminating limit group is solid. Also, the ungraded resolution may be degenerate, so we separate the finitely many possible types of ungraded resolutions associated with a rigid or solid specialization (p_{0}, w_{0}, h_{0}, g_{0}) of W P GH according to the stratas of the corresponding singular locus (see section 12 of [Se1]), and continue with each singular strata separately. With the associated ungraded well-structured resolution, we may associate its completion. Given a tuple (p_{0}, w_{0}, h_{0}, g_{0}), which is a rigid or a solid specialization of W P HG we collect all the formal solutions {x_{(p_{0},w_{0},h_{0},g_{0})}(s, z, y, a)} for which the words corresponding to the equations in the system Σ(x_{(p_{0},h_{0},g_{0})}(s, z, y, a), y, w_{0}, p_{0}, a) are the trivial words in some closure of the completion of the ungraded resolution associated with the given specialization. Using the construction presented in section 3 of [Se2], and viewing the subgroup < p, w, h > as parameters, from the entire collection of formal solutions for all possible specializations (p_{0}, w_{0}, h_{0}, g_{0}) which are rigid or solid specializations of W P HG we can construct a graded formal Makanin-Razborov diagram, so that any formal solution defined over a closure of an ungraded resolution associated with a rigid or solid specialization of W P HG, factors through one of the graded formal Makanin-Razborov resolutions. Note that by the construction of formal graded limit groups and their associated graded formal Makanin-Razborov diagrams, the collection of maximal graded formal limit groups and their graded formal Makanin-Razborov diagrams associated with a given terminating limit group W P HG, depends only on the strictly solid family from which the specialization of W P HG is taken, and not on the particular specialization taken from this strictly solid family.

Let GF L_{1}(x, z, y, g_{1}, h_{1}, w, p, a), …, GF L_{r}(x, z, y, g_{1}, h_{1}, w, p, a) be the maximal graded formal limit groups constructed from the collection of formal solutions associated with the graded limit group W P HG. By section 3 of [Se2], with each of the graded formal limit groups there is an associated graded formal Makanin-Razborov diagram with respect to the parameter subgroup < g_{1}, h_{1}, w, p >. By theorems 3.7 and 3.8 in [Se2], each of the graded formal resolutions in the graded formal Makanin-Razborov diagram associated with each of the graded formal limit groups: GF L_{1}(x, z, y, g_{1}, h_{1}, w, p, a), …, GF L_{r}(x, z, y, g_{1}, h_{1}, w, p, a) (with respect to the subgroup < g_{1}, h_{1}, w, p >) terminates in either a group of the form:

W P HGRgd(h_{2}^{R}, g_{1}, h_{1}, w, p, a) , ∗_{<r,g_{1},h_{1},w,p,a>} GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

where W P HGRgd(h_{2}^{R}, g_{1}, h_{1}, w, p, a) is a graded (not formal!) limit group which is rigid with respect to the parameter subgroup < g_{1}, h_{1}, w, p >, and:

GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

is a graded formal closure of the graded resolution W P HGF Res(y, g_{1}, h_{1}, w, p, a) associated with the graded limit group W P HG. We will denote a terminating rigid graded limit group W P HGRgd(h_{2}^{R}, g_{1}, h_{1}, w, p, a) by W P HGH^{R}. Alternatively, the terminating graded formal limit group of a graded formal resolution in one of the graded formal Makanin-Razborov diagrams associated with the graded formal limit groups:

GF L_{1}(x, z, y, g_{1}, h_{1}, w, p, a), …, GF L_{r}(x, z, y, g_{1}, h_{1}, w, p, a)

is of the form:

W P HGSld(h_{2}^{S}, g_{1}, h_{1}, w, p, a) ∗_{<r,g_{1},h_{1},w,p,a>} GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

where W P HGSld(h_{2}^{S}, g_{1}, h_{1}, w, p, a) is a graded (not formal!) limit group which is solid with respect to the parameter subgroup < g_{1}, h_{1}, w, p >, and:

GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

is a graded formal closure of the graded resolution W P HGF Res(y, g_{1}, h_{1}, w, p, a) associated with the graded limit group W P HG. We will denote the terminating solid graded limit group W P HGSld(h_{2}^{S}, g_{1}, h_{1}, w, p, a) by W HGH^{S}.

At this point we continue as in the first step, i.e., we analyze the set of y’s for which for all the formal solutions x = x(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), at least one of the equations in the system Ψ(x(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), y, w, p, a) = 1 holds. Since we have assumed that all the limit groups W P HG are of minimal (graded) rank, we will be able to use the analysis of quotient resolutions presented in section 1 of [Se4], to analyze the resolutions containing the entire set of the remaining y’s. In the next section we will modify this part of the iterative procedure in order to prove theorem 1.4 in the general case, using the analysis of quotient resolutions that appears in sections 2-5 of [Se4].

Let:

W P HGRgd(h_{2}^{R}, g_{1}, h_{1}, w, p, a) ∗_{<r,g_{1},h_{1},w,p,a>} GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

or:

W P HGSld(h_{2}^{S}, g_{1}, h_{1}, w, p, a) ∗_{<r,g_{1},h_{1},w,p,a>} GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

be the terminating rigid or solid limit group (with respect to the parameter subgroup < g_{1}, h_{1}, w, p >) of the graded formal resolution W P HGF Res(x, s, z, y, g_{1}, h_{1}, w, p, a). As we pointed out in the first step, the (formal) modular groups associated with each ungraded formal resolution that is covered by the graded formal resolution:

W P HGF Res(x, s, z, y, g_{1}, h_{1}, w, p, a)

that terminates in the rigid or solid graded limit group W P HGH, so that the rigid or solid specialization of W P HGH in which the ungraded resolution terminates does not belong to the singular locus, are compatible with the graded formal modular groups of the graded formal resolution W P HGF Res. Therefore, using the ”tower” of graded formal modular groups associated with the graded formal resolution W P HGF Res, a ”tower” that lies ”above” the terminating rigid or solid graded limit group W P HGH, we can associate a (usually infinite) system of equations (in the variables (s, z, y, h_{2}, g_{1}, h_{1}, w, p) and coefficients in F_{k}) corresponding to each of the equations in the system Ψ(x, y, p, a) = 1, imposed on all specializations of the tuple (x, y, p, a) in the same (formal) modular block as a given specialization of the tuple (h_{2}, g_{1}, h_{1}, w, p, a). By Guba’s theorem [Gu] this infinite system of equations is equivalent to a finite system of equations β(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) = 1.

At this stage we need to collect all the remaining y’s, i.e., those values of the variables y that satisfy one of the systems of equations β(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) =

1. As we did in our iterative procedure for validation of a sentence in section 1 of [Se4], our aim is to collect all the remaining y’s for all p_{0} ∈ EAE(p), and all possible tuples (w_{0}, p_{0}), in finitely many graded resolutions which will be of complexity which is not bigger than the corresponding graded resolution W P HRes(y, h_{1}, w, p, a). To be able to collect all the remaining y’s in graded resolutions of not bigger complexity we need to apply the same techniques used for that purpose in sections 1 of [Se4], modified slightly to be suitable for the graded set-up.

The collection of all specializations of the variables y that can be extended to a (shortest form) specialization that factors through the graded formal resolution W P HGF Res(x, s, z, y, g_{1}, h_{1}, w, p, a), and satisfies the system of equations β(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) = 1 factors through a canonical collection of maximal graded limit groups:

QGlim_{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), …, QGlim_{q}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

Let W P HRes(y, h_{1}, w, p, a) be the graded resolution (with respect to the parameter subgroup < h_{1}, w, p, a >) of y’s that remained after the first step we have started the second step with. Our analysis of the well-structured resolutions of the graded limit groups:

QGlim_{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), …, QGlim_{q}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)

is conducted in parallel, hence, we will omit the index and through the continuation denote the quotient graded limit group under consideration QGlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

Let z_{base} be the union of generating sets of the (graded) limit groups associated with all the levels of the graded formal closure W P HGF Cl(s, z, y, c, g_{1}, h_{1}, w, p, a) except the top level. We will call this union of generators together with the elements h_{2}, g_{1}, h_{1}, w and p the basis of the graded formal closure W P HGF Cl(s, z, y, c, g_{1}, h_{1}, p, a). Following the construction of the strict Makanin-Razborov diagram ([Se2], 1.10), we construct the (canonical) strict graded Makanin-Razborov diagram of the graded limit group QGlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) viewed as a graded limit group with respect to the parameter subgroup < z_{base}, h_{2}, g_{1}, h_{1}, w, p >. Let:

W P HGHRes_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), …, W P HGHRes_{v}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

be the well-structured graded Makanin-Razborov resolutions that appear in the strict graded Makanin-Razborov diagram of the (graded) limit group

QRlim(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

with respect to the parameter subgroup < z_{base}, h_{2}, g_{1}, h_{1}, w, p >, where each graded resolution is terminating in either a rigid or a solid graded limit group (with respect to the parameter subgroup < z_{base}, h_{2}, g_{1}, h_{1}, w, p >). We will treat the graded resolutions:

W P HGHRes_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), …, W P HGHRes_{v}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

in parallel, so for the continuation we will restrict ourselves to one of them which we denote W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) for brevity. Let

W P HGHRlim(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

be the graded limit group corresponding to the graded resolution:

GRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a).

Let Glim_{j}(s, z, y, z_{base}, h_{2}, g_{2}, h_{1}, w, p, a) be a graded limit group that appears in the j-th level of the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a). Naturally, there exists a canonical map:

τ_{j} : W P HRlim(y, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a). Let Λ_{j} be the graded quadratic decomposition of Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), i.e., the graded cyclic decomposition of Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) obtained from the graded abelian JSJ decomposition of Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) by collapsing all the edges connecting two non-QH subgroups. Let Q be a quadratically hanging subgroup in the graded abelian JSJ decomposition of W P HRlim(y, h_{1}, w, p, a) and let S be the corresponding (punctured) surface. Since the boundary elements of Q are mapped by τ_{j} to either the trivial element or elliptic elements in Λ_{j}, the (possibly trivial) cyclic decomposition inherited by τ_{j}(Q) from the cyclic decomposition Λ_{j}, can be lifted to a (possibly trivial) cyclic decomposition of the QH-subgroup Q of W P HRlim(y, h_{1}, w, p, a), which corresponds to some decomposition of the (punctured) surface S along a (possibly trivial) collection of disjoint non-homotopic s.c.c. . Let Γ_{j}(Q) be the corresponding cyclic decomposition of the QH subgroup Q, and let Γ_{j}(S) be a maximal associated collection of non-homotopic essential s.c.c. on S. Note that by construction every s.c.c. from the defining collection of Γ_{j}(S) is mapped by τ_{j} to either a trivial element or to an elliptic element in Λ_{j}.

Lemma 1.6.

(i) every s.c.c. from the collection Γ_{j}(S) is mapped by τ_{j} to either the trivial element or to a non-trivial elliptic element in Λ_{j}.

(ii) every non-separating s.c.c. on the surface S is mapped to a non-trivial ele-

ment by the homomorphism τ_{j}. (iii) Let Q′ be a quadratically hanging subgroup in Λ_{j}, and let S′ be the corre-

sponding (punctured) surface. If τ_{j} maps non-trivially a connected subsurface of S \ Γ_{j}(S) into Q′ then genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|. Furthermore, in this case τ_{j} maps the fundamental group of a subsurface of S into a finite index subgroup of Q′.

Proof: Identical to the proof of lemma 1.4 in [Se4].

□

Part (iii) of lemma 1.6 bounds the topological complexity of those QH subgroups Q′ that appear in the graded abelian JSJ decompositions associated with the various levels of the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) into which a QH subgroup Q that appears in the graded abelian JSJ decomposition of W P HRlim(y, h_{1}, w, p, a) is mapped non-trivially. To show that lemma 1.6 can be applied to bound the topological complexity of all the QH subgroups Q′ that appear in the graded abelian decompositions associated with the various levels of W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) we need the following proposition.

Proposition 1.7. Let Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) be the graded limit group that appears in the j-th level of the graded resolution:

W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) and let Q′ be a QH-subgroup that appears in the graded JSJ decomposition associated with Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) and S′ be its corresponding punctured surface. Let τ_{j} be the natural map:

τ_{j} : W P HRlim(y, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a).

Then there exists a QH subgroup Q in the graded abelian JSJ decomposition of W P HRlim(y, h_{1}, w, p, a) with corresponding punctured surface S, so that a subsurface S_{1} of the punctured surface S is mapped by τ_{j} into a finite index subgroup of a conjugate of Q′. In particular: genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|.

Proof: Identical with the proof of proposition 1.6 in [Se4].

□

Definition 1.8. Let Q be a quadratically hanging subgroup in the graded abelian JSJ decomposition of W P HRlim(y, h_{1}, w, p, a), and let S be its corresponding (punctured) surface. The QH subgroup Q (and the corresponding surface S) is called surviving if for some level j, there exists some quadratically hanging subgroup Q′ in Λ_{j}, the graded abelian JSJ decomposition of Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), with corresponding surface S′, so that τ_{j} maps Q non-trivially into Q′, genus(S′) = genus(S) and χ(S′) = χ(S).

By definition, if Q is a non-surviving QH subgroup in the graded abelian JSJ decomposition of W P HRlim(y, h_{1}, w, p, a), then every QH subgroup Q′ in any level of the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) into which a subsurface of Q is mapped non-trivially, has either a strictly lower genus or a strictly smaller Euler characteristic than that of the QH subgroup Q. This would eventually ”force” the complexity of the graded resolution:

W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

to be bounded by the complexity of the resolution W P HRes(y, h_{1}, w, p, a) once one is able to ”isolate” the surviving surfaces. This is the purpose of the following theorem.

Theorem 1.9. Let Q_{1}, …, Q_{r} be the surviving QH subgroups in the graded abelian JSJ decomposition of W P HRlim(y, h_{1}, w, p, a). Then the graded resolution:

W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

can be replaced by finitely many graded resolutions, each composed from two consecutive parts. The first part is a graded resolution of QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the parameter subgroup < z_{base}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r} >, which we denote W P HGHRes(s, z, y, (z_{base}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r}), a). The second part is a one step resolution that maps the rigid (solid) terminal graded limit group of W P HGHRes(s, z, y, (z_{base}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r}), a) to the rigid (solid) terminal graded limit group of the resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a). The two consecutive parts of the graded resolution have the following properties:

(1) the graded decomposition corresponding to the second part of the resolution contains a vertex stabilized by the terminal rigid (solid) graded limit group of the resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) connected to r′ surviving QH subgroups Q_{i_{1}}, …, Q_{i_{r′}}, for some r′ ≤ r, and 1 ≤ i_{1} < i_{2} < … < i_{r′} ≤ r.

(2) if the terminal graded limit group of the graded resolution:

W P HGHRes(s, z, y, (z_{base}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r}), a)

is rigid (solid) so is the terminal graded limit group of the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a). Furthermore, if they are both solid, their principal graded decompositions are in one to one correspondence. i.e., the decomposition differ only in the stabilizer of one vertex, the vertex stabilized by < z_{base}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r} > in the first part, and by < z_{base}, h_{2}, g_{1}, h_{1}, w, p > in the second part.

Proof: Identical to the proof of theorem 1.8 of [Se4].

□

We continue the analysis of the limit group QGlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) by reducing the set of defining parameters sequentially. Recall that in order to obtain the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) we used the subgroup < z_{base}, h_{2}, g_{1}, h_{1}, w, p > as the parameter subgroup, where z_{base} were the union of generating sets of the limit groups that appear in all levels of the graded formal closure W P HGHF Cl(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) except those defining the highest level. For the next step of the analysis we take z_{base}^{2} as the defining parameters, where z_{base}^{2} form the subset of the variables z_{base} that generate the limit groups that appear in all the levels of the graded formal closure W P HGHF Cl(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) except for the two highest levels. Let T_{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) be the terminal rigid or solid graded limit group in the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the parameter subgroup < z_{base}, h_{2}, g_{1}, h_{1}, w, p >. From the collection of rigid (solid) specializations of T_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), that are obtained (using our shortening procedure) from shortest form specializations that factor through the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), we construct the graded strict Makanin-Razborov diagram of T_{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), viewed as a graded limit group with respect to the parameter subgroup < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p >. Let:

W P HGHRes_{1}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a), …, W P HGHRes_{m}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

be the resolutions that appear in the strict graded Makanin-Razborov diagram of the (graded) limit group T_{1}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the parameter subgroup < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p >, where each graded resolution is terminating in either a rigid or a solid graded limit group (with respect to the parameter subgroup < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p >).

We will treat the graded resolutions W P HGHRes_{i}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) in parallel, so for the continuation we will restrict ourselves to one of them which we denote W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) for brevity. Let

Glim(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

be the graded limit group corresponding to the graded resolution

W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a).

If the subgroup Glim(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of the subgroup T_{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) we need to modify the graded resolution:

W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

so that it becomes a strict graded Makanin-Razborov resolution terminating with the limit group Glim(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a). Note that by modifying the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), we may need to replace the quotient limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) or one of the groups Glim_{j}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) that appear in one of the levels of the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) by a proper quotient of itself.

Let Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) be a graded limit group that appears in the j-th level of the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a). Let Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) be the limit group generated by < z_{base}, h_{2}, g_{1}, h_{1}, w, p, a > in the graded formal closure W P HGHF Cl(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). Naturally, there exists a canonical map:

τ_{j} : Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a).

Let Λ_{j} be the graded quadratic decomposition of Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a), i.e., the graded cyclic decomposition of Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) obtained from the abelian graded JSJ decomposition of Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) by collapsing all the edges connecting two non-QH subgroups. Let Q be a quadratically hanging subgroup in the abelian graded JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) and let S be the corresponding (punctured) surface. Since the boundary elements of Q are mapped by τ_{j} to elliptic elements in Λ_{j}, the (possibly trivial) cyclic decomposition inherited by τ_{j}(Q) from the cyclic decomposition Λ_{j} can be lifted to a (possibly trivial) cyclic decomposition of the QH subgroup Q of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), which corresponds to some decomposition of the (punctured) surface S along a (possibly trivial) collection of disjoint non-homotopic s.c.c. Let Γ_{j}(Q) be the corresponding cyclic decomposition of the QH subgroup Q, and let Γ_{j}(S) be a maximal associated collection of non-homotopic essential s.c.c. on S. Note that by construction every s.c.c. from the defining collection of Γ_{j}(S) is mapped by τ_{j} to either a trivial element or to an elliptic element in Λ_{j}.

Lemma 1.10.

(i) every s.c.c. from the collection Γ_{j}(S) is mapped by τ_{j} to either the trivial element or to a non-trivial elliptic element in Λ_{j}.

(ii) every non-separating s.c.c. on the surface S is mapped to a non-trivial ele-

ment by the homomorphism τ_{j}. (iii) Let Q′ be a quadratically hanging subgroup in Λ_{j}, and let S′ be the corre-

sponding (punctured) surface. If τ_{j} maps non-trivially a connected subsurface of S \ Γ_{j}(S) into Q′ then genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|. Furthermore, in this case τ_{j} maps the fundamental group of a subsurface of S into a finite index subgroup of Q′.

Proof: Identical to the proof of lemma 1.4 in [Se4].

□

The precise statement of proposition 1.7 is not valid for the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a). If Q′ is a QH subgroup that appears in an abelian decomposition associated with one of the levels of the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a), then it is not true that there exists some QH subgroup Q in the principal graded JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), so that the fundamental group of a subsurface of the punctured surface corresponding to Q is mapped non-trivially into Q′. However, we can still define surviving surfaces.

Definition 1.11. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), and let S be its corresponding (punctured) surface. The QH subgroup Q (and the corresponding surface S) is called surviving if for some level j, there exists some quadratically hanging subgroup Q′ in Λ_{j}, the JSJ decomposition of Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a), with corresponding surface S′, so that τ_{j} maps Q non-trivially into Q′, genus(S′) = genus(S) and χ(S′) = χ(S).

To control the ”complexity” of the graded resolution:

W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

we need to ”isolate” the surviving QH subgroups. This can be done in a similar way to theorem 1.9.

Theorem 1.12. Let Q_{1}, …, Q_{r} be the surviving QH subgroups in the JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a). Then the graded resolution:

W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

can be replaced by finitely many graded resolutions, each composed from two consecutive parts. The first part is a graded resolution of T_{1}(s, z, y, a) with respect to the parameter subgroup < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r} >, which we denote W P HGHRes(s, z, y, (z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r}), a). The second part is a one step resolution that maps the rigid (solid) terminal graded limit group of

W P HGHRes(s, z, y, (z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r}), a)

to the rigid (solid) terminal graded limit group of the resolution

W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a).

The two consecutive parts of the graded resolution have the following properties:

(1) the graded decomposition corresponding to the second part of the resolution contains a vertex stabilized by the terminal rigid (solid) graded limit group of the resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) connected to the r′ surviving QH subgroups Q_{i_{1}}, …, Q_{i_{r′}}, for some r′ ≤ r, and 1 ≤ i_{1} < i_{2} < … < i_{r′} ≤ r.

(2) if the terminal graded limit group of the graded resolution:

W P HGHRes(s, z, y, (z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r}), a) is rigid (solid) so is the terminal graded limit group of the graded resolution GRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a). Furthermore, if they are both solid, their principal graded decompositions are in one to one correspondence. i.e., the decomposition differ only in the stabilizer of one vertex, the vertex stabilized by < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, Q_{1}, …, Q_{r} > in the first part, and by the subgroup < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p > in the second part.

Proof: identical to the proof of theorem 1.8 in [Se4].

□

Given lemma 1.10 and theorem 1.12, to complete the analysis of the structure of the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) we still need an appropriate analogue of proposition 1.7, i.e., we need to associate every QH subgroup that appears in one of the abelian graded JSJ decompositions of the different levels of the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) with a QH subgroup of either the graded JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) or a QH subgroup that appears in the abelian graded JSJ decomposition of the terminal subgroup T_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) of the graded resolution:

W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a)

in case T_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) is solid. We divide the final analysis of W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) into two cases depending on

T_{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)

being rigid or solid.

Proposition 1.13. Suppose that the terminal subgroup T_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) of the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) is rigid. Let Q′ be a QH subgroup that appears in the j-th level abelian graded JSJ decomposition of the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a), and let S′ be its corresponding surface. Then:

(i) there exists a QH subgroup Q in the graded abelian JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), with an associated surface S, so that the map τ_{j} : Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) maps the fundamental group of a subsurface of S onto a subgroup of finite index of Q′.

(ii) genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|.

(iii) genus(S′) = genus(S) and |χ(S′)| = |χ(S)| if and only if Q is a surviving QH subgroup. In this last case, j is the bottom level of the graded resolution GRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a), and Q is mapped to the vertex stabilized by < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a > by the maps:

τ_{j} : Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

in all the levels above the bottom one.

Proof: Identical to the proof of proposition 1.12 in [Se4].

□

Proposition 1.14. Suppose that the terminal subgroup T_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) of the graded resolution W P HGHRes(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) is solid. Let Q′ be a QH subgroup that appears in the j-th level abelian graded JSJ decomposition of the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a). Then:

(i) there exists a QH subgroup Q with an associated surface S, which is either:

1) a QH subgroup in the (graded) JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), so that the map:

τ_{j} : Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

maps the fundamental group of a subsurface of S onto a subgroup of finite index of Q′.

2) a QH subgroup in the (graded) JSJ decomposition of (the solid graded limit group) T_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a), so that the map:

η_{j} : T_{1}(s, z, y, z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

maps the fundamental group of a subsurface of S onto a subgroup of finite index of Q′.

(ii) genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|.

(iii) if Q is a QH subgroup in the JSJ decomposition of Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) and genus(S′) = genus(S) and |χ(S′)| = |χ(S)| then Q is a surviving QH subgroup. In this last case, j is the bottom level of the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a), and Q is mapped to the vertex stabilized by < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a > by the maps:

τ_{j} : Zlim(z_{base}, h_{2}, g_{1}, h_{1}, w, p, a) → Glim_{j}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

in all the levels above the bottom one.

Proof: Identical to the proof of proposition 1.13 in [Se4].

□

We continue the analysis of the limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) by further reducing the group of parameters sequentially. Recall that in order to obtain the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) we first used the subgroup < z_{base}, h_{2}, g_{1}, h_{1}, w, p > as the parameter subgroup, where z_{base} were a union of generating sets of limit groups that appear in all the levels of the graded formal closure W P HGHF Cl(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) except those defining the highest level, and then used the subgroup < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p > as the parameter subgroup, where z_{base}^{2} form the subset of elements z_{base} that generate the limit groups that appear in all the levels of the graded formal closure W P HGHF Cl(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) except for the two highest levels. To continue the analysis of the resolutions of the limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), we set the variables z_{base}^{ℓ} to be the union of generating sets of the limit groups that appear in all levels of the graded formal closure W P HGHF Cl(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) except for the ℓ highest levels. Let T_{2}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) be the terminal rigid or solid graded limit group in the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the parameter subgroup < z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p >. We continue the graded resolution W P HGHRes(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a) by viewing

T_{2}(s, z, y, z_{base}^{2}, h_{2}, g_{1}, h_{1}, w, p, a)

as a graded limit group with respect to the parameter subgroup < z_{base}^{3}, h_{2}, g_{1}, h_{1}, w, p >, the obtained terminal rigid or solid graded limit group as a graded limit group with respect to the parameter subgroup < z_{base}^{4}, h_{2}, g_{1}, h_{1}, w, p > and so on, until we exclude all the z variables that appear in the different levels of the graded formal closure W P HGHF Cl(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). Clearly, lemma 1.10 and theorem 1.12 remain valid for all the steps of the procedure. Note that the final resolution we obtain terminates in a graded limit group which is rigid or solid with respect to the parameter subgroup < h_{2}, g_{1}, h_{1}, w, p >. We denote this terminating limit group T (s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). Also, note that the described procedure produces (canonically) finitely many such resolutions of the limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) or of a proper quotient of it.

To get the graded resolutions W P HGHRes(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) we have considered the subgroup < h_{2}, g_{1}, h_{1}, w, p > as the parameter subgroup. To complete the analysis of the set of the remaining y’s we need to further decrease the parameter subgroup. If the graded resolution W P H(y, h_{1}, w, p, a) terminates in a rigid graded limit group (with respect to the parameter subgroup < h_{1}, w, p >), then we end the procedure for the construction of the quotient resolutions with the graded resolution W P HGHRes(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). If the graded resolution W P H(y, h_{1}, w, p, a) terminates in a solid limit group (with respect to < h_{1}, w, p >, we denote by < ĥ_{2}, ĝ_{1} > the various vertex and edge groups in the terminating graded abelian decomposition of the terminating limit group of the graded resolution W P HG(s, z, y, g_{1}, h_{1}, w, p, a) (with respect to < h_{1}, w, p >), W P HGH(h_{2}, g_{1}, h_{1}, w, p, a), excluding the distinguished vertex stabilized by < h_{1}, w, p >. We continue the resolution W P HGH(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), by analyzing the multi-graded resolutions of the terminating subgroup T (s, z, y, h_{2}, g_{1}, w, p, a) with respect to the parameter subgroup < h_{1}, w, p > and the subgroups < ĥ_{2}, ĝ_{1} >. We denote these multi graded resolutions:

T MGRes_{1}(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a), …, T MGRes_{d}(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a).

The analysis of the various levels in the multi-graded resolutions T MGRes_{i} is identical to the analysis of the various levels in the resolution W P HGH(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

With each of the multi-graded resolutions T MGRes_{i}(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a) we associate a multi-graded resolution composed from the graded resolution

W P HGH(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)

followed by the given multi-graded resolution T MGRes_{i}. We denote the obtained multi-graded resolution: W P HGH(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a), and its terminal multigraded limit group W P HGHG(g_{2}, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a). With W P HGHG(g_{2}, h_{1}, w, p, ĥ_{1}, ĝ_{1}, a) we naturally associate its multi-graded abelian JSJ decomposition with respect to the parameter subgroup < h_{1}, w, p > and the subgroups < ĥ_{2}, ĝ_{1} >. As in a general graded or multi-graded resolution, we say that a specialization factors through the multi-graded resolution W P HGH(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a), if it is obtained from a rigid or solid specialization of the terminal multi-graded limit group W P HGHG(g_{2}, h_{1}, w, p, a, ĥ_{2}, ĝ_{1}, a) using elements of the multi-graded modular groups associated with the various levels of the multi-graded resolution

W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a).

The canonical (finite) collection of multi-graded resolutions:

W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a)

produced by the procedure described above, contain the entire set of the remaining y’s. To be able to continue with our iterative procedure and collect all the formal solutions defined over closures of the various multi-graded resolutions we have constructed, we need these multi-graded resolutions to be well-structured.

Proposition 1.15. The multi-graded resolutions: W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a) constructed by our procedure are well-structured (multi-graded) resolutions.

Proof: Identical with the proof of theorem 1.14 of [Se4].

□

Having constructed finally the multi-graded resolutions W P HGH(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a) that contain all the ”remaining” specializations of the variables y, we are ready to define their complexity.

Definition 1.16. Let W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a) be one of the (wellstructured) multi-graded resolutions constructed above. By construction, the complexities of the completions of all non-degenerate ungraded resolutions covered by the multi-graded resolution W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a) is identical (see definition 1.15 of [Se4] for the definition of the complexity of an ungraded completed resolution). We set the complexity of the multi-graded resolution:

W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a)

denoted Cmplx(W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a)), to be the complexity of the completion of a non-degenerate ungraded resolution covered by the multi-graded resolution W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{1}, ĝ_{1}, a).

In order to ensure the termination of our ”trial and error” procedure, we need the complexities of the multi-graded resolutions obtained in its different steps to decrease.

Theorem 1.17. Let W P HRes(y, h_{1}, w, p, a) be one of the resolutions obtained in the first step of our iterative procedure, and suppose the graded resolution W P HRes(y, h_{1}, w, p, a) do not correspond to the entire free group F_{y} we have started with. Let:

W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a)

be one of the multi-graded resolutions obtained in the second step of our iterative procedure staring with the graded resolution W P HRes(y, h_{1}, w, p, a). Then:

Cmplx(W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a)) ≤ Cmplx(W P HRes(y, h_{1}, w, p, a)) and in case of equality the two graded resolutions are compatible, i.e., the abelian decompositions and the corresponding variables that are associated with the two resolutions are in one to one correspondence.

Furthermore, suppose that a specialization p_{0} of the defining parameter p satisfies p_{0} ∈ EAE(p), and w_{0} is a specialization of the variables w for which the sentence:

∀y ∃x Σ(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ(x, y, w_{0}, p_{0}, a) ≠ 1

is a truth sentence over the free group F_{k}. Then there exists a specialization h_{0}^{1} of the variables h_{1} for which:

(i) every ungraded resolution W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a) covered by one of the graded resolutions W P HRes(y, h_{1}, w, p, a) does not correspond to the entire free group F_{y}.

(ii) let W P HG(g_{1}, h_{1}, w, p, a) be the (rigid or solid) terminal graded limit group of one of the graded the graded resolutions W P HRes(y, h_{1}, w, p, a), and let (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}, a) be a rigid or solid specialization of W P HG(g_{1}, h_{1}, w, p, a). Then there exists a specialization h_{0}^{2} of the variables h_{2} so that every ungraded resolution W P HGH(s, z, y, h_{0}^{1}, w_{0}, p_{0}, ĥ_{0}^{2}, ĝ_{0}^{1}) satisfies:

Cmplx(W P HGHRes(s, z, y, h_{0}^{1}, w_{0}, p_{0}, ĥ_{0}^{2}, ĝ_{0}^{1}, a)) < Cmplx(W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a)).

Proof: By the proof of theorem 1.16 in [Se4]:

Cmplx(W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a)) ≤ Cmplx(W P HRes(y, h_{1}, w, p, a))

and in case of equality the two graded resolutions are compatible, i.e., the abelian decompositions and the corresponding variables that are associated with the two resolutions are in one to one correspondence.

Suppose the specialization p_{0} of the defining parameter p satisfies p_{0} ∈ EAE(p), and w_{0} is a specialization of the variables w for which the sentence:

∀y ∃x Σ(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ(x, y, w_{0}, p_{0}, a) ≠ 1

is a truth sentence over the free group F_{k}. By the construction of the formal resolutions W P F Res(x, y, w, p, a) constructed in the first step of our iterative procedure, every formal solution x = x_{(w_{0},p_{0})}(y, a) (for which Σ(x_{(w_{0},p_{0})}(y, a), y, w, p, a) = 1 in the free group F_{k} ∗ F_{y}) factors through at least one of the formal resolutions W P F Res(x, y, w, p, a). By theorem 1.2 of [Se2] there exists a formal solution x = x_{(w_{0},p_{0})}(y, a) for which Σ(x_{(w_{0},p_{0})}(y, a), y, w, p, a) = 1 in the free group F_{k} ∗ F_{y} and there exists some specialization y_{0} of the variables y for which:

Ψ(x_{(w_{0},p_{0})}(y_{0}, a), y_{0}, w_{0}, p_{0}, a) ≠ 1.

Suppose this last formal solution factors through the formal resolution W P F Res(x, y, w, p, a) that terminates in the rigid or solid graded limit group (with respect to the parameter subgroup < w, p >) W P H(h_{1}, w, p), and let (h_{0}^{1}, w_{0}, p_{0}) be the rigid or solid specialization corresponding to the ungraded resolution containing the formal solution x = x_{(w_{0},p_{0})}(y, a). Since there exists a specialization y_{0} of the variables y for which: Ψ(x(y_{0}, a), y_{0}, w_{0}, p_{0}, a) ≠ 1, every ungraded resolution W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a) covered by one of the graded resolutions W P HRes(y, h_{1}, w, p, a) does not correspond to the entire free group F_{y} and we get part (i) of the theorem.

Let W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a) be an ungraded resolution covered by one the graded resolution W P HRes(y, h_{1}, w, p, a) with rigid or solid terminal graded limit group W P HG(g_{1}, h_{1}, w, p, a), and suppose (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) is the specialization of W P HG(g_{1}, h_{1}, w, p, a) corresponding to the ungraded resolution W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a). By the construction of the formal resolutions W P HGF Res(x, s, z, y, g_{1}, h_{1}, w, p, a) constructed in the second step of our iterative procedure, every formal solution x = x_{(g_{0}^{1},h_{0}^{1},w_{0},p_{0})}(s, z, y, a), for which Σ(x_{(g_{0}^{1},h_{0}^{1},w_{0},p_{0})}(s, z, y, a), y, w, p, a) = 1 in some closure of the ungraded resolution W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a) corresponding to the specialization (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) factors through at least one of the formal resolutions W P HGF Res(x, s, z, y, g_{1}, h_{1}, w, p, a). By theorem 1.18 of [Se2] there exists a formal solution x = x_{(g_{0}^{1},h_{0}^{1},w_{0},p_{0})}(s, z, y, a) for which Σ(x_{(g_{0}^{1},h_{0}^{1},w_{0},p_{0})}(s, z, y, a), y, w, p, a) = 1 over the corresponding closure of the ungraded resolution W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a) that corresponds to the specialization (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) of W P HG(g_{1}, h_{1}, w, p, a), and for some specialization (s_{0}, z_{0}, y_{0}, h_{0}^{1}, w_{0}, p_{0}) that factors through the closure Cl(W P HRes)(s, z, y, h_{0}^{1}, w_{0}, p_{0}, a): Ψ(x_{(g_{0}^{1},h_{0}^{1},w_{0},p_{0})}(s_{0}, z_{0}, y_{0}, a), y_{0}, w_{0}, p_{0}, a) ≠ 1. Suppose that this last formal solution factors through the formal resolution:

W P HGF Res(x, y, g_{1}, h_{1}w, p, a)

that terminates in the rigid or solid graded limit group (with respect to the parameter subgroup < g_{1}, h_{1}, w, p >) W P HGH(h_{2}, g_{1}, h_{1}, w, p), and let (h_{0}^{2}, g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) be the rigid or solid specialization corresponding to the ungraded resolution containing the formal solution x = x_{(g_{0}^{1},h_{0}^{1},w_{0},p_{0})}(s, z, y, a). Since there exists a specialization (s_{0}, z_{0}, y_{0}, h_{0}^{1}, w_{0}, p_{0}) that factors through the closure Cl(W P HRes)(s, z, y, h_{0}^{1}, w_{0}, p_{0}, a) for which: Ψ(x_{(g_{0}^{1},h_{0}^{1},w_{0},p_{0})}(s_{0}, z_{0}, y_{0}, w_{0}, p_{0}, a) ≠ 1, theorem 1.16 of [Se4] implies that every ungraded resolution W P HGHRes(s, z, y, h_{0}^{1}, w_{0}, p_{0}, ĥ_{0}^{2}, ĝ_{0}^{1}, a) covered by one of the multi-graded resolutions W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a) satisfies:

Cmplx(W P HGHRes(s, z, y, h_{0}^{1}, w_{0}, p_{0}, ĥ_{0}^{2}, ĝ_{0}^{1}, a)) < Cmplx(W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a))

which proves part (ii) of the theorem.

□

Given the rigid or solid limit groups W P HG(g_{1}, h_{1}, w, p, a) and the graded resolutions associated with each of their rigid or solid specializations with corresponding non-degenerate ungraded resolutions, we have collected all the formal solutions, defined over some closures of these graded resolutions, in a finite collection of graded formal resolutions W P HGF Res(x, y, g_{1}, h_{1}, w, p, a), each terminating in a rigid or solid graded limit group W P HGH(h_{2}, g_{1}, h_{1}, w, p, a). Given the set of rigid or solid specializations of W P HGH(h_{2}, g_{1}, h_{1}, w, p, a) with corresponding non-degenerate ungraded formal closures, we used the ”towers” of graded formal modular groups associated with the graded formal resolution W P HGHF Res(h_{2}, g_{1}, h_{1}, w, p, a), to get a system of equations imposed on the remaining modular block of (the remaining) y’s associated with each of the rigid or solid specializations of the limit group W P HGH(h_{2}, g_{1}, h_{1}, w, p, a). Given this system of equations we have used an iterative procedure to construct a finite set of (well-structured) multi-graded resolutions W P HGHRes(s, z, y, h_{1}, w, p, a, ĥ_{2}, ĝ_{1}, a) each terminating in a rigid or solid multi-graded limit group W P HGHG(g_{2}, h_{1}, w, p, a, ĥ_{2}, ĝ_{1}, a). These graded resolutions contain the entire set of the remaining y’s for each specialization of the tuple (w, p). By theorem 1.17 the complexity of each of the multi-graded resolutions W P HGHRes(s, z, y, h_{1}, w, p, ĥ_{2}, ĝ_{1}, a) is bounded above by the complexity of the graded resolution W P HRes(y, h_{1}, w, p, a) it was constructed from, and if p_{0} ∈ EAE(p) there must exist some specialization w_{0} and a specialization h_{0}^{1} of the variables h_{1}, so that for every solid or rigid specialization (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}, a) of W P HG(g_{1}, h_{1}, w, p, a), there exists a specialization h_{0}^{2} of the variables h_{2} so that the complexity of every ungraded resolution W P HGH(s, z, y, h_{0}^{1}, w_{0}, p_{0}, ĥ_{0}^{2}, ĝ_{0}^{1}) has strictly smaller complexity than the complexity of the ungraded resolution W P HRes(y, h_{0}^{1}, w_{0}, p_{0}, a).

The continuation of our ”trial and error” procedure is defined recursively, where at each step we continue only with those multi-graded resolutions that have strictly smaller complexity than the multi-graded resolutions they are associated with in the previous step. Given a multi-graded resolution W P (HG)^{n−1}HRes with terminating rigid or solid multi-graded limit group W P (HG)^{n}, and the set of rigid or solid specializations of a limit group W P (HG)^{n} with non-degenerate associated ungraded resolutions, we collect all the formal solutions defined over a closure of an ungraded resolution associated with each of the rigid or solid specializations of the limit group W P (HG)^{n}, and obtain finitely many formal multi-graded resolutions W P (HG)^{n}F Res, each terminating in a rigid or solid multi-graded limit group W P (HG)^{n}H.

Given one of the multi-graded formal resolution W P (HG)^{n}F Res and its terminating rigid or solid multi-graded limit group W P (HG)^{n}H, we use the ”tower” of graded formal modular groups associated with the graded formal resolution to get a system of equations imposed on the remaining set of y’s associated with each of the specializations of the terminating limit group W P (HG)^{n}H. Given this system of equations we use the iterative procedure presented above to construct a finite set of (well-structured) multi-graded which we denote W P (HG)^{n}HRes that terminate in a rigid or solid multi-graded limit group W P (HG)^{n+1}, that contain the set of y’s remaining after the first n + 1 steps of our ”trial and error” procedure. Theorem 1.17 remains valid for all the steps of the iterative procedure, so the complexity of each of the multi-graded resolutions W P (HG)^{n}HRes is bounded above by the complexity of the multi-graded resolution W P (HG)^{n−1}HRes it was constructed from, and if p_{0} ∈ EAE(p) there must exist some specialization w_{0} for which there exists a strict reduction in the corresponding complexities.

Like in the analysis of an AE sentence, the (strict) reduction in the complexity of the multi-graded resolutions containing the sets of the remaining y’s stated in theorem 1.17, guarantees the termination of our ”trail and error” procedure in the minimal (graded) rank case.

Theorem 1.18. Suppose that if an initial graded limit group λW P GL_{j}(y, h_{1}, w, p, a) admits an epimorphism τ onto a free group F_{k} ∗ F where τ(< p >) < F_{k}, then F is the trivial group. Then the iterative ”trial and error” procedure presented above terminates after finitely many steps.

Proof: Identical to the proof of theorem 1.19 of [Se4].

□

The outcome of the ”trial and error” procedure gives us a tree of stratified sets which encodes all the (finitely many) possible sequences of forms of formal solutions that are needed in order to validate that a certain specialization p_{0} of the defining parameters p is indeed in the set EAE(p). This stratification is the basis for our analysis of the structure of the set EAE(p). To explain the motivation for the way we analyze the structure of the set EAE(p), we need the natural notion of a witness.

Definition 1.19. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let EAE(p) be the set defined by the predicate:

EAE(p) = ∃w ∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1.

A specialization w_{0} of the variables w, is said to be a witness for a specialization p_{0} of the defining parameters p, if the following sentence:

∀y ∃x Σ(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ(x, y, w_{0}, p_{0}, a) ≠ 1

is a truth sentence. Clearly, if there exists a witness for a specialization p_{0} then p_{0} ∈ EAE(p), and every p_{0} ∈ EAE(p) has a witness.

By definition, in order to show that a specialization p_{0} of the defining parameters p is in the set EAE(p), we need to find a witness w_{0} for the specialization p_{0}. By the construction of the tree of stratified sets, a witness w_{0} for a specialization p_{0}, proves that p_{0} ∈ EAE(p) using a certain ”proof system” which is built from a finite sequence of (families of) formal solutions, which correspond to boundedly many paths along the tree of stratified sets. Given p_{0} ∈ EAE(p), we are not able to say much about a possible witness for p_{0} using the information we have collected so far. However, with each ”proof system”, i.e., with each path of (families of) formal solutions that goes along the tree of stratified sets, one can associate a certain Diophantine set. The bound on the form and number of all possible ”proof systems” associated with all possible witnesses suggested by the tree of stratified sets, forces every possible witness for p_{0} to belong to one of the finitely many Diophantine sets associated with the (finite) collection of all proof systems. Furthermore, in the sequel we will show that if a specialization p_{0} ∈ EAE(p), and it can be shown that p_{0} ∈ EAE(p) using a witness w_{0} with a corresponding proof system, then every ”generic” specialization of w that belongs to (a closure of) some modular block associated with the Diophantine set which is associated with the specific proof system is a witness for p_{0} using the same proof system as w_{0}. This will reduce the analysis of the set EAE(p) to the analysis of the Diophantine sets and the modular blocks associated with each ”proof system”, and eventually will enable us to show that the set EAE(p) is in the Boolean algebra of AE sets, which concludes the proof of theorem 1.4 in the minimal (graded) rank case.

Definition 1.20. Let p_{0} ∈ EAE(p) be a specialization of the defining parameters p, and let w_{0} be a witness for p_{0}. By the construction of the tree of stratified sets, one can associate a proof system with the couple (w_{0}, p_{0}), which corresponds to a (finite) collection of paths in the tree of stratified sets. Note that there may be several proof systems associated with a given couple (w_{0}, p_{0}), but the construction of the tree of stratified sets guarantees that the number of proof systems associated with the couple (w_{0}, p_{0}) is globally bounded. We will say that a given proof system associated with the couple (w_{0}, p_{0}) is of depth d, if all the paths associated with the proof system terminate after d steps (levels) of the tree of stratified sets.

For presentation purposes, we will start by demonstrating our approach for the analysis of the set EAE(p), by analyzing those specializations of the defining parameters p that have witnesses with proof systems of depth 1. We will continue by analyzing the specializations of the defining parameters p for which there are witnesses with proof systems of depth at most 2, and then present the analysis of the entire set EAE(p) under the minimal (graded) rank assumption.

Lemma 1.21. Let T_{1}(p) ⊂ EAE(p) be the subset of all specializations p_{0} ∈ EAE(p) of the defining parameters p, that have witnesses with proof system of depth 1. Then T_{1}(p) is an EA set.

Proof: T_{1}(p) is precisely the set U_{1}(p), that is proven to be an EA set in lemma 1.5.

□

Lemma 1.21 proves that the set of specializations p_{0} of the defining parameters p that have a witness with a proof system of depth 1 is an EA set. Before analyzing the entire set EAE(p) we analyze the set of specializations p_{0} that have witnesses with a proof system of depth 2. The analysis of specializations p_{0} ∈ EAE(p) having witnesses with proof systems of depth 2 is much more complicated than the analysis of witnesses with proof systems of depth 1, and will be presented in this section under the minimal (graded) rank assumption. In the next section we will use the general approach for validation of a sentence presented in [Se4] to drop the minimal rank assumption.

Theorem 1.22. Let T_{2}(p) ⊂ EAE(p) be the subset of all specializations p_{0} ∈ EAE(p) of the defining parameters p, that have witnesses with proof system of depth 2. Then T_{2}(p) is in the Boolean algebra of AE sets.

Proof: In this section we present a proof of theorem 1.22 under the minimal (graded) rank assumption, i.e., we will assume that if any of the graded limit groups λW P GL(y, h_{R}, w, p, a) or λW P GL(y, h_{S}, w, p, a) admit an epimorphism τ onto a free group F_{k} ∗ F where τ(< p >) < F_{k}, then F is the trivial group. Before we start with the proof of theorem 1.22 under the minimal (graded) rank assumption we need the notion of a valid P S statement.

Definition 1.23. Suppose that a specialization p_{0} ∈ EAE(p) has a witness w_{0} with a proof system of depth 2 (i.e., p_{0} ∈ T_{2}(p)). The structure of the tree of stratified sets guarantees the existence of a rigid or strictly solid families of specializations (h_{0}^{1}, w_{0}, p_{0}) of one of the rigid or solid limit groups W P H(h, w, p, a) with the following properties:

(i) For every rigid or solid limit group W P HG(g_{1}, h_{1}, w, p, a) there are at most (globally) boundedly many rigid or strictly solid families of specializations of the form (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) of W P HG(g_{1}, h_{1}, w, p, a), where the strictly solid families are with respect to the given set of closures associated with (some of the other) limit groups W P HG(g_{1}, h_{1}, w, p, a) (strictly solid families with respect to a given closure are presented in definition 2.10 in [Se3]). The elements (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}, a) that appear in a proof system, contains representatives for all the boundedly many classes in the various solid families that do not factor through the covering closure associated with (some of) the other (deeper) limit groups W P HG.

(ii) The specialization (h_{0}^{1}, w_{0}, p_{0}) is rigid or solid of the corresponding limit group W P H, and it does not factor through any of the modular blocks that factor through one of the limit groups λW P GL(y, h, w, p, a) and correspond to the entire free group F_{y}.

(iii) For each of the (boundedly many) rigid or strictly solid families of specializations (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) there exist a finite collection of rigid or strictly solid families of specializations (h_{0}^{2}, g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) of the rigid or strictly solid limit groups W P HGH(h_{2}, g_{1}, h_{1}, w, p, a), so that the (ungraded) resolutions corresponding to the specializations (h_{0}^{2}, g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) form a covering closure of the (ungraded) resolution corresponding to the specialization (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}).

(iv) For each of the (boundedly many) rigid or strictly solid families of specializations (h_{0}^{2}, g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) there exists no specialization g_{0}^{2} of the variables g_{2} so that the specialization (g_{0}^{2}, h_{0}^{2}, g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) factors through one of the (rigid or solid) limit groups W P HGHG(g_{2}, h_{1}, g_{1}, h_{1}, w, p, a).

We call a specialization of the form:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that satisfies conditions (i)-(iv) above, where the integer d(ps) depends on the fixed proof system, a valid P S statement.

The tree of stratified sets guarantees that there are finitely many proof systems of depth 2. Once we fix a proof system of depth 2, we have fixed the rigid or solid limit group W P H(h, w, p, a), the number of rigid or strictly solid families of specializations of each of the limit groups W P HG(g_{1}, h_{1}, w, p, a), and the number of rigid or solid specializations of each of the limit groups (h_{2}, g_{1}, h_{1}, w, p, a). We start the analysis of the set T_{2}(p) enumerating all the possible proof systems, and for each proof system we collect all possible (configurations of) valid P S statements: ((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) (the integer d(ps) depends on the fixed proof system), which by the standard arguments presented in section 5 of [Se1], factor through a (canonical) collection of maximal limit groups P SHGH_{1}, …, P SHGH_{m}, which we call P S (proof system) limit groups.

By construction, for each p_{0} ∈ T_{2}(p) there exists some witness w_{0} and a corresponding proof system, so that a specialization of the form:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that is associated with the specialization p_{0}, the witness w_{0} and that proof system, is a valid P S statement (i.e., it satisfies conditions (i)-(iv) of definition 1.23), and factors through a limit group P SHGH_{j}. Naturally, we will try to understand the set of valid P S statements:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factor through a given P S limit group P SHGH_{j}. Our main goal will be to show that these valid P S statements are ”generic” in some modular blocks associated in the sequel with each of the P S limit groups P SHGH. Before we start with the (technically involved) analysis in the general case, we analyze those P S limit groups which are rigid or solid with respect to the parameter subgroup P =< p >.

Proposition 1.24. Suppose that a P S limit group P SHGH is rigid or solid with respect to the parameter subgroup P =< p >, and if it is solid suppose that the subgroup W P =< w, p > is a subgroup of the distinguished vertex group in the graded JSJ decomposition of P SHGH (i.e., the vertex stabilized by the subgroup AP =< a, p >). The set of specializations p_{0} that have a witness w_{0}, and a rigid or solid specialization

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

of P SHGH which is a valid P S statement, which we denote P S(p), is in the Boolean algebra of AE sets.

Proof: Since we have assumed that the P S limit group is either rigid or solid with respect to the parameter subgroup P =< p >, with each specialization p_{0} of the defining parameters P there exist boundedly many rigid or strictly solid families of specializations:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

of the limit group P SHGH. By section 3 of [Se3] we already know that the set of specializations of the defining parameters P for which there are precisely s rigid or strictly solid specializations of the P S limit group P SHGH is in the Boolean algebra of AE sets. We define the set P S(p) as a disjunction of conjunctions of an EA and an AE predicates. The EA predicate in each conjunction of an EA and AE predicates verifies the following conditions:

(1) there exist (at least) s rigid or strictly solid families of specializations:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

of the P S limit group P SHGH.

(2) for each of the s rigid or strictly solid families of specializations of the P S limit group P SHGH, we specify if the specialization (h_{0}^{1}, w_{0}, p_{o}) is a rigid or strictly solid or flexible specialization of the rigid or solid limit group W P H, if each specialization (g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}, a) is a rigid or strictly solid or flexible specialization of the corresponding rigid or solid limit group W P HG(g_{1}, h_{1}, w, p, a), and if each couple of rigid or strictly solid specializations (g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}, a) belong to the same rigid or strictly solid family.

(3) for each of the s rigid or strictly solid families of specializations, we specify the (minimum) numbers of rigid or strictly solid families of specializations (g_{1}, h_{0}^{1}, w_{0}, p_{0}, a) of each of the rigid or solid limit groups W P HG(g_{1}, h_{1}, w, p, a).

(4) for each of the s rigid or strictly solid specializations, for which each of the specializations (g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}, a) is a rigid or a strictly solid specialization of the corresponding rigid or solid limit group W P HG(g_{1}, h_{1}, w, p, a), each couple of rigid or strictly solid specializations (g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}, a) does not belong to the same rigid or strictly solid family, and the (minimum) numbers of rigid or strictly solid families of specializations of each of the limit groups W P HGH(g_{1}, h_{1}, w, p, a) is precisely as required by the fixed proof system, we specify if the specializations (h_{j}^{2}, g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}) are rigid or solid specializations of the corresponding (rigid or solid) limit groups W P HGH. and for each tuple (h_{j}^{2}, g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}) if there exists some specialization g_{j}^{2} of the variables g_{2} so that the tuple (g_{j}^{2}, h_{j}^{2}, g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}) factors through one of the limit groups W P HGHG(g_{2}, h_{2}, g_{1}, h_{1}, w, p, a).

The AE predicate is complementary to the EA one, i.e., it checks if the minimal numbers specified by the EA numbers for the numbers of rigid or strictly solid families of specializations of the P S limit group P SHGH, and the the limit groups W P HG, are also maximal numbers. In addition, it checks if the ungraded resolutions (closures) associated with the specified specializations (h_{j}^{2}, g_{i}^{1}, h_{0}^{1}, w_{0}, p) form a covering closure of the ungraded resolutions associated with the specializations (g_{i}^{1}, h_{0}^{1}, w_{0}, p) (which is clearly an AE predicate).

Since we have fixed a proof system, the global bound on the number of rigid or strictly solid families of solutions of a rigid or solid limit group guarantees that the possible numbers of rigid or strictly solid families of solutions of the P S limit groups P SHGH, and the limit group W P HG, are globally bounded, hence, once we fix a proof system, to define the set of parameters p_{0}, for which there exists a witness w_{0} with the specified proof system we need a finite disjunction of the following conjunctions of an EA and an AE predicates. Since there are finitely many proof systems of depth 2, the set P S(p) is in the Boolean algebra of AE sets.

□

Proposition 1.24 proves theorem 1.22 in case the P S limit groups P SHGH are rigid or solid with no flexible quotients, and the subgroup W P =< w, p > is a subgroup of the distinguished vertex group in the graded JSJ decomposition of P SHGH. In this special case, the number of possible witnesses w_{0} associated with each specialization p_{0} of the defining parameter p is finite and globally bounded. In the general case, the number of possible witnesses associated with each specialization p_{0} of the defining parameters p is infinite, hence, there is no direct way to present the set T_{2}(p) using a predicate which is in the Boolean algebra of AE sets. Our goal in the analysis of the set T_{2}(p) in the general case, is to find finitely many modular blocks associated with each of the P S limit groups P SHGH, modular blocks in which a generic specialization of the variables w is a witness for the corresponding specialization p_{0} of the defining parameters P. To achieve this goal, i.e., to find the finitely many modular blocks associated with each of the P S limit groups P SHGH, we present a ”trial and error” procedure, similar to the one used to construct the tree of stratified sets. The output of the iterative ”trial and error”, i.e., the finitely many modular blocks associated with each P S limit group P SHGH, are later used to derive a predicate in the Boolean algebra of AE predicates that describes the set T_{2}(p).

Let P =< p > be the group of defining parameters. With each of the limit groups P SHGH_{i} we associate its canonical graded Makanin-Razborov diagram (with respect to the parameter subgroup P), which contains finitely many graded resolutions which we denote P SHGHRes_{j}, and each graded resolution P SHGHRes_{j} is defined over the rigid or solid limit group P T_{j}(t, p, a). We will treat the limit groups P SHGH_{i} and their graded resolutions P SHGHRes_{j} and rigid or solid limit groups P T_{j}(t, p, a) in parallel, hence, we will omit the indices of the limit group and its graded resolution. In the sequel, we will treat each strata in the singular locus of the graded resolutions P SHGHRes separately, and do it in parallel. We start our analysis with the definition of Non-Rigid and Non-Solid P S limit groups.

Definition 1.25. Let:

Comp(P SHGHRes)(v, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p, a)

be the completion of the graded resolution P SHGHRes. We look at the entire collection of all test sequences that factor through the graded completion Comp(P SHGHRes):

{v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{n}, a)}

(see definitions 1.20 and 3.1 in [Se2] for the notion of a test sequence of a completed graded resolution).

We start by looking at those graded test sequences for which the specialization (h_{1}(n), w_{n}, p_{n}) or for some index j for which the specialization (g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}) or the specialization (h_{j}^{2}(n), g_{j}^{1}(n), w_{n}, p_{n}) is supposed to be rigid according to our fixed proof system, but these specializations are not rigid (i.e., they are flexible) for every index n. The collection of all these ”non-rigid PS” (graded) test sequences factor through a (canonical) collection of maximal Non-Rigid P S limit groups NRgdP S_{1}, …, NRgdP S_{q}. The analysis of graded formal limit groups presented in section 3 of [Se2] associates (canonically) with each Non-Rigid PS limit group, NRgdP S_{i}, a graded formal Makanin-Razborov diagram, and each such graded formal resolution is in fact a one level graded resolution, which is a graded formal closure of the graded resolution P SHGHRes, GF Cl(P SHGHRes). Clearly, no specialization:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factors through the resolution P SHGH, and which is a valid P S statement with respect to our fixed proof system, factors through one of the NRgdP S limit groups NRgdP S_{1}, …, NRgdP S_{q}.

Defining Non-Rigid P S limit groups, we continue by defining Non-Solid P S limit groups. We look at those graded test sequences for which the specialization (h_{1}(n), w_{n}.p_{n}) or for some index j the specialization (g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}) or the specialization (h_{j}^{2}(n), g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}) is supposed to be strictly solid (with respect to the given covering closures) according to our fixed proof system, but these specializations are not strictly solid with respect to the given set of closures for every index n. For each specialization:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

in such a ”non-solid PS” (graded) test sequence, we add variables that ”demonstrate” that the specific specialization (g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}) is indeed not strictly solid, i.e., it factors through a closure associated with some flexible quotient of the corresponding solid limit group P SHG(g_{1}, h_{1}, w, p, a). The collection of all these ”non-solid PS” (graded) test sequences together with the extra variables that ”demonstrate” they are indeed ”non-solid PS” test sequences, factor through a (canonical) collection of maximal Non-Solid P S limit groups NSldP S_{1}, …, NSldP S_{r}. The analysis of graded formal limit groups presented in section 3 of [Se2] associates (canonically) with each Non-Solid PS limit group NSldP S_{i} a graded formal Makanin-Razborov diagram, and each such graded formal resolution terminates with a graded formal closure of the graded resolution P SHGHRes, GF Cl(P SHGHRes). Clearly, no specialization:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factors through the resolution P SHGH, and which is a valid P S statement with respect to our fixed proof system, factors through none of the graded formal closures GF Cl(P SHGHRes) that is associated with one of the non-solid limit groups NSldP S_{1}, …, NSldP S_{r}.

The graded formal closures associated with the collection of non-rigid and nonsolid PS limit groups, determine those ”generic” specializations that factor through the various P S resolutions P SHGHRes but fail to be valid P S statements with respect to the (fixed) proof system because certain specializations that are required to be rigid or strictly solid actually factor through flexible quotients of the corresponding rigid or solid limit groups. ”Generic” specializations that factor through the P S resolutions P SHGHRes can fail to be valid P S statements in other ways as well. In particular, it can be that there exist additional rigid or strictly solid specializations of the limit groups P SHG(g_{1}, h_{1}, w, p, a) that are not specified by the given specializations. The ”generic” specializations for which there exists ”surplus” in rigid or strictly solid specializations are collected in extra P S (graded) limit groups and graded resolutions.

Definition 1.26. Let:

Comp(P SHGHRes)(v, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p, a)

be the completion of the graded resolution P SHGHRes. We look at the entire collection of all test sequences that factor through the graded completion Comp(P SHGHRes):

{v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{n}, a)}.

By theorems 2.3 and 2.11 for each specialization: (h_{0}^{1}, w_{0}, p_{0}, a) there exist boundedly many rigid or strictly solid families of specializations (g_{0}^{1}, h_{0}^{1}, w_{0}, p_{0}) of each of the rigid or solid limit groups W P HGH(g_{1}, h_{1}, w, p, a). We look at those graded test sequences for which for every index n there are rigid or strictly solid families of specializations of the form (g_{0}^{1}, h_{1}(n), w_{n}, p_{n}) which are not in the rigid or strictly solid families of each of the specializations: (g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}) for every index j. To each specialization:

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{n}, a))

we add variables for the additional rigid specializations or the almost shortest specializations (see definition 2.6 in [Se3]) in the additional strictly solid families of specializations. Since by theorems 2.3 and 2.11 there exists a global bound on the number of additional rigid or strictly solid families of specializations, the number of additional variables we need to add is globally bounded.

The collection of all these ”extra PS” (graded) test sequences (including the added variables) factor through a (canonical) collection of maximal Extra P S limit groups ExtraP S_{1}, …, ExtraP S_{ℓ}. The analysis of graded formal limit groups presented in section 3 of [Se2] associates (canonically) with each Extra PS limit group ExtraP S_{i} a graded formal Makanin-Razborov diagram, and each such graded formal resolution is in fact a one level graded resolution, which is a graded formal closure of the graded resolution P SHGHRes, GF Cl(P SHGHRes). We denote such an extra P S (graded) resolution

ExtraP SRes(u, v, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p, a)

(the variables u are the variables added for the extra rigid and solid specializations).

The extra P S limit groups and their associated graded formal closures collect all the ”generic” specializations (i.e., all the test sequences) of the P S limit groups P SHGH for which there exist rigid or strictly solid families in addition to those specified by the generic specializations. For a general specializations of the P S limit groups P SHGH, i.e., a specialization which is not necessarily ”generic”, it may as well be that the additional rigid or strictly solid specializations, collected by the extra P S limit groups and their associated graded formal closures, do become flexible or do coincide with the rigid or strictly solid families of the various specializations (g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}). To collect all the specializations that factor through one of the extra P S graded formal closures associated with one of the extra P S limit groups ExtraP S, in which such a ”collapse” of the additional rigid and strictly solid families occurs, we need to define the collapsed extra P S (graded) limit groups and their associated (collapsed extra P S) graded resolutions. We start by listing all possible collapsings of a specialization of an Extra P S resolution.

Definition 1.27. Let ExtraP SRes be one of the extra P S graded resolutions associated with one of the extra P S graded limit groups ExtraP S_{i}. Note that by construction, ExtraP SRes is a graded formal closure of the P S resolution P SHGHRes we have started the analysis with. We will say that a shortest form specialization (see definition 1.2 in [Se4]) that factors through the extra P S graded resolution ExtraP SRes is collapsed if the variables added for each of the additional rigid or strictly solid family of specializations (i.e., the ones that were not specified by the proof system) satisfy one of the following:

(1) A specialization of the variables added for one of the additional rigid specializations becomes flexible.

(2) A specialization of the variables added for one of the additional rigid specializations becomes equal to one of rigid specializations specified by the proof system, i.e., with one of the specializations g_{j}^{1} in the specialization:

(u, v, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}).

(3) A specialization of the variables added for one of the additional strictly solid families of specializations factors through one of the given closures associated with a flexible quotient associated with the corresponding solid limit group.

(4) A specialization of the variables added for one of the additional strictly solid families of specializations belongs to one of strictly solid families of specializations specified by the proof system, i.e., with a family of one one of the strictly solid families of specializations g_{j}^{1} in the specialization:

(u, v, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}).

Note that by definition there are only finitely many ways a (shortest form) specialization that factors through the Extra P S resolution, ExtraP SRes, can become a collapsed specialization. We will call each way a specialization can become collapsed a collapse form.

Defining the finitely many possibilities for collapse forms, we collect all the shortest form specializations that are collapsed specializations in finitely many graded resolutions which we call collapsed Extra P S (graded) resolutions.

Definition 1.28. Let ExtraP SRes be one of the extra P S graded resolutions associated with one of the extra P S graded limit groups ExtraP S_{i}. Note that by construction, ExtraP SRes is a graded formal closure of the P S resolution P SHGHRes we have started the analysis with. With the Extra P S resolution we fix one of collapse forms associated with it. With each collapsed specialization that factors through the extra P S resolution:

(u_{0}, v_{0}, (h_{1}^{2}(0), g_{1}^{1}(0)), …, (h_{d(ps)}^{2}(0), g_{d(ps)}^{1}(0)), h_{1}(0), w_{0}, p_{0}, a))

which is collapsed in the particular collapse form we have fixed, we associate specializations of additional variables {c}, c_{0}, that demonstrate that the specializations of the additional variables added for one of the additional strictly solid family of specializations are either flexible or belongs to one of strictly solid family of specializations specified by the proof system, according to our fixed collapse form. We pick the specialization c_{0} to be the shortest among all possible ones. By our standard method presented in section 5 of [Se1], the collection of specializations:

(c_{0}, u_{0}, v_{0}, (h_{1}^{2}(0), g_{1}^{1}(0)), …, (h_{d(ps)}^{2}(0), g_{d(ps)}^{1}(0)), h_{1}(0), w_{0}, p_{0}, a))

which are collapsed and demonstrate the collapse of the specializations of variables added for additional strictly solid families, factor through a (canonical) collection of finitely many maximal limit groups which we call collapsed extra P S (graded) limit groups, denoted CollapseExtraP S_{i}.

Note that each collapsed specialization restricts, in particular, to a shortest form specialization of the completed resolution Comp(P SHGHRes), so given one of the finitely many collapsed extra P S limit groups, CollapseExtraP S_{i}, we are able to use the iterative procedure for the analysis of quotient (multi-graded) resolutions presented in this section (1.5-1.17) and based on section 1 of [Se4], to get a (canonical) collection of finitely many (multi-graded) resolutions that contain the entire collection of (collapsed) specializations:

(c_{0}, u_{0}, v_{0}, (h_{1}^{2}(0), g_{1}^{1}(0)), …, (h_{d(ps)}^{2}(0), g_{d(ps)}^{1}(0)), h_{1}(0), w_{0}, p_{0}, a))

that factor through the extra P S resolution ExtraP SRes and the collapsed extra P S limit group CollapseExtraP S_{i}. We call such a resolution a collapsed extra P S (multi-graded) resolutions, and denote it: CollapseExtraP SRes_{j}.

The collapsed extra P S resolutions were obtained using the iterative procedure for the analysis of quotient resolutions presented earlier in this section. This iterative procedure was constructed in the first section of [Se4] towards controlling the complexity of the obtained quotient resolutions. The collapsed extra limit groups CollapseExtraP S are not quotients of the completed (graded) resolution Comp(P SHGH), since we have added the variables {u} to demonstrate extra rigid and strictly solid families of solutions to obtain the extra P S resolution ExtraP SRes, and additional variables {c} to demonstrate their collapse according to the fixed collapse form. The variables {u} were added to demonstrate extra rigid and strictly solid solutions, hence, the extra P S resolutions, ExtraP SRes, are in fact a graded formal closure of the graded resolution P SHGHRes we have started with. Hence, apart from adding various roots to abelian vertex groups in the graded abelian JSJ decompositions associated with the various levels of the completion Comp(P SHGHRes), the extra P S resolution ExtraP SRes may differ from the completion Comp(P SHGHRes) only in the graded abelian JSJ decomposition associated with the bottom level, in case the terminating limit group of the graded resolution ExtraP SRes is solid. The variables {c} were added to demonstrate that the specializations of the variables added for extra strictly solid families of solutions, actually factor through one of the given closures associated with flexible quotients of the corresponding solid limit group, or they belong to the same solid family as one of the solid families specified by the proof system. By the minimal graded rank assumption, each of the limit groups W P HG, and W P HGH do not admit a map onto a free group F_{k} ∗ F in which the factor F is non-trivial and the subgroup AP =< a, p > is mapped onto F_{k}. Clearly, there are natural maps from the completion, Comp(P SHGHRes), to each of the limit groups that appear along the graded resolution CollapseExtraP SRes. By the way the iterative procedure for the analysis of quotient resolutions that was used to construct the collapsed extra P S resolutions collapseExtraP SRes is defined it follows that:

(i) If in a graded limit group associated with one of the levels of the graded resolution CollapseExtraP SRes admits a graded free decomposition, i.e., a non-trivial free decomposition in which the subgroup AP =< a, p > is contained in one of the factors, then necessarily the entire image of the completion Comp(P SHGHRes) is contained in the factor that contains the subgroup AP. Hence, if we modify the other factor to be the identity, the shortest form specializations of the completion Comp(P SHGHRes) that factor through CollapseExtraP SRes will not be changed. Therefore, we may assume that no limit group along the resolution CollapseExtraP SRes admit a graded free decomposition.

(ii) let Q be a QH vertex group that appears in one of the graded abelian JSJ decompositions associated with the various levels along the graded resolution CollapseExtraP SRes except, perhaps, the bottom level in case the terminating limit group of the graded resolution CollapseExtraP SRes is solid. Then there exists a subgroup of finite index in (a conjugate of) the QH subgroup Q which is in the image of the completion Comp(P SHGHRes).

(iii) Suppose that all the QH vertex groups in the completion Comp(P SHGH) are surviving QH subgroups and let A be an abelian vertex group in one of the graded abelian JSJ decompositions associated with the various levels along the graded resolution CollapseExtraP SRes. Then there exist conjugates of subgroups in the image of the completion Comp(P SHGHRes), which generate a subgroup of finite index in A.

Since otherwise, either the group P SHGH admits a map onto a free group F_{k} ∗ F with a non-trivial F, so that AP =< a, p > is mapped onto F, in contradiction to our assumptions, or the specializations of the variables c can be modified so that a QH vertex group will be eliminated or the rank of an abelian vertex group will be decreased, or a QH vertex group can be pushed to the bottom level of the graded resolution CollapseExtraP SRes.

Given these properties of the collapsed extra resolutions collapseExtraP SRes we need to slightly modify the way we measure their complexity in order to be able to control it in terms of the complexity of the completion Comp(P SHGHRes) we have started with.

Definition 1.29. Let:

CollapseExtraP SRes(c, u, v(h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p, a)

be one of the (well-structured, graded) collapsed extra P S resolutions constructed by the iterative procedure for the analysis of quotient resolutions from the extra P S resolution ExtraP SRes. Let SubCollapseExtraP SRes be the subresolution induced by the subgroup K =< v, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p, a > from the collapsed extra P S resolution CollapseExtraP SRes. i.e., if the terminating limit group of CollapseExtraP SRes is rigid then SubCollapseExtraP SRes is the resolution CollapseExtraP SRes itself, and if it is solid then SubCollapseExtraP SRes is obtained from CollapseExtraP SRes by eliminating all vertex groups for which the canonical image of the completion Comp(P SHGHRes) does not intersect any of their conjugates, except, perhaps, in an edge group. By construction, the complexities of the completions of all non-degenerate ungraded resolutions covered by the multi-graded resolution SubCollapseExtraP SRes are identical (see definition 1.15 of [Se4] for the definition of the complexity of an ungraded completed resolution). We set he complexity of the (well-structured, multi-graded) collapsed extra P S resolution CollapseExtraP SRes, denoted Cmplx(CollapseExtraP SRes), to be the complexity of the completion of a non-degenerate ungraded resolution covered by the multi-graded resolution SubCollapseExtraP SRes.

Once we have defined the complexity of the collapsed extra P S resolutions properly, the iterative procedure for the construction of quotient resolutions enable us to control the complexity of these resolutions by the complexity of the completion Comp(P SHGHRes) we have started with.

Theorem 1.30. Let CollapseExtraP SRes be one of the multi-graded resolutions obtained from the extra P S resolution ExtraP SRes which is obtained from the P S resolution P SHGH. Then:

Cmplx(CollapseExtraP SRes) ≤ Cmplx(Comp(P SHGH))

and in case of equality the two graded resolutions are compatible, i.e., the abelian decompositions and the corresponding variables that are associated with the completion Comp(P SHGH) and the subresolution SubCollapseExtraP SRes are in one to one correspondence. Hence, in case of equality the collapsed extra P S resolution CollapseExtraP SRes is a graded formal closure of the graded resolution P SHGHRes. In the equality case, we call the collapse extra P S resolution, CollapseExtraP SRes, a formal collapse extra P S resolution.

Proof: The resolutions CollapseExtraP SRes were constructed by the iterative procedure for the analysis of quotient resolutions, hence, the theorem follows using the same arguments used to prove theorem 1.17, which is identical to the argument used to prove theorem 1.16 of [Se4].

□

So far we have collected all the test sequences that factor through the P S resolutions P SHGHRes, and for which the specializations of (at least) one of the (tuples of) variables that were supposed to be rigid are in fact flexible (or trivial), or the specializations of two (tuples of) variables that are supposed to be rigid coincide, in the NRgdP S (graded) resolutions NRgdP SRes. We have collected all the test sequences that factor through the P S resolutions P SHGHRes, and for which the specializations of (at least) one of the (tuples of) variables that were supposed to be strictly solid are in fact flexible or trivial, or the specializations of two (tuples of) variables that are supposed to be (disjoint) strictly solid families belong to the same strictly solid family, in the NSldP S (graded) resolutions NSldP SRes. We have collected all the test sequences that factor through the P S resolutions P SHGHRes, and for which there exists additional rigid or strictly solid families of specializations which are not specified by the P S specializations themselves in the extra P S resolutions ExtraP SRes. Finally, we have collected all the shortest form specializations that factor through the extra P S resolutions ExtraP SRes, and for which all the additional specializations of the (tuples of) variables corresponding to the additional rigid or strictly solid specializations in the (finitely many) collapsed extra P S resolutions, CollapseExtraP SRes.

Before we start with the analysis of the set T_{2}(p), and use the bundles of resolutions we have constructed to show it is indeed in the Boolean algebra of AE sets, we need to collect all the the test sequences that factor through the P S resolutions P SHGHRes, and for which for at least one of the tuples: (h_{j}^{2}(n), g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}, a) there exist some specialization g_{j}^{2}(n) so that the (combined) specialization:

(g_{j}^{2}(n), h_{j}^{2}(n), g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}, a)

factors through (at least) one of the limit groups W P (HG)^{2}, or for which there exists a specialization g_{1}(n) so that the (combined) specialization: (g_{1}(n), h_{1}(n), w_{n}, p_{n}, a) factors through (at least) one of the limit groups W P HG that is not associated with one of the paths associated with our fixed proof system.

Definition 1.31. Let:

Comp(P SHGHRes)(v, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p, a)

be the completion of the graded resolution P SHGHRes. we look at the entire collection of all test sequences that factor through the graded completion Comp(P SHGHRes):

{v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{n}, a)}.

We look at those graded test sequences for which for some index j there exists some specialization g_{j}^{2}(n) so that the combined specialization:

(g_{j}^{2}(n), h_{j}^{2}(n), g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n})

that is not supposed to factor through any of the limit groups W P (HG)^{2} according to our fixed proof system (which has depth 2), factors through at least one of these limit groups, or for which there exists some specialization g_{1}(n) so that the combined specialization: (g_{1}(n), h_{1}(n), w_{n}, p_{n}) factors through a limit group W P HG which is not associated with one of the paths associated with our fixed proof system. The collection of all these (graded) test sequences factor through a (canonical) collection of maximal left P S limit groups LeftP S_{1}, …, LeftP S_{m}. The analysis of graded formal limit groups presented in section 3 of [Se2] associates (canonically) with each Left PS limit group LeftP S_{i} a graded formal Makanin-Razborov diagram, and each such graded formal resolution is in fact a one level graded resolution, which is a graded formal closure of the graded resolution P SHGHRes, GF Cl(P SHGHRes), that we denote LeftP SRes, and call Left P S resolution. Clearly, no specialization ((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) that factors through the resolution P SHGH, and which is a valid P S statement with respect to our fixed proof system, factors through one of the LeftP S limit groups LeftP S_{1}, …, LeftP S_{m}, and their associated Left P S resolutions.

So far, with each P S resolution, P SHGHRes, we have associated a finite collection of non-rigid, non-solid, left, extra P S, collapse extra P S, and formal collapse extra P S resolutions. Except for collapse extra P S resolutions that are not of maximal possible complexity, all the other (graded) resolutions associated with a P S resolution, P SHGHRes, are graded closures of it. Hence, as with any closure of a resolution (see definition 1.16 in [Se2]), with each such resolution associated with a P S resolution, P SHGHRes, one can naturally associate a coset of a finite index subgroup of a free abelian group Z^{m}, where m is the sum of the ranks of the free non-cyclic abelian vertex groups that appear along the resolution minus the ranks of the edge groups connected to these abelian vertex groups (see definition 1.16 in [Se2]). We set gind, the global index, to be the least common multiple of the indices of the finite index subgroups associated with the various resolutions that are associated with the P S resolution P SHGHRes.

In the sequel, we will need to consider cases in which a certain closure is or is not a covering closure when the pegs of some of the abelian groups associated with the resolutions associated with our given P S resolutions are or are not proper powers. To prepare tools that will enable us to consider such cases and associate appropriate predicates with them, we go over the finite collection of formal collapse P S resolutions associated with the P S resolution P SHGHRes. With a formal collapse P S resolution, CollapseExtraP SRes, we naturally associate its corresponding (graded) limit group. To that limit group we add new generators, that are set to be roots of pegs of abelian vertex groups along the formal collapse P S resolutions, that force these pegs to have roots of orders divisible by the global index gind. Since there are finitely many abelian vertex groups along a collapse P S resolution, and since gind has finitely many divisors, there are finitely many possibilities for the tuple of generators we need to add to the limit group associated with a formal collapse P S resolution, hence, we finally end up with a finite collection of limit groups associated with each of the formal collapse P S resolutions associated with a P S resolution, P SHGHRes. We analyze each of the obtained limit groups, in the same way we analyzed a collapse P S limit group. Therefore, with each of the obtained limit groups we associate a finite collection of (graded) resolutions, that we (also) call collapse extra P S resolutions. By theorem 1.30, the complexity of each of these resolutions is bounded by the complexity of the P S resolution, P SHGHRes, we have started with. If the complexity of one of the obtained resolutions is equal to the complexity of P SHGHRes, we call that resolution, formal collapse extra P S resolution.

The P S limit groups P SHGH and their associated resolutions, together with the non-rigid and non-solid P S limit groups and their associated graded resolutions, the extra P S resolutions and the collapsed extra P S resolutions, and the left P S limit groups and their associated resolutions, enable us to present the main principle which is the key in our general approach to the entire quantifier elimination process. Conceptually, we show that if a valid P S statement factors through one of the resolutions P SHGHRes then either there exists a valid P S statement that factors through one of the non-formal collapsed extra P S resolutions associated with P SHGHRes, and these have strictly smaller complexity than the P S resolution, P SHGHRes, we have started this branch with, or there exists a ”generic” valid P S statement, i.e., a test sequence of valid P S statements that factor through the P S resolution P SHGHRes.

Theorem 1.32. Let P SHGHRes be one of the P S graded resolutions, and suppose that there exists a valid P S statement:

((h_{1}^{2}, …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factors through the completion of the resolution P SHGHRes with respect to our fixed proof system. Then one of the following holds:

(1) there exists a test sequence of valid P S statements (with respect to our fixed proof system):

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

that factor through the completion of the P S resolution P SHGHRes (note that the specialization p_{0} of the defining parameters p are fixed along the entire test sequence).

(2) there exists a specialization:

(c_{0}, u_{0}, v_{0}, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a))

which is shortest form with respect to one of the collapsed extra P S resolutions CollapseExtraP SRes associated with the resolution P SHGHRes, and for which:

Cmplx(CollapseExtraP SRes) < Cmplx(Comp(P SHGHRes)).

Proof: Suppose that there exists a valid P S statement:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factors through the completion of the resolution P SHGHRes with respect to our fixed proof system, and for no specialization (c_{0}, u_{0}, v_{0}) the (combined) specialization:

(c_{0}, u_{0}, v_{0}, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a))

is a shortest form specialization that factors through one of the (non-formal) collapsed extra P S resolutions CollapseExtraP SRes associated with the resolution P SHGHRes, i.e. those for which:

Cmplx(CollapseExtraP SRes) < Cmplx(Comp(P SHGHRes))

If the collection of the ungraded resolutions covered by the Non-Rigid P S resolutions, NRgdP SRes, the ungraded resolutions covered by the Non-Solid P S resolutions, and the ungraded resolutions covered by the left P S resolutions, associated with the graded P S resolution P SHGHRes form a covering closure of all the ungraded resolutions covered by the graded resolution P SHGHRes:

P SHGHRes((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p_{0}, a)

then no valid P S statement of the form:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

factors through the completion of the resolution P SHGHRes.

Furthermore, if the collection of the ungraded resolutions covered by the Non-Rigid P S resolutions, NRgdP SRes, the Non-Solid P S resolutions, NSldP SRes, and the left P S resolutions, LeftP SRes, described above, together with the set of ungraded resolutions covered by the extra P S resolutions, ExtraP SRes, from which we take out the ungraded resolutions covered by the formal collapsed extra P S resolutions, form a covering closure of all the ungraded resolutions covered by the graded resolution P SHGHRes:

P SHGHRes((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p_{0}, a)

then for every valid P S statement of the form:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factors through the completion of the resolution P SHGHRes, there must exist a specialization: (c_{0}, u_{0}, v_{0}) for which the specialization:

(c_{0}, u_{0}, v_{0}, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a))

is a shortest form specialization that factors through one of the non-formal collapsed extra P S resolutions, CollapseExtraP SRes.

Since we have assumed that there exists a valid P S statement that factors through the completion of the resolution P SHGHRes, and for which there is no specialization (c_{0}, u_{0}, v_{0}) for which the corresponding (shortest form) specialization factors through one of the non-formal collapsed extra P S resolutions CollapseExtraP SRes, the collection of ungraded resolutions covered by the Non-Rigid P S resolutions NRgdP SRes, the Non-Solid P S resolutions NSldP SRes, the left P S resolutions, LeftP SRes, and the set of ungraded resolutions covered by the extra P S resolutions ExtraP SRes minus those that are covered by the formal collapsed extra P S resolutions, collapseExtraP SRes, does not form a covering closure of the ungraded resolutions covered by the P S resolution P SHGHRes:

P SHGHRes((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h^{1}, w, p_{0}, a).

Therefore, there must exist a test sequence:

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

of specializations that factor through the completion of the P S resolution P SHGHRes, so that for every index n the corresponding specialization does not factor through any of the Non-Rigid, Non-Solid or the left P S resolutions, NRgdP SRes, NSldP SRes and LeftP SRes, and if it factors through an extra P S resolution ExtraP SRes, then it factors through a formal collapsed extra P S resolution, CollapseExtraP SRes, associated with it. Furthermore, since there exists a valid P S statement that factors through the P S resolution, P SHGHRes, and not through any of the non-formal collapsed P S resolutions associated with it, we can further assume that for every index n, and every specialization from the test sequence, the collection of (ungraded) resolutions associated with the elements h_{j}^{2}(n) form a covering closure of the (ungraded) resolutions associated with the specializations g_{j}^{1}(n). Hence, for every index n, the corresponding specialization:

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

is a valid P S statement, so we have found a test sequence of valid P S statements and the theorem follows.

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Theorem 1.32 is the key point in our analysis of the set T_{2}(p). Conceptually, theorem 1.32 reduces the analysis of the set T_{2}(p) to the set of specializations of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through the various resolutions P SHGHRes.

Proposition 1.33. Let P SHGHRes be one of the (non-trivial) P S graded resolutions. Let T SP S(p) be the set of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements:

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{d(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

that factor through the completion of the P S resolution P SHGHRes, Comp(P SHGHRes). Then T SP S(p) is in the Boolean algebra of AE sets.

Proof: As we have already pointed out in the proof of theorem 1.32, a specialization p_{0} of the defining parameters P =< p > satisfies: p_{0} ∈ T SP S(p), if and only if the collection of ungraded resolutions covered by the Non-Rigid P S resolutions NRgdP SRes, the Non-Solid P S resolutions, the left P S resolutions, together with the set of ungraded resolutions covered by the extra P S resolutions, ExtraP SRes, from which we take out the ungraded resolutions covered by the formal collapsed extra P S resolutions, associated with the ungraded resolution P SHGHRes, does not form a covering closure of all the ungraded resolutions covered by the graded resolution P SHGHRes:

P SHGHRes((h_{1}^{2}, , g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{1}, w, p_{0}, a)

and there exists a test sequence that factors through this ungraded resolution and not through the non-rigid, non-solid, Left and extra PS resolutions (minus the associated formal collapse extra P S resolutions), for which for every index n, and every specialization from the test sequence, the collection of (ungraded) resolutions associated with the elements h_{j}^{2}(n) form a covering closure of the (ungraded) resolutions associated with the specializations g_{j}^{1}(n). Therefore, to prove the proposition, we need to find a predicate in the Boolean algebra of AE predicates that defines the set of specializations p_{0} for which the ungraded resolutions defined above and associated with the specialization p_{0} satisfy the properties listed above.

By construction, there are finitely many P S graded resolutions P SHGHRes, NRgdP SRes, NSldP SRes, LeftP SRes, ExtraP SRes, and CollapseExtraP SRes, and each of these graded resolutions terminates in either a rigid or solid graded limit group (with respect to defining parameters P =< p >). i If a rigid or solid limit groups W P HGH is associated with a solid limit group W P HG, then with W P HGH there is an associated (graded) closure of the graded resolution associated with the solid limit group W P HG. With each such (graded) closure there is an associated coset of a finite index subgroup of a free abelian group which is the direct sum of the abelian vertex groups that appear along the graded resolution associated with W P HG. Let rind be the least common multiple of the indices of all these finite index subgroups. With each P S resolution P SHGHRes there are associated rigid and solid specializations of the various subgroups W P HG that were indicated by the proof system. With each P S resolution P SHGHRes we further add graded resolutions that have the same structure as the original P S resolution P SHGHRes, and indicate if specializations of pegs of abelian groups connected to W P HG (in the graded resolution associated with it) have roots of orders that divide the least common multiple rind.

Given these additional ”rooted” resolutions, for any specialization p_{0} of the defining parameters P =< p >, there are finitely many combinations for the number of ungraded resolutions covered by a P S resolutions P SHGHRes, and the numbers of ungraded resolutions that are covered by the other graded resolutions associated with a P S resolution, P SHGHRes. A specialization p_{0} is in the set T SP S(p) if and only if the ungraded resolutions covered by these resolutions and associated with it are divided according to a subset of these combinations and not according to the complement of this subset of combinations. The combination of ungraded resolutions covered by the above graded resolutions and associated with p_{0}, is equivalent to the combination of rigid and strictly solid families of specializations of the terminating rigid and solid limit groups of these resolutions. The set of specializations p_{0} associated with a given combination of rigid and strictly solid families of specializations of (finitely many) rigid and solid limit groups is can be defined using a conjunction of an EA and an AE predicates. Hence, the set T SP S(p) is in the Boolean algebra of AE sets.

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At this stage we have all the tools needed for showing that the set T_{2}(p) is in the Boolean algebra of AE sets. By construction, if p_{0} ∈ T_{2}(p) then there must exist a valid P S statement of the form:

((h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factors through one the of the P S resolutions P SHGHRes constructed with respect to all proof systems of depth 2. By proposition 1.33, the sets T SP S(p) associated with the various P S resolutions P SHGHRes, i.e., the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of the P S resolutions P SHGHRes, is in the Boolean algebra of AE sets. By theorem 1.32 if there exists a valid P S statement that factors through a P S resolution P SHGHRes, then either there exists a test sequence of valid P S statements that factors through that P S resolution, or there must exist a specialization: (c_{0}, u_{0}, v_{0}), so that the (combined) specialization:

(c_{0}, u_{0}, v_{0}, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

factors through one the of the extra collapsed P S resolutions, CollapseExtraP SRes associated with the P S resolution P SHGHRes, and for which:

Cmplx(CollapseExtraP SRes) < Cmplx(Comp(P SHGHRes)).

We continue with each of the (non-formal) collapsed extra P S resolutions, CollapseExtraP SRes, i.e. those that satisfy:

Cmplx(CollapseExtraP SRes) < Cmplx(Comp(P SHGHRes))

in parallel. Exactly as we did with each of the P S resolutions P SHGHRes, we associate (canonically) with such a resolution, CollapseExtraP SRes, its Non-Rigid and Non-Solid P S resolutions, left P S resolutions, extra P S resolutions, and collapsed extra P S resolutions. By proposition 1.33 applied to the various resolutions CollapseExtraP SRes, the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of the resolutions CollapseExtraP SRes, are in the Boolean algebra of AE sets. By proposition 1.32 applied to the various resolutions CollapseExtraP SRes, if there exists a valid P S statement that factors through a resolution CoolapseExtraP SRes, then either there exists a test sequence of valid P S statements that factors through that extra collapse P S resolution, or there must exist a specialization: (c_{1}, u_{1}, v_{1}), so that the (combined) specialization:

(c_{1}, u_{1}, v_{1}, c_{0}, u_{0}, v_{0}, (h_{1}^{2}, g_{1}^{1}), …, (h_{d(ps)}^{2}, g_{d(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

factors through one the of the extra collapsed P S resolutions associated with the resolution CollapseExtraP SRes and for which their complexity is strictly smaller than the complexity of CollapseExtraP SRes.

Continuing iteratively with the associated collapsed extra resolutions of strictly smaller complexity, we obtain a terminating iterative procedure by the proof of theorem 1.18 which is identical to the proof of theorem 1.19 of [Se4]. The iterative procedure we have constructed has to terminate with either a rigid limit group with respect to the defining parameters P =< p >, or a solid limit groups in which the subgroup W P =< w, p > is contained in the distinguished vertex group in the associated graded abelian JSJ decomposition. By iteratively applying proposition 1.32 to the various resolutions that appear along the iterative procedure, if there exists a valid P S statement that factors through any of these resolutions, then either there exists a test sequence of valid P S statements that factors through one of these resolutions, or there must exist a valid P S statement that factors through the one of the terminating rigid or solid limit groups. By iteratively applying proposition 1.33 to the various resolutions constructed along the iterative procedure, the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of these resolutions are in the Boolean algebra of AE sets. By proposition 1.24 the set of specializations p_{0} of the defining parameters P =< p > for which there exists a valid P S statement that factors through one of the terminating rigid or solid P S limit groups is in the Boolean algebra of AE sets. The entire set T_{2}(p) is precisely the set of specializations p_{0} of the defining parameters P =< p > for which there exists a valid P S statement, hence, the set T_{2}(p) is in the Boolean algebra of AE sets, and the proof of theorem 1.22 in the minimal (graded) rank case is completed.

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At this stage we are finally ready to show that the entire set EAE(p) is in the Boolean algebra of AE sets. The tree of stratified sets has a finite depth, which (by definition) bounds the depth of all possible proof systems associated with the tree of stratified sets. For each integer d, we set T_{d}(p) to be the set of specializations p_{0} of the defining parameters P =< p > for which there exists a valid P S statement for some proof system of depth d. Clearly: EAE(p) = T_{1}(p) ∪ … ∪ T_{d_{0}}(p) where d_{0} is the depth of tree of stratified sets, hence, to show that the set EAE(p) is in the Boolean algebra of AE sets it is enough to show that each of the sets T_{d}(p) is in the Boolean algebra of AE sets. By the structure of the tree of stratified sets, and the global bounds on possible disjoint rigid and strictly solid families associated with each of the stratas in these tee, given a fixed depth d there exist finitely many possible proof systems of depth d. Given a fixed proof system of depth d, which we denote P S, we collect all its associated valid P S statements in a canonical collection of finitely many P S limit groups which we denote: P S(HG)^{d−1}H. With each P S limit group P S(HG)^{d−1}H we associate it canonical (Makanin-Razborov) resolutions P S(HG)^{d−1}HRes. With each of these P S resolutions we associate (canonically) the set of Non-Rigid and Non-Solid P S resolutions, the Left P S resolutions, the extra P S resolutions, and the collapsed extra P S resolutions. In exactly the same way we handled the set T_{2}(p) we continue iteratively with those collapsed extra P S resolutions that have strictly smaller complexity to the next step. The iterative procedure we construct terminates after finitely many steps since the complexity of the obtained resolutions strictly decreases in each step (theorem 1.18). Theorem 1.32 applied iteratively to the resolutions along the iterative procedure, implies that if there exists a valid P S statement that factors along the resolution P S(HG)^{d−1}HRes, then either there exists a test sequence of valid P S statements that factor along one of the resolutions along the iterative procedure, or there exists a valid P S statement that factors along one of the terminal rigid or solid P S limit groups. Theorems 1.33 and 1.24 applied to the various resolutions that appear along the iterative procedure, and the terminal rigid or solid P S limit groups finally implies that each of the sets T_{d}(p) is in the Boolean algebra of AE sets, hence, the entire set EAE(p) is in the Boolean algebra of AE sets, and the proof of theorem 1.4 in the minimal (graded) rank case is completed.

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The proof of theorem 1.4 shows that a set EAE(p) defined using a conjunction of a system of equalities and a system of inequalities:

EAE(p) = ∃w ∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1

is in the Boolean algebra of AE sets. The generalization of that proof to a set EAE(p) defined using a (finite) disjunction of conjunctions of a system of equalities and a system of inequalities:

EAE(p) = ∃ ∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1)

is rather straightforward. Indeed, the only change required is in the construction of the tree of stratified sets. In proving theorem 1.4, i.e., when the predicate defining the set EAE(p) is the conjunction of a system of equalities and a system of inequalities, we have constructed the tree of stratified sets iteratively, where in each step we have first collected all the formal solutions defined over closures of the resolutions of the remaining y’s from the previous step, and then applied the collections of formal solutions to the set of the remaining y’s from the previous step, and analyzed the set of y’s for which (at least) one of the inequalities from our given system is in fact an equality when we substitute the families of formal solutions we have collected, using an iterative procedure for the analysis of quotient resolutions.

When the set EAE(p) is defined using a (finite) disjunction of conjunctions of a system of equalities and a system of inequalities we work in parallel with each of the conjunctions in each step of the iterative procedure that constructs the tree of stratified sets. In each step of this iterative procedure, we do the following for the indices j, 1 ≤ j ≤ r, in parallel. We first collect all the formal solutions of the system of equalities Σ_{j}(x, y, w, p, a) defined over closures of the resolutions of the remaining y’s from the previous step, and then apply the collections of these formal solutions to the set of the remaining y’s from the previous step, and analyze the set of y’s for which (at least) one of the inequalities from the system Ψ_{j}(x, y, w, p, a) is in fact an equality when we substitute the families of formal solutions we have collected, using the iterative procedure for the analysis of quotient resolutions. The termination of this modified procedure for the construction of the tree of stratified sets follows using the same argument used in proving theorem 1.4 (theorem 1.18). Given the tree of stratified sets, the analysis of the set EAE(p) is identical to the analysis described in proving theorem 1.4. This finally shows that the set EAE(p) is in the Boolean algebra of AE sets, also when EAE(p) is defined using a (finite) disjunction of conjunctions of a system of equalities and a system of inequalities, hence, concludes the proof of theorem 1.3 in the minimal (graded) rank case.

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§2. The Tree of Stratified Sets (general case)

To obtain quantifier elimination of elementary predicates over a free group, our goal is showing that the Boolean algebra of AE sets is invariant under projections. For presentation purposes, we presented our approach to the analysis of the projection of the Boolean algebra of AE sets assuming the (graded) limit groups appear in our procedure are of minimal (graded) rank in the first section of this paper. In this section we combine the approach presented in the previous section, together with the concepts and techniques that appear in section 4 of [Se4], to generalize the construction of the tree of stratified sets, presented in the previous section assuming the limit groups in question are of minimal (graded) rank. The tree of stratified sets constructed in this section, is combined with a generalization of the ”sieve” method for identifying witnesses presented in the next sections, to finally prove the invariance of the Boolean algebra ofAE sets under projection in the general case.

By lemmas 1.1 and 1.2, the analysis of the projection of the Boolean algebra of AE sets reduces to the analysis of the projection of AE sets, hence, in order to prove the invariance of the Boolean algebra of AE sets under projection it is enough to prove theorem 1.3, i.e., to show that the projection of an AE set, an EAE set, is in the Boolean algebra of AE sets. As we did in the previous section, we will start by proving theorem 1.4, i.e., we will show that the projection of an AE set, an EAE set, is in the Boolean algebra of AE sets, in the case r = 1 (see theorem 1.3 for this notation). The generalization to arbitrary r is fairly straightforward, and is presented afterwards.

Throughout this section we will use the notions used in the previous section and in section 4 of [Se4], hence, we assume that the reader is already familiar with both. Since we are proving theorems 1.3 and 1.4, we will use the notation used in the statement of these theorems. Recall that in stating theorem 1.4 we defined the set AE(w, p) to be:

AE(w, p) = ∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1

and its projection EAE(p) to be:

EAE(p) = ∃w∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1

and our goal is to show that the set EAE(p) is in the Boolean algebra of AE sets. We start in the same way we started in the minimal (graded) rank case.

Let F_{y} be the free group F_{y} =< y_{1}, …, y_{ℓ} >, and let:

ψ_{1}(x, y, w, p, a) = 1, …, ψ_{q}(x, y, w, p, a) = 1

be the defining equations of the system Ψ(x, y, w, p, a) = 1. By ([Se2],1.2) for every p_{0} ∈ EAE(p), there exists some (witness) w_{0}, and a formal solution x = x_{w_{0},p_{0}}(y, a), so that the words corresponding to the defining equations of the system Σ(x_{w_{0},p_{0}}(y, a), y, w_{0}, p_{0}, a) are trivial in the free group F_{k} ∗ F_{y} =< a, y >, and the sentence:

∃y ψ_{1}(x_{w_{0},p_{0}}(y, a), y, w_{0}, p_{0}, a) ≠ 1 ∧ … ∧ ψ_{q}(x_{w_{0},p_{0}}(y, a), y, w_{0}, p_{0}, a) ≠ 1

is a truth sentence in F_{k}.

By the construction of graded formal limit groups presented in section 3 of [Se2], viewing W P =< w, p > as the parameters subgroup, one can associate with the free group F_{y} and the system of equations Σ(x, y, w, p, a) = 1, a (canonical) finite collection of graded formal limit groups GF L_{1}(x, y, w, p, a), …, GF L_{r}(x, y, w, p, a), so that every formal solution x = x_{w_{0},p_{0}}(y, a) of the system Σ(x, y, w, p, a) factors through at least one of the resolutions of the graded formal Makanin-Razborov diagrams of the graded formal limit groups GF L_{1}(x, y, w, p, a), …, GF L_{r}(x, y, w, p, a).

Viewing W P =< w, p > as the parameters subgroup, each graded formal resolution in the graded formal Makanin-Razborov diagrams of the graded formal limit groups GF L_{1}(x, y, w, p, a), …, GF L_{r}(x, y, w, p, a) terminates in either a rigid formal limit group of the form W P Rgd(h_{R}, w, p, a) ∗ F_{y}, where W P Rgd(h_{R}, w, p, a) is a rigid (not formal!) graded limit group (with respect to W P), or in a solid formal limit group of the form W P Sld(h_{S}, w, p, a) ∗ F_{y}, where W P Sld(h_{S}, w, p, a) is a solid (not formal!) graded limit group. Note that by ([Se3],2.3), for each specialization (w_{0}, p_{0}) there exists a global bound on the number of rigid solutions of the form (h_{R}, w_{0}, p_{0}, a) of any of the rigid graded limit groups W P Rgd(h_{R}, w, p, a), and by ([Se3],2.7), for each specialization (w_{0}, p_{0}) there exists a global bound on the number of solid families of solutions of the form (h_{S}, w_{0}, p_{0}, a) of any of the solid graded limit groups W P Sld(h_{S}, w, p, a).

Let W P Rgd(h_{R}, w, p, a) be one of the terminating rigid graded limit groups in the taut graded formal Makanin-Razborov diagram with respect to the defining parameters W P =< w, p >. The modular groups associated with each (ungraded) resolution that terminates in a rigid specialization of the rigid graded limit group W P Rgd(h_{R}, w, p, a) that lies outside the singular locus associated with W P Rgd(h_{R}, w, p, a) (see section 11 in [Se1]), are compatible with the graded formal resolution that terminates in the rigid graded formal limit group W P Rgd(h_{R}, w, p, a)∗ F_{y}, i.e., the ”tower” of modular groups in such formal graded resolution is independent of the particular rigid specialization of W P Rgd(h_{R}, w, p, a). Therefore, we separate the various stratas in the singular locus of the graded limit groups W P Rgd(h_{R}, w, p, a), and use the ”tower” of modular groups that lie ”above” each of the rigid formal graded limit groups W P Rgd(h_{R}, w, p, a) ∗ F_{y}, to associate a (usually infinite) system of equations (in the variables (h_{R}, y, w, p) and coefficients in F_{k}) corresponding to the system of equations Ψ(x, y, w, p, a) = 1. By Guba’s theorem [Gu] this infinite system of equations is equivalent to a finite system of equations λ_{R}(h_{R}, y, w, p, a) = 1. Similarly, with each (strata of the) terminating solid formal graded limit group W P Sld(h_{S}, w, p, a) ∗ F_{y} we can associate a system of equations λ_{S}(h_{S}, y, w, p, a) = 1.

From now on we work with each of the graded formal resolutions W P GF Res_{i}(x, y, w, p, a), its terminating rigid formal graded limit group W P Rgd_{i}(h_{R}, w, p, a) ∗ F_{y} or solid graded formal limit group W P Sld_{i}(h_{S}, w, p, a) ∗ F_{y} in parallel, so we may restrict our intention to one of these graded formal resolutions, omit its index and denote it W P GF Res(x, y, w, p, a). Note that each of these formal resolutions is with respect to the parameter subgroup W P =< w, p >. Suppose the graded formal resolution W P GF Res(x, y, w, p, a) terminates in the rigid graded formal limit group W P Rgd(h_{R}, w, p, a) ∗ F_{y}. Let

λW P RGL_{1}(h_{R}, y, w, p, a), …, λW P RGL_{d}(h_{R}, y, w, p, a)

be the canonical collection of maximal graded limit groups (with respect to the parameter subgroup W P =< w, p >) corresponding to the set of specializations (h_{R}, y, w, p, a) for which (h_{R}, w, p, a) is a rigid specialization of W P Rgd(h_{R}, w, p, a) that lies outside the singular locus, (h_{R}, y, w, p, a) factors through the graded formal resolution W P GF Res(x, y, w, p, a), and λ_{R}(h_{R}, y, w, p, a) = 1. Similarly, if the graded formal resolution W P GF Res(x, y, w, p, a) terminates in the solid formal graded limit group W P Sld(h_{S}, w, p, a)∗F_{y}, we associate with the solid graded limit group W P Sld(h_{S}, w, p, a) and the system of equations λ_{S}(h_{S}, y, w, p, a) the canonical collection of maximal graded limit groups

λW P SGL_{1}(h_{S}, y, w, p, a), …, λW P SGL_{d}(h_{S}, y, w, p, a)

corresponding to the set of specializations (h_{S}, y, w, p, a) for which (h_{S}, w, p, a) is a solid specialization of W P Sld(h_{S}, w, p, a) that lies outside the singular locus, (h_{S}, y, w, p, a) factors through the graded formal resolution W P GF Res(x, y, w, p, a), and λ_{S}(h_{S}, y, w, p, a) = 1.

At this point we need to collect the ”remaining” set of specializations of the variables y for each value of our parameters (w, p). Suppose that the terminating graded limit group of the formal graded resolution W P GF Res(x, y, w, p, a) is the rigid formal graded limit group Rgd(h_{R}, w, p, a)∗F_{y} or the solid graded limit group Sld(h_{S}, w, p, a) ∗ F_{y}. With each of the graded limit groups:

λW P RGL_{1}(h_{R}, y, w, p, a), …, λW P RGL_{d}(h_{R}, y, w, p, a)

or λW P RGL_{1}(h_{S}, y, w, p, a), …, λW P RGL_{d}(h_{S}, y, w, p, a) (depending on whether the graded formal resolution W P GF Res(x, y, w, p, a) terminates in a rigid or solid limit group with respect to W P =< w, p >) we associate its taut graded Makanin-Razborov diagram with respect to the parameter subgroup < h_{R}, w, p > or < h_{S}, w, p > in correspondence. Each graded resolution λW P GRes(y, h_{R}, w, p, a) (or λW P GRes(y, h_{S}, p, a)) in one of the taut graded diagrams of the graded limit groups: λW P RGL_{1}(h_{R}, y, w, p, a), …, λW P RGL_{d}(h_{R}, y, w, p, a) (or

λW P RGL_{1}(h_{S}, y, w, p, a), …, λW P RGL_{d}(h_{S}, y, w, p, a))

terminates in either a rigid graded limit group (with respect to < h_{R}, w, p > or < h_{S}, w, p >) which we denote λW P Rgd(g_{R}, h_{R}, w, p, a) (or λW P Rgd(g_{R}, h_{S}, w, p, a)) or a solid graded limit group which we denote λW P Sld(g_{S}, h_{R}, w, p, a) (or

λW P Sld(g_{S}, h_{S}, w, p, a)).

With each graded resolution λW P GRes(y, h_{R}, w, p, a) or λW P GRes(y, h_{S}, w, p, a) we associate the finite set of taut resolutions and their terminal rigid or solid limit groups associated with the various stratas of the singular locus associated with the resolution. We add the resolutions associated with the singular loci to the list of resolutions containing the entire set of the remaining y’s.

To construct the graded formal resolutions GF Res(x, y, w, p, a), we have collected all formal solutions x_{(w,p)}(y, a), for which all the words corresponding to the equations in Σ(x_{(w,p)}(y, a), y, w, p, a) represent the trivial words in F_{a,y} =< a, y >. By theorem 1.18 of [Se2], if p_{0} ∈ EAE(p) then there must exist some ”witness” w_{0} and a formal solution x_{(w_{0},p_{0})}(y, a) so that the maximal limit groups corresponding to each of the equations in the system Ψ(x_{(w_{0},p_{0})}(y, a), y, w, p, a) = 1 are all proper quotients of the free group < a, y > = F_{k} ∗ F_{y}. Hence, for every p_{0} ∈ EAE(p) there must exist some witness w_{0}, and a rigid specialization (hr_{0}, w_{0}, p_{0}) of one of the rigid limit groups W P Rgd(h_{R}, w, p, a) or a solid specialization (hs_{0}, w_{0}, p_{0}) of one of the solid limit groups W P Sld(h_{S}, w, p, a), so that every ungraded resolution W P GRes(y, hr_{0}, w_{0}, p_{0}, a) (or W P GRes(y, hs_{0}, w_{0}, p_{0}, a)) does not correspond to the entire set of y’s but rather to a resolution of a limit group which is a proper quotient of the free group < a, y > = F_{k} ∗ F_{y}.

Therefore, the outcome of the preliminary step of our ”trial and error” procedure is a decrease in the complexity (definitions 1.15 and 3.2 of [Se4]) of the ungraded resolutions of y’s associated with each p_{0} ∈ AE(p), at least for one specialization (hr_{0}, w_{0}, p_{0}) or (hs_{0}, w_{0}, p_{0}). Each of the next steps of the procedure is meant to sequentially decrease either the complexity of the ”remaining” ungraded resolutions or Zariski closures of certain sets of specializations associated with the staggered data-structures we construct, as we did in the iterative procedure for validation of a sentence ([Se4],4). As in the minimal (graded) rank case, once the iterative procedure terminates, we present second iterative procedure, that uses the outcome of the first iterative procedure to sequentially approximate the set EAE(p) by sets which are all in the Boolean algebra of AE sets. Finally, we show that the approximations we construct in the second iterative procedure get identical with the set EAE(p) after finitely many steps. Since the approximations are all in the Boolean algebra of AE sets, this will imply that an EAE set is indeed in the Boolean algebra of AE sets, which finally proves theorem 1.4.

For the continuation of the iterative procedure we will denote (for brevity) each of the limit groups W P Rgd(h_{R}, w, p, a) or W P Sld(h_{S}, w, p, a) by W P H(h, w, p, a), and each of the limit groups:

λW P Rgd(g_{R}, h_{R}, w, p, a), λW P Rgd(g_{R}, h_{S}, w, p, a), λW P Sld(g_{S}, h_{R}, w, p, a),

λW P Sld(g_{S}, h_{S}, w, p, a)

by W P HG(g, h, w, p, a). Our treatment of these limit groups will be conducted in parallel, so we don’t keep the indices associated with each of these (finite collection of) limit groups. Also, the rest of our ”trail and error” procedure does not depend in an essential way on the type (rigid or solid) of the terminating graded limit groups in the preliminary two steps, hence, we do not keep notation for the type of each of these terminating limit groups.

We continue to the next step of our iterative procedure for the analysis of the set EAE(p) only with graded resolutions λW P GRes(y, h_{R}, w, p, a) or λW P GRes(y, h_{S}, p, a) for which each ungraded resolution associated with a specialization of the corresponding terminating rigid or solid limit group W P HG does not correspond to the entire limit group F_{y} ∗ F_{k}, but rather to a resolution of a proper quotient of it. For each tuple (g_{0}, h_{0}, w_{0}, p_{0}), which is either a rigid or a solid specialization of such terminating limit group W P HG there is an ungraded (well-separated) resolution associated. The associated ungraded resolution depends only on the solid family of the specialization in case the corresponding terminating limit group is solid. Also, the ungraded resolution may be degenerate, so we separate the finitely many possible types of ungraded resolutions associated with a rigid or solid specialization (g_{0}, h_{0}, w_{0}, p_{0}) of W P GH according to the stratas of the corresponding singular locus (see section 11 of [Se1]). With the associated ungraded well-separated resolution, we may associate its completion. Given a tuple (g_{0}, h_{0}, w_{0}, p_{0}), which is a rigid or solid specialization of W P HG, we collect all the formal solutions {x_{(g_{0},h_{0},w_{0},p_{0})}(s, z, y, a)} for which the words corresponding to the equations in the system Σ(x_{(g_{0},h_{0},w_{0},p_{0})}(s, z, y, a), y, w_{0}, p_{0}, a) are the trivial words in some closure of the completion of the ungraded resolution associated with the given specialization. Using the construction presented in section 3 of [Se2], and viewing the subgroup < h, w, p > as parameters, from the entire collection of formal solutions for all possible specializations (g_{0}, h_{0}, w_{0}, p_{0}) which are rigid or solid specializations of W P HG we can construct a graded formal Makanin-Razborov diagram, so that any formal solution defined over a closure of an ungraded resolution associated with a rigid or solid specialization of W P HG, factors through one of the graded formal Makanin-Razborov resolutions.

Let GF L_{1}(x, z, y, g_{1}, h_{1}, w, p, a), …, GF L_{r}(x, z, y, g_{1}, h_{1}, w, p, a) be the maximal graded formal limit groups constructed from the collection of formal solutions associated with the graded limit group W P HG. By section 3 of [Se2], with each of the graded formal limit groups there is an associated graded formal Makanin-Razborov diagram with respect to the parameter subgroup < g_{1}, h_{1}, w, p >. By theorems 3.7 and 3.8 of [Se2], each of the graded formal resolutions in the graded formal Makanin-Razborov diagram associated with each of the graded formal limit groups: GF L_{1}(x, z, y, g_{1}, h_{1}, w, p, a), …, GF L_{r}(x, z, y, g_{1}, h_{1}, w, p, a) (with respect to < g_{1}, h_{1}, w, p >) terminates in either a group of the form:

W P HGRgd(h_{2}^{R}, g_{1}, h_{1}, w, p, a) ∗_{<r,g_{1},h_{1},w,p,a>} GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

where W P HGRgd(h_{2}^{R}, g_{1}, h_{1}, w, p, a) is a graded (not formal!) limit group which is rigid with respect to the parameter subgroup < g_{1}, h_{1}, w, p >, and

GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

is a graded formal closure of the (well-separated) graded resolution

W P HGF Res(y, g_{1}, h_{1}, w, p, a)

associated with the graded limit group W P HG. We will denote a terminating rigid graded limit group W P HGRgd(h_{2}^{R}, g_{1}, h_{1}, w, p, a) by W P HGH^{R}. Alternatively, the terminating graded formal limit group of a graded formal resolution in one of the graded formal Makanin-Razborov diagrams associated with the graded formal limit groups GF L_{1}(x, z, y, g_{1}, h_{1}, w, p, a), …, GF L_{r}(x, z, y, g_{1}, h_{1}, w, p, a) is of the form:

W P HGSld(h_{2}^{S}, g_{1}, h_{1}, w, p, a) ∗_{<r,g_{1},h_{1},w,p,a>} GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

where W P HGSld(h_{2}^{S}, g_{1}, h_{1}, w, p, a) is a graded (not formal!) limit group which is solid with respect to the parameter subgroup < g_{1}, h_{1}, w, p >, and

GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

is a graded formal closure of the graded resolution W P HGF Res(y, g_{1}, h_{1}, w, p, a) associated with the graded limit group W P HG. We will denote the terminating solid graded limit group W P HGSld(h_{2}^{S}, g_{1}, h_{1}, w, p, a) by W HGH^{S}.

At this point we need to associate with each of the graded formal closures, a (canonical) finite collection of graded auxiliary resolutions, and their corresponding graded auxiliary decompositions, graded modular groups and graded auxiliary limit groups in a similar way to the auxiliary resolutions associated with a closure presented in definition 4.1 of [Se4].

Definition 2.1. Let a graded formal resolution terminate in either a rigid graded formal limit group W P HF GRgd(s, z, y, h_{2}^{R}, g_{1}, h_{1}, w, p, a) or a solid graded formal limit group W P HF GSld(s, z, y, h_{2}^{S}, g_{1}, h_{1}, w, p, a), and let

GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a)

be its terminating graded formal closure. Suppose that the graded formal closure GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a) contains ℓ levels. Let Rlim(y, h_{1}, w, p, a) be the subgroup generated by < y, h_{1}, w, p, a > in the limit group associated with the graded formal closure, and let Rlim(z_{i}, h_{1}, w, p, a > be its image in the (graded) limit group associated with the i-th level in the graded formal closure GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a). With the terminal rigid or solid graded formal limit group, we associate a taut multi-graded Makanin-Razborov diagram with respect to the non-QH, non-abelian vertex groups and edge groups in the graded abelian JSJ decomposition of subgroup < y, h_{1}, w, p, a > with respect to the parameter subgroup < h_{1}, w, p >. We call each of the resolutions in this multi-graded diagram a (graded) auxiliary resolution, and its terminating solid or rigid limit group a (graded) auxiliary limit group, which we denote: AuxGF Cl(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). Note that a graded auxiliary limit group is a quotient of W P HF GRgd(s, z, y, h_{2}^{R}, g_{1}, h_{1}, w, p, a) or W P HF GSld(s, z, y, h_{2}^{S}, g_{1}, h_{1}, w, p, a) we have started with, in correspondence. With a graded auxiliary limit group we further associate it’s multi-graded decompositions with respect to the various non-QH, non-abelian vertex groups and edge groups in the graded abelian decompositions of the various subgroups < z_{i}, h_{1}, w, p, a >, where the elements z_{i} generate the (graded) limit group associated with the i-th level in the given graded resolution, which we call the auxiliary (multi) graded decompositions. To the list of the above auxiliary resolutions, limit groups, and multi-graded decompositions, we add the list of multi-graded resolutions corresponding to the singular locus of all possible graded auxiliary decompositions. We call such a resolution a graded auxiliary resolution as well. With each such resolution we associate its corresponding graded auxiliary limit group and auxiliary graded decomposition.

With an auxiliary resolution we associate limit groups which we call auxiliary domain. With the terminal rigid or solid limit group of the (graded) auxiliary resolution, we associate its graded taut Makanin-Razborov diagram with respect to the parameter subgroup < h_{2}, g_{1}, h_{1}, w, p >. Each terminal rigid or solid limit group of a graded resolution in this taut graded Makanin-Razborov diagram is called auxiliary domain and denoted AuxDom(W P HGH)(d, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose we are given a graded formal resolution defined over one of the rigid or solid graded limit groups W P HGH, its terminal rigid or solid graded formal limit group W P HF GRgd(s, z, y, h_{2}^{R}, g_{1}, h_{1}, w, p, a) or W P HF GSld(s, z, y, h_{2}^{S}, g_{1}, h_{1}, w, p, a), its associated graded formal closure GF Cl(r, s, z, c, y, g_{1}, h_{1}, w, p, a), one of its graded auxiliary resolution, and its associated graded auxiliary limit group

AuxGF Cl(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose that x(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) is a formal solution that factors through the given graded formal resolution and through its associated graded auxiliary resolution. We can clearly use the modular groups associated with both the various levels of the graded auxiliary resolution, and the various levels of the given graded formal resolution, to construct a family of formal solutions x_{α}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) defined over the graded auxiliary limit group AuxGF Cl(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). In the steps of the iterative procedure for analyzing an EAE set we present, we always start with this setup, which includes a graded formal resolution and its terminal rigid or solid graded formal limit group, its associated graded formal closure, one of its graded auxiliary resolutions, its associated graded auxiliary limit group, and a family of formal solutions associated with the graded auxiliary resolution and with the graded formal resolution we have started with.

To analyze the remaining set of y’s, we construct an iterative procedure that produces a sequence of multi-graded well-separated resolutions and their completions. The iterative procedure for analyzing an EAE set is based on the procedures presented in the first section and the iterative procedure for validation of a sentence presented in section 4 of [Se4]. As in section 4 of [Se4] we will associate finitely many (multi-graded) staggered data-structures, developing resolutions, auxiliary resolutions and anvils with each of the graded quotient resolutions constructed along the procedure. As we did in the previous section, in each step of the procedure after analyzing the quotient resolutions that contain the remaining set of y’s, we collect all the formal solutions defined over closures of these quotient resolutions in finitely many multi-graded formal limit groups, and continue to the next step of the procedure with the resolutions that appear in the finite collection of resolutions that appear in the taut multi-graded formal Makanin-Razborov diagrams of these multi-graded formal limit groups.

Let GF L(x, z, y, g_{1}, h_{1}, w, p, a) be one of the graded formal limit groups collecting the formal solutions defined over the graded resolution λW P HRes(y, g_{1}, h_{1}, w, p, a), and let W P HF GRes(x, z, y, g_{1}, h_{1}, w, p, a) be one of the graded formal resolutions in the taut graded formal Makanin-Razborov diagram of GF L(x, z, y, g_{1}, h_{1}, w, p, a) that terminates in a rigid graded formal limit group W P HF GRgd(s, z, y, h_{2}^{R}, g_{1}, h_{1}, w, p, a) or the solid graded formal limit group W P HF GSld(s, z, y, h_{2}^{S}, g_{1}, h_{1}, w, p, a). With the terminating rigid or solid formal limit group we associate one of the graded auxiliary resolutions, its associated graded auxiliary limit group:

AuxGF Cl(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)

graded auxiliary modular groups, and graded auxiliary decompositions. With all these we associate an anvil and set it to be the graded auxiliary limit group: AuxGF Cl(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), and denote it:

Anv(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

With the anvil we associate a family of formal solutions {x_{α}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)}, obtained using the graded modular groups associated with the graded auxiliary resolution and the graded formal resolution we have started with, so that for each specialization (s_{0}, z_{0}, y_{0}, h2_{0}, g1_{0}, h1_{0}, w_{0}, p_{0}, a) that factors through and is taut with respect to the graded auxiliary resolution:

Σ(x_{α}(s_{0}, z_{0}, y_{0}, h2_{0}, g1_{0}, h1_{0}, w_{0}, p_{0}, a), y_{0}, w_{0}, p_{0}, a) = 1.

At this stage we continue as in the preliminary step, i.e., we need to analyze the set of y’s for which for the entire family of formal solutions: {x_{α}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)} associated with the anvil: ψ_{j}(x_{α}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), y, w, p, a) = 1 for some of the equations ψ_{j} in the system Ψ(x, y, w, p, a) used to define the set EAE(p). The entire collection of specializations (s_{0}, z_{0}, y_{0}, h2_{0}, g1_{0}, h1_{0}, w_{0}, p_{0}, a), y_{0}, w_{0}, p_{0}, a) that factor through and are taut with respect to the anvil Anv(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) and satisfy at least one of the equations ψ_{j}(x_{α}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), y, w, p, a) = 1 for every index α, is contained in a finite set of maximal graded limit groups:

QRlim_{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), …, QRlim_{u}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

Since our analysis of these quotient limit groups is conducted in parallel, we will continue with one of them and omit its index.

In section 1 we have started the procedure for validation of a sentence with a graded (minimal rank) modular block, and showed that the complexities of the various (quotient) graded modular blocks associated with it which we need to continue to the next step of the iterative procedure, are strictly smaller than the complexity of the original modular block. In the general case, we are not able to get a reduction in the complexity of the obtained graded modular blocks in each step of our iterative procedure. To ”force” the ”size” of the set of the remaining specializations y_{0} associated with each specialization of the variables < w, p > to actually decrease, we need to associate with each graded modular block information about certain (multi-graded) resolutions and abelian decompositions associated with it together with Zariski closures of some subgroups associated with the graded modular block. To carry all the information attached to a graded modular block, we associate a (graded) staggered data-structure, and (canonical) resolutions which we call graded developing resolutions, with each of the quotient resolutions. We construct the graded staggered data-structure and graded developing resolutions iteratively, in a similar way to their construction in the iterative procedure for validation of a sentence presented in section 4 of [Se4]. As in the iterative procedure for validation of a sentence, we divide the construction of the graded developing resolution and the associated anvil into several cases, depending on the structure of the graded resolution W P HGRes(y, h_{1}, w, p, a) associated with the anvil Anv(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) we start the initial step with, and the structure of the multi-graded quotient resolutions constructed along the initial step. We describe the initial step of our iterative procedure, then the general step of the iterative procedure, and finally prove the termination of the iterative procedure for the analysis of an EAE set.

I: The initial step

(1) Let Q(y, h_{1}, w, p, a) be the restricted limit group generated by < y, h_{1}, w, p, a > in the limit group associated with the graded resolution W P HGRes(y, h_{1}, w, p, a) associated with the anvil, Anv(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). Q(y, h_{1}, w, p, a) is a quotient of the limit group associated with the resolution

λW P HGRes(y, h_{1}, w, p, a)

associated with the anvil, Anv(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), we started the preliminary step with. Let Q^{1}(y, h_{1}, w, p, a) be the limit group generated by < y, h_{1}, w, p, a > in the quotient limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). If Q^{1}(y, h_{1}, w, p, a) is a proper quotient of the subgroup Q(y, h_{1}, w, p, a), we continue this branch of the iterative procedure by replacing the quotient limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) with the quotient graded resolutions obtained by starting the preliminary step of the procedure with the limit group Q^{1}(y, h_{1}, w, p, a) instead of the limit group associated with the graded resolution λW P HGRes(y, h_{1}, w, p, a) we started the preliminary step with. Since the graded resolution λW P HGRes(y, h_{1}, w, p, a) is not of maximal possible complexity, i.e., each ungraded resolution does not correspond to the entire free group F_{y} we have started the iterative procedure with, and the maximal complexity graded resolutions were collected separately in the preliminary step of the procedure, in analyzing graded resolutions of Q^{1}(y, h_{1}, w, p, a) we need to consider only those graded resolutions which are not of maximal possible complexity, i.e., only those graded quotient resolutions for which each ungraded resolution does not correspond to the entire free group F_{y}.

(2) At this stage we may assume that Q^{1}(y, h_{1}, w, p, a) is isomorphic to Q(y, h_{1}, w, p, a). We set the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} to be the non-abelian, non-QH vertex groups in the graded abelian decomposition associated with the top level of the anvil Anv(W P HGH)(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). Note that the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the anvil is, by definition, a subgroup of Base_{2,1}^{1}. With the quotient limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) and its subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} we associate its multi-graded taut Makanin-Razborov diagram which contains the multi-graded quotient resolutions:

MGQRes_{1}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), …, MGQRes_{q}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a).

If for some QH vertex group Q in the abelian decomposition associated with the top level of the graded resolution λW P GRes(y, h_{1}, w, p, a), the sequence of abelian decompositions Q inherits from a multi-graded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is not compatible with the specified collection of s.c.c. on Q that are mapped into the trivial element in the second level of the graded resolution λW P GRes(y, h_{1}, w, p, a), we omit the multi-graded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) from our list of multi-graded resolutions of the quotient limit group

QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

We continue with each of the remaining multi-graded quotient resolutions in parallel, hence, we omit the index of the specific resolution we continue with.

By proposition 4.2 of [Se4] the complexity of each of the multi-graded abelian decompositions associated with the various levels of the multigraded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is bounded by the complexity of the graded abelian decomposition associated with the top level of the graded resolution λW P Res(y, h_{1}, w, p, a) associated with the anvil we have started with. In this part we also assume that the multigraded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is not of maximal possible complexity, i.e., it does not have a single level with a (multi-graded) abelian decomposition identical with the abelian decomposition associated with the top level of λW P GRes(y, h_{1}, w, p, a). The case of maximal complexity will be treated in the next part of the initial step of the procedure. We treat such a multi-graded quotient resolution as we treated a quotient resolution in part (2) of the initial step of the iterative procedure for validation of a sentence. To handle multi-graded resolutions that are not of maximal complexity, we need the following two basic observations.

Proposition 2.2. Let MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be one of the multigraded quotient resolutions constructed above. By construction, the limit group Q^{1}(y, h_{1}, w, p, a) is mapped into the limit group associated with each of the levels of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Let Q_{term}^{1}(y, h_{1}, w, p, a) be the image of Q(y, h_{1}, w, p, a) in the terminal (rigid or solid) limit group of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Then the terminal limit group of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) can be replaced by two finite collections of multi-graded limit groups, a (possibly empty) finite collection of terminal multi-graded limit groups in which Q_{term}^{1}(y, h_{1}, w, p, a) is a proper quotient of Q^{1}(y, h_{1}, w, p, a), and a (possibly empty) finite collection of terminal rigid or solid graded limit groups that are rigid or solid with respect to the parameter subgroup < h_{1}, w, p >.

Proof: The argument is similar to the argument used to prove part (ii) of proposition 4.3 of [Se4]. By part (1) we may assume that Q^{1}(y, h_{1}, w, p, a) is isomorphic to Q(y, h_{1}, w, p, a). Let MGT erm(s, z, y, h_{1}, w, p, a) be the terminal rigid or solid limit group of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). With the terminal limit group MGT erm(s, z, y, h_{1}, w, p, a) we associate the resolutions that appear in its graded taut Makanin-Razborov diagram with respect to the parameter subgroup < h_{1}, w, p >. We treat these resolutions in parallel, so let GT Res(s, z, y, h_{1}, w, p, a) be one of these resolutions. If the graded resolution GT Res(s, z, y, h_{1}, w, p, a) has only one level, and corresponds to either rigid or solid specializations of the rigid or solid graded limit group MGT erm(s, z, y, h_{1}, w, p, a) (with respect to the parameter subgroup < h_{1}, w, p, a >), the conclusion of the proposition follows for that graded resolution. Hence, for the rest of the argument we may assume that the graded resolution GT Res(s, z, y, h_{1}, w, p, a) does not correspond to rigid or solid specializations of MGT erm(s, z, y, h_{1}, w, p, a). Let Λ_{top} be the graded abelian decomposition associated with the top level of the graded resolution GT Res(s, z, y, h_{1}, w, p, a). Q_{term}(y, h_{1}, w, p, a) inherits a (graded) abelian decomposition from Λ_{top}. Since Q_{term}(y, h_{1}, w, p, a) is isomorphic to Q(y, h_{1}, w, p, a), the non-QH, non-abelian vertex groups in the graded abelian JSJ decomposition of Q_{term}(y, h_{1}, w, p, a) have to be elliptic in the graded abelian decomposition inherited by Q_{term}(y, h_{1}, w, p, a) from Λ_{top}. Hence, if any of the subgroups Base_{2,j}^{1} is not elliptic in the abelian decomposition Λ_{top}, then for each specialization (y_{0}, h_{0}^{1}, w_{0}, p_{0}, a), for which there exists a specialization (s_{0}, z_{0}, y_{0}, h_{0}^{1}, w_{0}, p_{0}, a) that factors through and is taut with respect to the graded resolution obtained from the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) followed by the resolution GT Res(s, z, y, h_{1}, w, p, a), the specialization (s_{0}, z_{0}, y_{0}, h_{0}^{1}, w_{0}, p_{0}, a) factors through a graded auxiliary resolution of higher complexity than the graded auxiliary resolution we have started with. In this case we can omit the resolution GT Res(s, z, y, h_{1}, w, p, a) from our list of graded resolutions of the terminal limit group MGT erm(s, z, y, h_{1}, w, p, a). Hence, for the continuation we may assume that for all j, 1 ≤ j ≤ v_{1}, the subgroups Base_{2,j}^{1} are elliptic in the abelian decomposition Λ_{top}. If any of the non-abelian, non-QH vertex groups or any of the edge groups in the multi-graded abelian decomposition associated with the terminal level of the multi-graded resolution

MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

is not elliptic in Λ_{top}, then for each specialization (y_{0}, h_{0}^{1}, w_{0}, p_{0}), for which there exists a specialization (s_{0}, z_{0}, y_{0}, h_{0}^{1}, w_{0}, p_{0}, a) that factors through and is taut with respect to the graded resolution obtained from the graded resolution

MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

followed by the resolution GT Res(s, z, y, h_{1}, w, p, a), the specialization (s_{0}, z_{0}, y_{0}, h_{0}^{1}, w_{0}, p_{0}, a) factors through a multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) of higher complexity.

Therefore, we may assume that all the non-QH, non-abelian vertex groups and all the edge groups in the multi-graded abelian decomposition associated with the terminal level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), and all the subgroups Base_{2,j}^{1} are elliptic in the abelian decomposition Λ_{top}. Hence, the modular groups associated with the abelian decomposition Λ_{top}, Mod(Λ_{top}), is a subgroup of the graded modular group associated with the terminal level of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Hence, the multi-graded modular group associated with the terminal level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) can be applied to map the terminal limit group MGT erm(s, z, y, h_{1}, w, p, a) into the limit group associated with the second level of the graded resolution GT Res(s, z, y, h_{1}, w, p, a). If the limit group associated with the second level of GT Res(s, z, y, h_{1}, w, p, a) is rigid or solid with respect to < h_{1}, w, p, a >, or if the image of Q_{term}(y, h_{1}, w, p, a) in that limit group is a proper quotient of Q_{term}(y, h_{1}, w, p, a), we replace the limit group MGT erm(s, z, y, h_{1}, w, p, a) with the limit group associated with the second level of GT Res(s, z, y, h_{1}, w, p, a) and the claim of the proposition follows. If the image of Q_{term}(y, h_{1}, w, p, a) in the limit group associated with the second level of GT Res(s, z, y, h_{1}, w, p, a) is isomorphic to Q_{term}(s, z, y, h_{1}, w, p, a), we apply the same argument to show that the graded modular groups associated with the terminal level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) can be used to map MGT erm(s, z, y, h_{1}, w, p, a) into the limit group associated with the third level of the graded resolution GT Res(s, z, y, h_{1}, w, p, a). Applying this argument inductively, finishes the proof of the proposition.

□

By lemma 2.2 we can either omit the multi-graded resolution

MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

from our list of multi-graded quotient resolutions, or we may assume that either the image of the subgroup Q^{1}(y, h_{1}, w, p, a) in the terminal graded limit group of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), Q_{term}^{1}(y, h_{1}, w, p, a), is a proper quotient of Q^{1}(y, h_{1}, w, p, a), or the terminal graded limit group of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is rigid or solid with respect to the parameter subgroup < h_{1}, w, p >. To continue handling the various multi-graded resolutions

MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

of the quotient limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), we also need the following proposition that is similar to proposition 4.4 of [Se4].

Proposition 2.3. Let MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be one of the multigraded quotient resolutions in our list of multi-graded quotient resolutions of

QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a).

Let Q^{1}(y, h_{1}, w, p, a) and Q^{1}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be the subgroups generated by < y, h_{1}, w, p, a > and < s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a > in correspondence, in the limit group associated with the multi-graded quotient resolution

MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a).

Let Q_{2}^{1}(y, h_{1}, w, p, a) and Q_{2}^{1}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be the images in the limit group associated with the second level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), GQlim_{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), of the subgroups: Q^{1}(y, h_{1}, w, p, a) and Q^{1}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), in correspondence. Then either Q_{2}^{1}(y, h_{1}, w, p, a) is a proper quotient of Q^{1}(y, h_{1}, w, p, a), or

Q_{2}^{1}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

is a proper quotient of Q^{1}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a).

Proof: The proposition is simply a basic property of a multi-graded resolution.

□

Suppose that the image of Q^{1}(y, h_{1}, w, p, a) in the limit group associated with the second level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), Q_{2}^{1}(y, h_{1}, w, p, a), is a proper quotient of Q^{1}(y, h_{1}, w, p, a). In this case we do the following. With the subgroup Q_{2}^{1}(y, h_{1}, w, p, a) we associate the graded resolutions that appear in its graded taut Makanin-Razborov diagram with respect to the subgroup < h_{1}, w, p >: GQRes_{1}(y, h_{1}, w, p, a), …, GQRes_{t}(y, h_{1}, w, p, a). We continue with each of the graded resolutions GQRes_{j}(y, h_{1}, w, p, a) in parallel. If the subgroup generated by < y, h_{1}, w, p, a > in the limit group associated with the resolution GQRes_{j}(y, h_{1}, w, p, a) is a proper quotient of Q_{2}^{1}(y, h_{1}, w, p, a), we replace the graded resolution GQRes_{j}(y, h_{1}, w, p, a) by starting part (2) of the the initial step with the graded resolution obtained from MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) by replacing its second limit group Q_{2}^{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) with the maximal limit groups obtained from all specializations that factor through both Q_{2}^{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) and the subgroup generated by < y, h_{1}, w, p, a > in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a). If the subgroup generated by

< s, z, y, h_{2}, g_{1}, h_{1}, w, p, a >

in the obtained (one level) resolution, QRlim′(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of QRlim(s, z, y, h_{2}, g_{1}, w, p, a), we replace the obtained resolution by starting the initial step of our iterative procedure with the limit group QRlim′(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) instead of the limit group

QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)

and since the resolution MGQRes is not of maximal complexity, in analyzing the limit group QRlim′(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) we need to consider only those resolutions in its multi-graded Makanin-Razborov diagram that are not of maximal possible complexity. Hence, for the rest of this part we may assume that the subgroup generated by < y, h_{1}, w, p, a > in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a) is isomorphic to the subgroup generated by < y, h_{1}, w, p, a > in Q_{2}^{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a). Suppose that a graded quotient resolution GQRes_{j}(y, h_{1}, w, p, a) is of maximal possible complexity, i.e., the limit group associated with it is of the form < h_{1}, w, p, a > ∗F_{y}. Since the limit group Q_{2}^{1}(y, h_{1}, w, p, a) is a proper quotient of Q^{1}(y, h_{1}, w, p, a), and since the limit group Q_{2}^{1}(y, h_{1}, w, p, a) is naturally mapped onto the limit group < h_{1}, w, p, a > ∗F_{y} associated with the graded resolution GQRes_{j}(y, h_{1}, w, p, a), the Hopf property for limit groups implies that the subgroup generated by < h_{1}, w, p, a > in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a) is a proper quotient of the limit group generated by < h_{1}, w, p, a > in Q^{1}(y, h_{1}, w, p, a). Hence, we can replace the resolution GQRes_{j}(y, h_{1}, w, p, a) by starting the preliminary step of the procedure with the subgroup < h_{1}, w, p, a > ∗F_{y}, where < h_{1}, w, p, a > is the subgroup generated by these elements in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a). Since resolutions of maximal possible complexity of the limit group Q^{1}(y, h_{1}, w, p, a) with respect to the parameter subgroup < h_{1}, w, p, a >, i.e., those corresponding to the entire free group F_{y}, were already analyzed in the preliminary step of the procedure, we can omit a graded resolution GQRes_{j}(y, h_{1}, w, p, a) from our list of graded quotient resolutions {GQRes_{j}(y, h_{1}, w, p, a)}. Hence, for the rest of this part we may assume that the graded resolution GQRes_{j}(y, h_{1}, w, p, a) we continue with is not of maximal possible complexity. If the subgroup generated by < y, h_{1}, w, p, a > in the limit group corresponding to the graded resolution CRes_{j}(y, h_{1}, w, p, a), where CRes_{j}(y, h_{1}, w, p, a) is the graded resolution obtained from the resolution induced by the subgroup < y, h_{1}, w, p, a > from the completion of the graded resolution

MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

followed by the graded resolution GQRes_{j}(y, h_{1}, w, p, a), is a proper quotient of Q^{1}(y, h_{1}, w, p, a), we replace the graded resolution GQRes_{j}(y, h_{1}, w, p, a) by starting the preliminary step of the procedure with the subgroup generated by < y, h_{1}, w, p, a > in the limit group corresponding to the resolution CRes_{j}(y, h_{1}, w, p, a), and treat only those graded resolutions of this limit group (with respect to the parameter subgroup < h_{1}, w, p, a >) that are not of maximal possible complexity, i.e., those graded resolutions that do not cover ungraded ones that correspond to the entire free group F_{y}. Hence, we may assume that for the rest of this part the subgroup generated by < y, h_{1}, w, p, a > in the limit group corresponding to CRes_{j}(y, h_{1}, w, p, a) is isomorphic to Q^{1}(y, h_{1}, w, p, a). In particular, we may assume that each of the graded resolutions GQRes_{j}(y, h_{1}, w, p, a) in question is not of maximal possible complexity.

We now treat each of the graded resolutions GQRes_{j}(y, h_{1}, w, p, a) and their associated resolution CRes_{j}(y, h_{1}, w, p, a) in a similar way to our treatment of multi-graded quotient resolutions in part (4) of the general step of the iterative procedure for validation of a sentence presented in section 4 of [Se4]. Let W P HGHG(g_{2}, h_{1}, w, p, a) be the terminal rigid or solid limit group of the graded resolution GQRes_{j}(y, h_{1}, w, p, a) (which is also the terminal rigid or solid limit group of its associated resolution CRes_{j}(y, h_{1}, w, p, a)). We start by collecting all the formal solutions defined over ungraded resolutions covered by the graded resolution CRes_{j}(y, h_{1}, w, p, a). This collection of formal solutions factors through a canonical collection of graded formal limit groups. With each graded formal limit group we associate its graded formal Makanin-Razborov diagram as we did in section 3 of [Se2]. We continue with each of the graded formal resolutions that appear in these diagrams in parallel. Let GF Res(x, t, y, g_{2}, h_{1}, w, p, a) be a graded formal resolution in one of these diagrams, and let W P HGHGH(h_{3}, g_{2}, h_{1}, w, p, a) be its terminating rigid or solid (not formal!) limit group. With the graded formal resolution GF Res(x, t, y, h_{3}, g_{2}, h_{1}, w, p, a) we associate the resolution

GRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a)

which is the graded (not formal!) resolution associated with the terminal formal limit group of the graded formal resolution GF Res(x, t, y, h_{3}, g_{2}, h_{1}, w, p, a) (i.e., the formal graded closure associated with the formal resolution GF Res). With GRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a) we associate the graded resolution

BRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a)

which is obtained from GRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a) by excluding its top level. Note that the terminal rigid or solid limit groups of those resolutions is W P HGHGH(h_{3}, g_{2}, h_{1}, w, p, a). With BRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a) we associate its (canonical) finite collection of graded auxiliary resolutions GAuxRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a) (that are multi-graded with respect to the non-abelian, non-QH vertex groups and edge groups in the graded abelian decomposition of Q(y, h_{1}, w, p, a)), their associated graded abelian decompositions and graded auxiliary limit groups AuxGF Cl(t, y, h_{3}, g_{2}, h_{1}, w, p, a). We continue with each of the graded auxiliary resolutions and their associated graded auxiliary decompositions and graded auxiliary limit groups in parallel.

If the subgroup generated by < y, h_{1}, w, p, a > in a graded auxiliary limit group AuxGF Cl(t, y, h_{3}, g_{2}, h_{1}, w, p, a) is a proper quotient of Q_{2}^{1}(y, h_{1}, w, p, a), we replace this graded auxiliary limit group and its associated auxiliary resolution by modifying the graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), by replacing its terminal graded limit group Q_{2}^{1}(s, z, y, g_{1}, h_{1}, w, p, a) with each of the (canonical) finite collection of limit groups that includes all specializations that factor through both Q_{2}^{1}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) and the limit group generated by < y, h_{1}, w, p, a > in the graded auxiliary limit group AuxGF Cl(t, y, h_{3}, g_{2}, h_{1}, w, p, a), and start part (2) of the initial step with each of the obtained graded resolutions. Hence, we may assume that the subgroup generated by < y, h_{1}, w, p, a > in the graded auxiliary limit group AuxGF Cl(t, y, h_{3}, g_{2}, h_{1}, w, p, a > is isomorphic to Q_{2}^{1}(y, h_{1}, w, p, a). In this case we set the developing resolution to be the resolution obtained from the levels of the resolution GRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a). With the developing resolution we associate the graded auxiliary resolution GAuxRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a) associated with the given graded auxiliary limit group, and the family of formal solutions x_{α}(t, y, h_{3}, g_{2}, h_{1}, w, p, a) defined over the developing resolution and parameterized by the graded modular groups associated with the graded auxiliary resolution

GAuxRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a)

and the graded formal resolution GF Res(x, t, y, g_{2}, h_{1}, w, p, a). We further set the anvil, Anv(MGQRes)(t, y, a), to be the group generated by the completion of the graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) with the auxiliary limit group, amalgamated along the corresponding copies of the subgroup Q_{2}^{1}(y, h_{1}, w, p, a). Since the associated family of formal solutions is defined over the developing resolution, and the developing resolution is canonically mapped into the anvil, the canonical family of formal solutions associated with the developing resolution is naturally defined over the anvil as well.

Suppose that Q_{2}^{1}(y, h_{1}, w, p, a) is isomorphic to Q^{1}(y, h_{1}, w, p, a). In this case we continue to the next level of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). If for some level j of the multigraded resolution, the image of Q^{1}(y, h_{1}, w, p, a) in the limit group associated with this level, Q_{j}^{1}(y, h_{1}, w, p, a), is a proper quotient of Q^{1}(y, h_{1}, w, p, a), then from the highest such level j, we can continue as in case Q_{2}^{1}(y, h_{1}, w, p, a) is a proper quotient of Q^{1}(y, h_{1}, w, p, a), and associate with the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) a finite collection of developing resolutions, anvils, graded auxiliary resolutions, and families of formal solutions defined over each of the developing resolutions and its associated anvil.

Finally, suppose that for every level j, the image of Q^{1}(y, h_{1}, w, p, a) in the limit group associated with the j-th level of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), Q_{j}^{1}(y, h_{1}, w, p, a), is isomorphic to Q^{1}(y, h_{1}, w, p, a). In this case, by lemma 2.2, the terminal limit group of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), Q_{term}^{1}(s, z, y, h_{1}, w, p, a), is rigid or solid with respect to the parameter subgroup < h_{1}, w, p >. We denote the terminal limit group of the multi-graded resolution MGQRes, W P HGHG(g_{2}, h_{1}, w, p, a). Note that this terminal limit group is rigid or solid with respect to the parameter subgroup < h_{1}, w, p >, and is isomorphic to Q_{term}^{1}(s, z, y, w, h_{1}, w, p, a). We set the graded resolution CRes(y, g_{2}, h_{1}, w, p, a), to be the resolution induced by the subgroup generated by < y, h_{1}, w, p, a >. We collect all the formal solutions defined over ungraded resolutions covered by the graded resolution CRes(y, g_{2}, h_{1}, w, p, a). This collection of formal solutions factors through a canonical collection of graded formal limit groups. With each graded formal limit group we associate its graded formal Makanin-Razborov diagram as we did in section 3 of [Se2]. We continue with each of the graded formal resolutions that appear in these diagrams in parallel. Let GF Res(x, t, y, g_{2}, h_{1}, w, p, a) be a graded formal resolution in one of these diagrams, and let W P HGHGH(h_{3}, g_{2}, h_{1}, w, p, a) be its terminating rigid or solid (not formal!) limit group. With the graded formal resolution GF Res(x, t, y, h_{3}, g_{2}, h_{1}, w, p, a) we associate the resolution

GRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a)

which is the graded (not formal!) resolution associated with the terminal formal limit group of the graded formal resolution GF Res(x, t, y, h_{3}, g_{2}, h_{1}, w, p, a) (i.e., it’s graded formal closure). Note that the terminal rigid or solid limit groups of those resolutions is W P HGHGH(h_{3}, g_{2}, h_{1}, w, p, a).

We set the developing resolution to be the resolution GRes(t, y, h_{3}, g_{2}, h_{1}, w, p, a). To set the finite collection of anvils, we first look at the group generated by (the completion of) MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) and the (terminal) limit group W P HGHGH(h_{3}, g_{2}, h_{1}, w, p, a) identified along the corresponding images of the subgroup < g_{2}, h_{1}, w, p, a >. With the obtained group we naturally associate a finite collection of maximal limit groups. With each such maximal limit group we associate its canonical graded Makanin-Razborov diagram with respect to the parameter subgroup < h_{1}, w, p, a >, a finite collection of auxiliary resolutions, and their associated auxiliary limit groups. Each of the obtained auxiliary limit groups is set to be an anvil, Anv(MGQRes). With the developing resolution we associate the the family of formal solutions x_{α}(t, y, h_{3}, g_{2}, h_{1}, w, p, a) defined over the developing resolution and parameterized by the graded modular groups associated with the graded auxiliary resolution and the graded formal resolution GF Res(x, t, y, g_{2}, h_{1}, w, p, a). Since the associated family of formal solutions is defined over the developing resolution, and the developing resolution is canonically mapped into the anvil, the canonical family of formal solutions associated with the developing resolution is naturally defined over the anvil as well.

(3) By part (1) we may assume that Q^{1}(y, h_{1}, w, p, a) is isomorphic to Q(y, h_{1}, w, p, a). Part (2) treats all the cases in which the multi-graded quotient resolution MGQRes(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is not of maximal possible complexity. Hence, the only case left in presenting the initial step of our procedure for analyzing the set EAE(p), is the case of a multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{2}, a) of maximal possible complexity. We treat this case as we treated it in the minimal rank case. In case the abelian decomposition associated with MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) and the top level of (the completion of) λW P GRes(y, h_{1}, w, p, a) are identical, we use the modular groups associated with the abelian decomposition associated with MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) to map the subgroup QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) into the subgroup of the anvil, Anv(W P HGH), QRlim_{2}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), associated with the second level of (the completion of) the resolution λW P GRes(y, h_{1}, w, p, a). We now set the subgroups Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1} to be the subgroups of the anvil, Anv(W P HGH), corresponding to the non-abelian, non-QH vertex groups in the multi-graded abelian decomposition associated with the second level of the completion of the graded resolution λW P GRes(y, h_{1}, w, p, a). By construction, these subgroups form a (canonical) free decomposition of the subgroup generated by: < s_{3}, z_{3}, …, s_{ℓ}, z_{ℓ}, h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with the anvil, Anv(W P HGH). With the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} we associate the auxiliary resolution, and auxiliary multi-graded decomposition we have fixed, restricts to an auxiliary resolution, and auxiliary multi-graded decompositions of the subgroups:

Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}

with respect to the subgroups: Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1}. At this point we analyze the quotient limit group QRlim_{2}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the subgroups Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1} exactly as we analyzed the quotient limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}. i.e., we associate with

QRlim_{2}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)

all its multi-graded quotient resolutions with respect to the subgroups:

Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1}

that are its subgroups, and analyze each of the obtained multi-graded quotient resolutions according to parts (1)-(3). If the multi-graded abelian decomposition associated with a multi-graded quotient resolution of

QRlim_{2}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a)

with respect to the subgroups: Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1} is of maximal possible complexity, i.e., if part (3) applies to an obtained quotient multi-graded resolution, we continue by analyzing multi-graded quotient resolutions of the quotient limit group QRlim_{3}(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the subgroups Base_{4,1}^{1}, …, Base_{4,r_{4}}^{1}. As long as the multi-graded abelian decompositions associated with the constructed graded quotient resolutions are of maximal possible complexity we continue analyzing multi-graded quotient resolutions associated with various level of the graded resolution λW P GRes(y, h_{1}, w, p, a) we have started with, until one of the parts (1)-(2) applies to at least one of the corresponding multi-graded quotient resolutions.

If there exists a level for which one of the parts (1)-(2) applies, we set a developing resolution, a graded formal resolution associated with it, and an anvil with the limit group associated with this level according to the part (1)-(2) that applies to it. With that anvil we associate a canonical finite collection of graded auxiliary resolutions and their associated terminal (rigid or solid) limit groups to be the associated graded auxiliary limit group, and the modular groups associated with the graded auxiliary resolution to be the graded auxiliary modular groups. Given an auxiliary resolution we set the developing resolution to be the resolution induced by the subgroup < y, h_{1}, w, p, a > from the parts of the resolution above that level followed by the image of the developing resolution associated with that level in the associated auxiliary limit group. We set the anvil, Anv(MGQRes)(t, y, a), to be the subgroup generated by the the auxiliary limit group associated with the auxiliary resolution and the subgroup QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), amalgamated along the copies of the limit group associated with the level to which one of the steps (1)-(2) was applied. With the developing resolution and its associated anvil, we naturally associate a family of formal solutions, x_{α}(t, y, h_{3}, g_{2}, h_{2}, g_{1}, h_{1}, w, p, a), parameterized by the elements of the auxiliary modular groups, and the graded formal modular groups associated with the graded formal resolution associated with the developing resolution and the anvil. With the anvil we also associated a rigid or solid limit group of the form W P HGHG(g_{2}, h_{2}, g_{1}, h_{1}, w, p, a) as we did according to the part (1)-(2) that applies to the multi-graded resolution associated with the corresponding level.

If all the abelian decompositions associated with the multi-graded resolutions used for the construction of the developing resolution are of maximal complexity, i.e., if none of the parts (1)-(2) applies to any of these multigraded resolutions, we reached a terminal state of our branching process. In this case, we do not include the obtained multi-graded resolution in our list of multi-graded resolutions and their associated anvils and developing resolutions with which we continue to the next step of the procedure. We just associate with the obtained multi-graded resolution its terminal rigid or solid limit group W P HGHG(g_{2}, h_{2}, g_{1}, h_{1}, w, p, a).

Definition 2.4. Starting with the limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), the procedure for the construction of the staggered data structure and developing resolutions produces finitely many multi-graded quotient resolutions of QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a) and some of its quotients. With each such multi-graded quotient resolution the procedure associates a developing resolution, and with each developing resolution it associates finitely many graded auxiliary resolutions, and with each graded auxiliary resolution its corresponding graded auxiliary limit group and graded auxiliary multi-graded decomposition, and the associated anvil. We call the entire collection of the original limit group λW P GRlim(y, h_{1}, w, p, a), the resolution we have started with λW P Res(y, h_{1}, w, p, a), its terminating limit group W P HG(g_{1}, h_{1}, w, p, a), the graded formal resolution GF Res(x, s, z, y, g_{1}, h_{1}, w, p, a), its terminal limit group W P HGH(h_{2}, g_{1}, h_{1}, w, p, a) and the associated auxiliary resolution, auxiliary limit group and auxiliary multi-graded decomposition, together with the quotient limit group QRlim(s, z, y, h_{2}, g_{1}, h_{1}, w, p, a), its chosen quotient and multi-graded quotient resolution, the associated graded and multi-graded limit groups produced along the initial step of the procedure, the developing resolution, the corresponding graded auxiliary resolution, graded auxiliary limit group and graded auxiliary decomposition, the associated anvil, and the family of formal solutions defined over the anvil and the developing resolution and parameterized by the graded auxiliary limit groups and the associated graded formal resolution, a staggered data-structure.

In order to force the iterative procedure for analyzing the set EAE(p) to terminate, we will use the families of formal solutions defined over the developing resolution and the anvil produced in each step of the iterative procedure. In the next step of the iterative procedure we will consider only those specializations that factor and are taut with respect to the resolutions associated with the anvil, and satisfy the equations corresponding to the family of formal solutions associated with the anvil.

II: The general step

In the preliminary and initial steps of the iterative procedure for analyzing the set EAE(p) we have finally obtained finitely many staggered data-structures and their associated graded formal resolutions, developing resolutions, graded auxiliary resolutions, and anvils. With each tuple of a graded formal resolution, developing resolution, a graded auxiliary resolution and anvil we have associated a family of formal solutions defined over the developing resolution, which is embedded into the anvil. After presenting the initial steps, we finally present the general step of the procedure for analyzing the set EAE(p), and then prove it terminates after finitely many steps.

We define the general step of the procedure inductively. For brevity we denote the multi-graded resolutions that were obtained in the previous steps of the procedure, MGQ^{m}Res(t, y, w, p, a), where m is the index of the step in which they were constructed. With each such multi-graded quotient resolution there is an associated graded formal resolution, developing resolution, graded auxiliary resolution, and an anvil, which we denote Anv(MGQ^{m}Res)(t, y, w, p, a). We start the general step of our iterative procedure for the analysis of the set EAE(p) with the (finite) collection of multi-graded quotient resolutions constructed in the previous step, and their associated graded formal resolutions, developing resolutions, graded auxiliary resolutions and anvils. The ultimate goal of the general step of the iterative procedure is to obtain a strict reduction in either the complexity of certain decompositions and resolutions or a strict reduction in the Zariski closures of certain limit groups associated with the anvils constructed in the previous steps of the procedure. The strict reduction in complexity and Zariski closures will finally guarantee the termination of the iterative procedure after finitely many steps.

Since we treat the anvils in parallel, we present the general (n-th) step of the procedure with one of the anvils, Anv(MGQ^{n−1}Res)(t, y, w, p, a). With each anvil we have associated a family of formal solutions x_{α}(t, y, w, p, a). Starting with the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a) we impose the family of formal solutions x_{α}(t, y, w, p, a) associated with the corresponding closure of the developing resolution. We look at the set of specializations (t_{0}, y_{0}, w_{0}, p_{0}, a) that factor through and are taut with respect to the resolutions associated with the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a) (which includes its associated developing resolution), for which for some index j: ψ_{j}(x_{α}(t_{0}, y_{0}, w_{0}, p_{0}, a), y_{0}, w_{0}, p_{0}, a) = 1 for all possible modular elements α.

The entire collection of specializations (t_{0}, y_{0}, w_{0}, p_{0}, a) that satisfy the corresponding system of equations is contained in a finite set of maximal (n-th) limit groups Q^{n}Rlim_{1}(t, y, w, p, a), …, Q^{n}Rlim_{u_{j}^{n}}(t, y, w, p, a). Since our analysis of these (n-th) quotient limit groups is conducted in parallel, we will omit the indices from these (n-th) quotient limit groups and denote them Q^{n}Rlim(t, y, w, p, a).

We construct the staggered data-structure and developing resolutions associated with the anvil and the n-th quotient limit group Q^{n}Rlim(w, y, a) iteratively, similarly to the way we have analyzed quotient resolutions in the general step of the procedure for validation of a sentence presented in section 4 of [Se4]. The analysis we carry out in the general step depends on the structure of the staggered data-structure, the developing resolutions, and the graded auxiliary resolutions constructed in the previous steps of the procedure. As in the preliminary and initial steps of the procedure, our aim is to obtain a strict decrease in either the Zariski closure or the complexity of the resolution associated with some level of the staggered data structure we construct.

(1) Let Q^{n−1}(y, h_{1}, w, p, a) be the restricted limit group generated by < y, h_{1}, w, p, a > in the anvil Anv(MGQ^{n−1}Res)(t, y, h_{1}, w, p, a). Let Q^{n}(y, h_{1}, w, p, a) be the limit group generated by < y, h_{1}, w, p, a > in the n-th quotient limit group Q^{n}Rlim(t, y, h_{1}, w, p, a). If Q^{n}(y, h_{1}, w, p, a) is a proper quotient of the subgroup Q^{n−1}(y, h_{1}, w, p, a), we continue this branch of the iterative procedure, by starting the preliminary step of the procedure with the graded limit group Q^{n}(y, h_{1}, w, p, a) instead of the graded limit group Q^{n−1}(y, h_{1}, w, p, a).

(2) At this stage we may assume that Q^{n}(y, h_{1}, w, p, a) is isomorphic to Q^{n−1}(y, h_{1}, w, p, a). Along the process used to construct the anvil, Anv(MGQ^{n−1}Res)(t, y, w, p, a), we enlarge the parameter subgroups each time the complexity of the abelian decomposition associated with the top level of the corresponding multigraded quotient resolution is being reduced. At step m, 1 ≤ m ≤ n − 1, we set the parameter subgroups to be: Base_{2,1}^{s(m)}, …, Base_{2,v_{s(m)}}^{s(m)} and the corresponding multi-graded quotient resolution to be:

MGQ^{m}Res(t_{m}, y, Base_{2,1}^{s(m)}, …, Base_{2,v_{s(m)}}^{s(m)}, a).

In this part of the general step we assume that the multi-graded quotient resolution MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a) is of maximal complexity.

Suppose that for some index s, 1 ≤ s ≤ s(n − 1) − 1, Q^{n}(t_{ℓ(s)}, y, a) is a proper quotient of Q^{ℓ(s)}(t_{ℓ(s)}, y, a), and let s be the minimal index for which this happens. Then we omit the n-th limit group Q^{n}Rlim(t, y, w, p, a) from our list of n-th quotient limit groups, and replace it by going back to the ℓ(s)-th step of the iterative procedure, and start it with the limit group Q^{n}(t_{ℓ(s)}, y, w, p, a) instead of the ℓ(s)-th limit group Q^{ℓ(s)}Rlim(t_{ℓ(s)}, y, w, p, a) used in the s-th step of the process that lead to the construction of the anvil, Anv(MGQ^{n−1}Res)(t, y, w, p, a). Since, by definition of the index ℓ(s), the parameter subgroups were enlarged at step ℓ(s) + 1, in analyzing the quotient limit group Q^{n}Rlim(t_{ℓ(s)}, y, a) we need to take into account only those multi-graded quotient resolutions which are not of maximal complexity, i.e., only those multi-graded quotient resolutions which do not have a single level with a multi-graded abelian decomposition with the same structure as the abelian decomposition associated with the top level of the multi-graded quotient resolution MGQ^{ℓ(s)−1}Res(t_{ℓ(s)−1}, y, Base_{2,1}^{s(ℓ(s)−1)}, …, Base_{2,v_{s(ℓ(s)−1)}}^{s(ℓ(s)−1)}, a) used in the process of the construction of the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a).

Suppose that for s(n−1)−1 (hence, for every index s, 1 ≤ s ≤ s(n−1)−1), Q^{n}(t_{ℓ(s)}, y, w, p, a) is isomorphic to Q^{ℓ(s)}(t_{ℓ(s)}, y, w, p, a). We set s(n) = s(n − 1). Let:

MGQ^{n}Res_{1}(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), …, MGQ^{n}Res_{q}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

be the multi-graded resolutions in the multi-graded taut Makanin-Razborov diagram of Q^{n}Rlim(t, y, w, p, a) with respect to the parameter subgroups Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}. We will treat the multi-graded quotient reso-

lutions MGQ^{n}Res_{j}(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) in parallel, hence, we omit their index. If there exists a QH vertex group Q in the abelian decomposition associated with the top level of the ℓ(s(n) − 1) quotient resolution

MGQ^{ℓ(s(n)−1)}Res(t_{ℓ(s(n)−1)}, y, Base_{2,1}^{s(n)−1}, …, Base_{2,v_{s(n)−1}}^{s(n)−1}, a)

for which the sequence of abelian decompositions inherited by Q from the various levels of the multi-graded resolution

MGQ^{n}Res(t_{n}, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

is not compatible with the specified collection of s.c.c. on the QH vertex group Q that are mapped to the trivial element in the second level of the ℓ(s(n) − 1) quotient resolution

MGQ^{ℓ(s(n)−1)}Res(t_{ℓ(s(n)−1)}, y, Base_{2,1}^{s(n)−1}, …, Base_{2,v_{s(n)−1}}^{s(n)−1}, a)

we omit the multi-graded resolution MGQ^{n}Res(t_{n}, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) from our list of multi-graded resolutions.

By proposition 4.2 of [Se4] the complexities of the abelian decompositions associated with the various levels of each of the n-th multi-graded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) are bounded by the complexity of the multi-graded quotient abelian decomposition associated with the top level of MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a), and if the complexity of the abelian decomposition associated with some level of MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) is equal to the complexity of the abelian decomposition associated with the top level of

MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a)

then the n-th multi-graded quotient resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

has only one level above the terminating solid or rigid limit group, and the structure of the abelian decomposition associated with this level is identical with the structure of the abelian decomposition associated with the top level of MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a). In this part of the n-th step of the procedure we will also assume that the complexities of the abelian decompositions associated with the various levels of the n-th multigraded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) are strictly smaller than the complexity of the abelian decomposition associated with the top level of the multi-graded quotient resolution

MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a).

We treat the n-th multi-graded quotient resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

according to the various parts of step n − 1 of the iterative procedure.

(3) At this stage we may assume that Q^{n}(y, h_{1}, w, p, a) is isomorphic to Q^{n−1}(y, h_{1}, w, p, a). At this part we assume that the n − 1 multi-graded quotient resolution MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a) is not of maximal possible complexity. Suppose that for some index s, 1 ≤ s ≤ s(n − 1), Q^{n}(w_{ℓ(s)}, y, a) is a proper quotient of Q^{ℓ(s)}(t_{ℓ(s)}, y, a), and suppose that s is the minimal index for which this happens. Then we omit the n-th limit group Q^{n}Rlim(t, y, w, p, a) from our list of n-th quotient limit groups, and replace it by going back to the ℓ(s)-th step of the iterative procedure, and start it with the limit group Q^{n}(t_{ℓ(s)}, y, a), the subgroup generated by < t_{ℓ(s)}, y, a > in the n-th quotient limit group Q^{n}Rlim(t, y, w, p, a), instead of the ℓ(s)-th limit group Q^{ℓ(s)}Rlim(t_{ℓ(s)}, y, w, p, a) used in the ℓ(s)-th step of the process that lead to the construction of the anvil, Anv(MGQ^{n−1}Res)(t, y, w, p, a). Since, by definition of the index ℓ(s), in case ℓ(s) < n − 1 the parameter subgroups were enlarged at step ℓ(s)+1, and in case ℓ(s) = n−1 the multi-graded quotient resolution MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a) is not of maximal possible complexity, in analyzing the quotient limit group

Q^{n}Rlim(t_{ℓ(s)}, y, w, p, a)

we need to take into account only those multi-graded quotient resolutions which are not of maximal complexity, i.e., only those multi-graded quotient resolutions which do not have a single level with a multi-graded abelian decomposition with the same structure as the abelian decomposition associated with the top level of the multi-graded quotient resolution

MGQ^{ℓ(s)−1}Res(t_{ℓ(s)−1}, y, Base_{2,1}^{s(ℓ(s)−1)}, …, Base_{2,v_{s(ℓ(s)−1)}}^{s(ℓ(s)−1)}, a)

used in the process of the construction of the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a).

(4) At this part we may assume that the multi-graded quotient resolution MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a) is not of maximal complexity, and that Q^{n}(t_{n−1}, y, w, p, a) is isomorphic to Q^{n−1}(t_{n−1}, y, w, p, a). We set s(n) = s(n−1)+1, and the parameter subgroups Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)} to be the non-abelian, non-QH vertex groups in the multi-graded abelian decomposition associated with the top level of the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a). Let:

MGQ^{n}Res_{1}(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), …, MGQRes_{q}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

be the resolutions in the taut multi-graded Makanin-Razborov diagram of Q^{n}Rlim(t, y, w, p, a) with respect to the parameter subgroups:

Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}.

We analyze the n-th multi-graded quotient resolutions

MGQ^{n}Res_{j}(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

in parallel, hence, we will omit their index. If there exists a QH vertex group Q in the abelian decomposition associated with the top level of the ℓ(s(n) − 1) quotient resolution

MGQ^{ℓ(s(n)−1)}Res(t_{ℓ(s(n)−1)}, y, Base_{2,1}^{s(n)−1}, …, Base_{2,v_{s(n)−1}}^{s(n)−1}, a)

for which the sequence of abelian decompositions inherited by Q from the various levels of the multi-graded resolution

MGQ^{n}Res(t_{n}, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

is not compatible with the specified collection of s.c.c. on the QH vertex group Q that are mapped to the trivial element in the second level of the ℓ(s(n) − 1) quotient resolution

MGQ^{ℓ(s(n)−1)}Res(t_{ℓ(s(n)−1)}, y, Base_{2,1}^{s(n)−1}, …, Base_{2,v_{s(n)−1}}^{s(n)−1}, a)

we omit the multi-graded resolution MGQ^{n}Res(t_{n}, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) from our list of multi-graded resolutions.

At this part we will also assume that the n-th multi-graded quotient resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

is not of maximal possible complexity, i.e., it does not have a single level with a (multi-graded) abelian decomposition identical with the abelian decomposition associated with the top level of the multi-graded quotient resolution MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a). The case of maximal complexity will be treated in the next part of the general step. To treat an n-th multi-graded quotient resolution which is not of maximal possible complexity we need the following observation, which is similar to lemma 2.2.

Lemma 2.5. Let MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) be an n-th multigraded quotient resolution that is not of maximal complexity. By construction, the limit group Q^{n}(t_{n−1}, y, w, p, a) is mapped into the limit group associated with each of the levels of the multi-graded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a). Let Q_{term}^{n}(t_{n−1}, y, w, p, a) be the image of Q^{n}(t_{n−1}, y, w, p, a) in the terminal (rigid or solid) limit group of MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a). Then the ter-

minal limit group of the multi-graded resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) can be replaced by a collection of finitely many terminal multi-graded limit groups in which Q_{term}^{n}(t_{n−1}, y, w, p, a) is a proper quotient of Q^{n}(t_{n−1}, y, w, p, a), and a finite collection of terminal rigid or solid graded limit groups which are rigid or solid with respect to the parameter subgroup < h_{1}, w, p >.

Proof: Identical to the proof of lemma 2.2.

□

By lemma 2.5 under the assumptions of part (4), we can either drop the n-th multi-graded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{v_{s(n)}}^{s(n)}, a) from our list of n-th multi-graded quotient resolutions, or we may assume that either Q_{term}^{n}(t_{ℓ(s(n)−1)}, y, w, p, a) is a proper quotient of Q^{n}(t_{ℓ(s(n)−1)}, y, w, p, a), or the terminal limit group of the multi-graded resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{v_{s(n)}}^{s(n)}, a)

is rigid or solid with respect to the parameter subgroup W P =< w, p >. At this point we need the following lemma that is similar to proposition 2.3.

Lemma 2.6. Let MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) be an n-th multigraded quotient resolution which is not of maximal possible complexity. By construction, the limit group Q^{n}(t, y, w, p, a) is mapped onto the limit group associated with each of the levels of the multi-graded quotient resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a).

Let Q_{2}^{n}(t_{n−1}, y, a), Q_{2}^{n}(t_{n}, y, a) be the images of the subgroups generated by Q^{n}(t_{n−1}, y, w, p, a), Q^{n}(t_{n}, y, w, p, a) in correspondence, in the limit group associated with the second level of the multi-graded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a). Then either Q_{2}^{n}(t_{n−1}, y, w, p, a) is a proper quotient of the subgroup Q^{n}(t_{n−1}, y, w, p, a), or Q_{2}^{n}(t_{n}, y, w, p, a) is a proper quotient of the subgroup Q^{n}(t_{n}, y, w, p, a).

Proof: The claim of the proposition simply follows from the basic properties of a multi-graded resolution.

□

Suppose that Q_{2}^{n}(y, h_{1}, w, p, a) is a proper quotient of Q^{n}(y, h_{1}, w, p, a). In this case we modify the procedure used in part (2) of the initial step of the procedure. We decrease the parameter subgroup to be < h_{1}, w, p >, and associate with Q_{2}^{n}(y, h_{1}, w, p, a) its taut graded Makanin-Razborov diagram with respect to the parameter subgroup < h_{1}, w, p >:

GQRes_{1}(y, h_{1}, w, p, a), …, GQRes_{t}(y, h_{1}, w, p, a).

We continue with each of the graded resolutions GQRes_{j}(y, h_{1}, w, p, a) in parallel. If the subgroup generated by < y, h_{1}, w, p, a > in the limit group associated with the resolution GQRes_{j}(y, h_{1}, w, p, a) is a proper quotient of Q_{2}^{n}(y, h_{1}, w, p, a), we replace the graded resolution GQRes_{j}(y, h_{1}, w, p, a) by starting part (4) of the the general step with the multi-graded resolution obtained from MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) by replacing its second limit group Q_{2}^{n}(t, y, w, p, a) with the maximal limit groups obtained from all specializations that factor through both Q_{2}^{n}(t, y, w, p, a) and the subgroup generated by < y, h_{1}, w, p, a > in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a). If the subgroup generated by < t, y, w, p, a > in the obtained (one level) resolution, QRlim′(t, y, w, p, a), is a proper quotient of QRlim(t, y, w, p, a), we replace the obtained resolution by starting the n-th step of our iterative procedure with the limit group QRlim′(t, y, w, p, a) instead of the limit group QRlim(t, y, w, p, a), and since the resolution MGQRes is not of maximal complexity, in analyzing the limit group QRlim′(t, y, w, p, a) we need to consider only those resolutions in its multi-graded Makanin-Razborov diagram that are not of maximal possible complexity. Hence, for the rest of this part we may assume that the subgroup generated by < y, h_{1}, w, p, a > in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a) is isomorphic to the subgroup generated by < y, h_{1}, w, p, a > in Q_{2}^{n}(t, y, w, p, a).

Suppose that a graded quotient resolution GQRes_{j}(y, h_{1}, w, p, a) is of maximal possible complexity, i.e., the limit group associated with it is of the form < h_{1}, w, p, a > ∗F_{y}. Since the limit group Q_{2}^{n}(y, h_{1}, w, p, a) is a proper quotient of Q^{n}(y, h_{1}, w, p, a), and since the limit group Q_{2}^{n}(y, h_{1}, w, p, a) is naturally mapped onto the limit group < h_{1}, w, p, a > ∗F_{y} associated with the graded resolution GQRes_{j}(y, h_{1}, w, p, a), the Hopf property for limit groups implies that the subgroup generated by < h_{1}, w, p, a > in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a) is a proper quotient of the limit group generated by < h_{1}, w, p, a > in Q^{n}(y, h_{1}, w, p, a). Hence, we can replace the resolution GQRes_{j}(y, h_{1}, w, p, a) by starting the preliminary step of the procedure with the subgroup < h_{1}, w, p, a > ∗F_{y}, where < h_{1}, w, p, a > is the subgroup generated by these elements in the limit group associated with GQRes_{j}(y, h_{1}, w, p, a). Since resolutions of maximal possible complexity of the limit group Q^{n}(y, h_{1}, w, p, a) with respect to the parameter subgroup < h_{1}, w, p, a >, i.e., those corresponding to the entire free group F_{y}, were already analyzed in the preliminary step of the procedure, we can omit a graded resolution GQRes_{j}(y, h_{1}, w, p, a) from our list of graded quotient resolutions {GQRes_{j}(y, h_{1}, w, p, a)}. Hence, for the rest of this part we may assume that the graded resolution GQRes_{j}(y, h_{1}, w, p, a) we continue with is not of maximal possible complexity.

If the subgroup generated by < y, h_{1}, w, p, a > in the limit group corresponding to the graded resolution CRes_{j}(y, h_{1}, w, p, a), where CRes_{j}(y, h_{1}, w, p, a) is the graded resolution obtained from the resolution induced by the subgroup < y, h_{1}, w, p, a > from the top level of the completion of the multigraded resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), followed by the graded resolution GQRes_{j}(y, h_{1}, w, p, a), is a proper quotient of Q^{n}(y, h_{1}, w, p, a), we replace the graded resolution GQRes_{j}(y, h_{1}, w, p, a) by starting the preliminary step of the procedure with the subgroup generated by < y, h_{1}, w, p, a > in the limit group corresponding to the resolution CRes_{j}(y, h_{1}, w, p, a), and treat only those graded resolutions of this limit group (with respect to the parameter subgroup < h_{1}, w, p, a >) that are not of maximal possible complexity, i.e., those graded resolutions that do not cover ungraded ones that correspond to the entire free group F_{y}. Hence, we may assume that for the rest of this part the subgroup generated by < y, h_{1}, w, p, a > in the limit group corresponding to CRes_{j}(y, h_{1}, w, p, a) is isomorphic to Q^{n}(y, h_{1}, w, p, a). In particular, we may assume that each of the graded resolutions GQRes_{j}(y, h_{1}, w, p, a) in question is not of maximal possible complexity.

We now treat each of the graded resolutions GQRes_{j}(y, h_{1}, w, p, a) and their associated resolution CRes_{j}(y, h_{1}, w, p, a) in a similar way to our treatment of multi-graded quotient resolutions in part (4) of the general step of the iterative procedure for validation of a sentence presented in section 4 of [Se4]. Let W P (HG)^{n+1}(g_{n+1}, h_{1}, w, p, a) be the terminal rigid or solid limit group of the graded resolution GQRes_{j}(y, h_{1}, w, p, a) (which is also the terminal rigid or solid limit group of its associated resolution CRes_{j}(y, h_{1}, w, p, a)). We start by collecting all the formal solutions defined over ungraded resolutions covered by the graded resolution CRes_{j}(y, h_{1}, w, p, a). This collection of formal solutions factors through a canonical collection of graded formal limit groups. With each graded formal limit group we associate its graded formal Makanin-Razborov diagram as we did in section 3 of [Se2]. We continue with each of the graded formal resolutions that appear in these diagrams in parallel. Let GF Res(x, t, y, g_{n+1}, h_{1}, w, p, a) be a graded formal resolution in one of these diagrams, and let W P (HG)^{n}H(h_{n+2}, g_{n+1}, h_{1}, w, p, a) be its terminating rigid or solid (not formal!) limit group. With the graded formal resolution GF Res(x, t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) we associate the resolution GRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a), which is the graded (not formal!) resolution associated with the terminal formal limit group of the graded formal resolution GF Res(x, t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a). With GRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) we associate the graded resolution BRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a), which is obtained from GRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) by excluding its top level. Note that the terminal rigid or solid limit groups of those resolutions is W P (HG)^{n+1}H(h_{n+2}, g_{n+1}, h_{1}, w, p, a). With BRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) we associate its (canonical) finite collection of graded auxiliary resolutions GAuxRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) (multi-graded with respect to the nonabelian, non-QH vertex groups and edge groups in the graded abelian decomposition of the subgroup Q(y, h_{1}, w, p, a)), their associated graded abelian decompositions and graded auxiliary limit groups

AuxGF Cl(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a).

We continue with each of the graded auxiliary resolutions and their associated graded auxiliary decompositions and graded auxiliary limit groups in parallel.

If the subgroup generated by < y, h_{1}, w, p, a > in a graded auxiliary limit group AuxGF Cl(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) is a proper quotient of Q_{2}^{n}(y, h_{1}, w, p, a), we replace this graded auxiliary limit group and its associated auxiliary resolution by modifying the multi-graded resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

by replacing its terminal graded limit group Q_{2}^{n}(t, y, w, p, a) with each of the (canonical) finite collection of limit groups that includes all specializations that factor through both Q_{2}^{n}(t, y, w, p, a) and the limit group generated by < y, h_{1}, w, p, a > in the graded auxiliary limit group

AuxGF Cl(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a)

and start part (4) of the general step with each of the obtained multigraded resolutions. Hence, we may assume that the subgroup generated by < y, h_{1}, w, p, a > in the graded auxiliary limit group

AuxGF Cl(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a)

is isomorphic to Q_{2}^{n}(y, h_{1}, w, p, a). In this case we set the developing resolution to be the resolution obtained from the levels of the resolution

GRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a).

With the developing resolution we associate the graded auxiliary resolution GAuxRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) associated with the given graded auxiliary limit group, and the family of formal solutions x_{α}(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) defined over the developing resolution and parameterized by the graded modular groups associated with the graded auxiliary resolution

GAuxRes(t, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a)

and the graded formal resolution GF Res(x, t, y, g_{n+1}, h_{1}, w, p, a). We further set the anvil, Anv(MGQ^{n}Res)(t, y, a), to be the group generated by the completion of the multi-graded resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

with the auxiliary limit group, amalgamated along the corresponding copies of the subgroup Q_{2}^{n}(y, h_{1}, w, p, a). Since the associated family of formal solutions is defined over the developing resolution, and the developing resolution is canonically mapped into the anvil, the canonical family of formal solutions associated with the developing resolution is naturally defined over the anvil as well.

Suppose Q_{2}^{n}(y, h_{1}, w, p, a) is isomorphic to Q^{n}(y, h_{1}, w, p, a), and Q_{2}^{n}(t_{n−1}, y, w, p, a) is a proper quotient of Q^{n}(t_{n−1}, y, w, p, a). We set s, 1 ≤ s ≤ s(n) − 1, to be the minimal index for which: Q_{2}^{n}(t_{ℓ(s)}, y, w, p, a) is a proper quotient of Q^{n}(t_{ℓ(s)}, y, w, p, a) = Q^{ℓ(s)}(t_{ℓ(s)}, y, w, p, a). Let:

MGQRes_{1}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), …, MGQRes_{d}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

be the resolutions in the taut multi-graded diagram of

Q_{2}^{n}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

with respect to the parameter subgroups Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}.

Given a multi-graded resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), we set the multi-graded quotient resolution CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) to be the multi-graded resolution obtained from the resolution induced by the subgroup < t_{ℓ(s)}, y, w, p, a > from the top level of the completion of the multi-graded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) followed by the multi-graded resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a). If the subgroup generated by < t_{ℓ(s)}, y, w, p, a > in the limit group associated with the resolution CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is a proper quotient of the ℓ(s) quotient limit group Q^{ℓ(s)}Rlim(t_{ℓ(s)}, y, w, p, a), we set q, 1 ≤ q ≤ s, to be the minimal index for which the subgroup generated by < t_{ℓ(q)}, y, w, p, a > in the limit group associated with the resolution CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is a proper quotient of the ℓ(q) quotient limit group Q^{ℓ(q)}Rlim(t_{ℓ(q)}, y, w, p, a). We now replace the multigraded quotient resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) of the limit group associated with the second level of the multi-graded resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) by starting the ℓ(q)-th step of our process with the limit group generated by < t_{ℓ(q)}, y, w, p, a > in (the closure of) CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) instead of the limit group Q^{ℓ(q)}Rlim(t_{ℓ(q)}, y, w, p, a) = Q^{n}Rlim(t_{ℓ(q)}, y, w, p, a) used in the ℓ(q)- th step of the procedure. If there exists a QH vertex group Q in the abelian decomposition associated with the top level of the ℓ(s − 1) quotient resolution

MGQ^{ℓ(s−1)}Res(t_{ℓ(s−1)}, y, Base_{2,1}^{s−1}, …, Base_{2,v_{s−1}}^{s−1}, a)

for which the sequence of abelian decompositions inherited by Q from the various levels of the multi-graded resolution CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is not compatible with the specified collection of s.c.c. on the QH vertex group Q that are mapped to the trivial element in the second level of the ℓ(s−1) quotient resolution MGQ^{ℓ(s−1)}Res(t_{ℓ(s−1)}, y, Base_{2,1}^{s−1}, …, Base_{2,v_{s−1}}^{s−1}, a), we omit the multi-graded resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) from our list of multi-graded resolutions.

Suppose that a multi-graded quotient resolution

MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

is of maximal possible complexity, i.e., that it has a single level with an abelian decomposition of the same structure as the abelian decomposition associated with the top level of the multi-graded quotient resolution MGQ^{ℓ(s)−1}Res(t_{ℓ(s)−1}, y, Base_{2,1}^{s(ℓ(s)−1)}, …, Base_{2,v_{s(ℓ(s)−1)}}^{s(ℓ(s)−1)}, a). Let Dt_{ℓ(s)} be the subgroup that is generated by the non-abelian, non-QH (distinguished) vertex groups in the anvil:

Anv(MGQ^{ℓ(s)−1}Res)(t_{ℓ(s)}, y, w, p, a).

Since the subgroup Q_{2}^{n}(t_{ℓ(s)}, y, w, p, a) is a proper quotient of Q^{n}(t_{ℓ(s)}, y, w, p, a), and since limit groups are residually finite, hence, satisfy the Hopf property, the image of the subgroup Dt_{ℓ(s)} in the (closure of the) multi-graded quotient resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is a proper quotient of the subgroup Dt_{ℓ(s)}. In this case of

MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

being a resolution of maximal possible complexity we do the following. We set the multi-graded quotient resolution CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) to be the multi-graded resolution obtained from the resolution induced by the subgroup < t_{ℓ(s)}, y, w, p, a > from the completion of the multigraded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) followed by the multi-graded resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a). We set q, 1 ≤ q ≤ s, to be the minimal index for which the subgroup generated by < t_{ℓ(q)}, y, w, p, a > in (the closure of) CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is a proper quotient of Q^{n}(t_{ℓ(q)}, y, w, p, a) = Q^{ℓ(q)}(t_{ℓ(q)}, y, w, p, a). We now replace the multi-graded quotient resolution

MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

of the limit group associated with the second level of

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

by starting the ℓ(q)-th step of our process with the limit group generated by < t_{ℓ(q)}, y, w, p, a > in (the closure of) CRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) instead of the limit group Q^{ℓ(q)}Rlim(t_{ℓ(q)}, y, w, p, a) = Q^{n}Rlim(t_{ℓ(q)}, y, w, p, a) used in the ℓ(q)-th step of the procedure.

By the above argument, we may consider only those multi-graded quotient resolutions MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) which are not of maximal possible complexity. In this case we analyze each of the multigraded quotient resolutions MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) as we did in step ℓ(s) of our iterative procedure. According to step ℓ(s), with the multi-graded resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) we have associated a canonical collection of developing resolutions and their associated terminal limit groups which we denote W P (HG)^{n+1}, associated graded formal resolutions and their associated terminal limit groups which we denote W P (HG)^{n+1}H, and their associated anvils. and families of formal solutions. As we did in part (4) of the general step of the procedure for validation of sentence ([Se4],4), given an anvil associated with the graded resolution MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), we associate with it its multi-graded taut Makanin-Razborov diagram with respect to the non-QH, non-abelian vertex groups and edge groups in the multi-graded abelian decomposition of the subgroup < t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a >. We call each resolution in this multi-graded taut Makanin-Razborov diagram, a graded auxiliary resolution, its terminal rigid or solid limit group, graded auxiliary limit group, and the modular groups associated with the graded auxiliary resolution, graded auxiliary modular groups. Given a graded auxiliary resolution associated with MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), we set a resolution CRes_{j}(u, y, w, p, a) obtained from the resolution induced by the subgroup < y, h_{1}, w, p, a > from the completion of the n-th multigraded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(r)}, …, Base_{2,v_{s(r)}}^{s(r)}, a) followed by the image of the developing resolution in the auxiliary limit group associated with the multi-graded resolution: MGQRes_{j}(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a). We set the developing resolution to be the resolution CRes_{j}(u, y, w, p, a).

We set the anvil, Anv(MGQ^{n}Res)(t, y, w, p, a), to be the group generated by the corresponding closure of the multi-graded resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

and the graded auxiliary limit group associated with

MGQRes(t_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

amalgamated along the corresponding images of the subgroup Q_{2}^{n}(t_{ℓ(s)}, y, a). With the anvil we associate graded auxiliary resolutions and limit groups with respect to the non-abelian, non-QH vertex groups and edge groups in the multi-graded abelian decomposition of the quotient limit group Q^{n}(t, y, h_{1}, w, p, a) we have started this step with. With the developing resolution we associate the family of formal solutions x_{α}(t, y, w, p, a) parameterized by the elements of the graded modular groups and the graded formal modular groups associated with the developing resolution. Since the developing resolution is mapped into the anvil, Anv(MGQ^{n}Res)(w, y, a), the family of formal solutions is naturally defined over the anvil as well.

Suppose that Q_{2}^{n}(y, h_{1}, w, p, a) is isomorphic to Q^{n}(y, h_{1}, w, p, a), and Q_{2}^{n}(t_{n−1}, y, h_{1}, w, p, a) is isomorphic to Q^{n}(t_{n−1}, y, h_{1}, w, p, a). In this case we continue to the next level of the multi-graded quotient resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a).

If for some level j of the multi-graded resolution, the image of Q^{n}(y, h_{1}, w, p, a) in the limit group associated with this level, Q_{j}^{n}(y, h_{1}, w, p, a), is a proper quotient of Q^{n}(y, h_{1}, w, p, a), or the image of the limit group Q^{n}(t_{n−1}, y, h_{1}, w, p, a) in the limit group associated with this level, Q_{j}^{n}(t_{n−1}, y, h_{1}, w, p, a), is a proper quotient of Q^{n}(t_{n−1}, y, h_{1}, w, p, a), then from the highest such level j, we can continue as in case Q_{2}^{n}(y, h_{1}, w, p, a) is a proper quotient of Q^{n}(y, h_{1}, w, p, a), or Q_{2}^{n}(t_{n−1}, y, h_{1}, w, p, a) is a proper quotient of Q^{n}(t_{n−1}, y, h_{1}, w, p, a), and associate with the multi-graded resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

a finite collection of developing resolutions, anvils, graded auxiliary resolutions, and families of formal solutions defined over each of the developing resolutions and its associated anvil.

Finally, suppose that for every level j, the image of Q^{n}(t_{n−1}, y, h_{1}, w, p, a) in the limit group associated with the j-th level of the multi-graded resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), Q_{j}^{n}(t_{n−1}, y, h_{1}, w, p, a), is isomorphic to Q^{n}(t_{n−1}, y, h_{1}, w, p, a). In this case, by lemma 2.5, the terminal limit group of the multi-graded resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

Q_{term}^{n}(t, y, h_{1}, w, p, a), is rigid or solid with respect to the parameter subgroup < h_{1}, w, p >.

We denote the terminal limit group of the multi-graded resolution MGQRes, W P H(GH)^{n+1}(g_{n+1}, h_{1}, w, p, a). Note that this terminal limit group is rigid or solid with respect to the parameter subgroup < h_{1}, w, p >. We set the graded resolution CRes(y, g_{n+1}, h_{1}, w, p, a), to be the resolution induced by the subgroup generated by < y, h_{1}, w, p, a > from the multi-graded resolution MGQRes. We collect all the formal solutions defined over ungraded resolutions covered by the graded resolution CRes(y, g_{n+1}, h_{1}, w, p, a). This collection of formal solutions factors through a canonical collection of graded formal limit groups. With each graded formal limit group we associate its graded formal Makanin-Razborov diagram as we did in section 3 of [Se2]. We continue with each of the graded formal resolutions that appear in these diagrams in parallel. Let GF Res(x, u, y, g_{n+1}, h_{1}, w, p, a) be a graded formal resolution in one of these diagrams, and let W P (HG)^{n+1}H(h_{n+2}, g_{n+1}, h_{1}, w, p, a) be its terminating rigid or solid (not formal!) limit group. With the graded formal resolution GF Res(x, u, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a) we associate the resolution GRes(u, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a), which is the graded (not formal!) resolution associated with the terminal formal limit group of the graded formal resolution GF Res(x, u, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a). Note that the terminal rigid or solid limit groups of those resolutions is W P (HG)^{n+1}H(h_{n+2}, g_{n+1}, h_{1}, w, p, a).

We set the developing resolution to be the resolution

GRes(u, y, h_{n+2}, g_{n+1}, h_{1}, w, p, a).

To set the finite collection of anvils, we first look at the group generated by the completion of GQ^{n}Res′(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) and the (terminal) limit group W P HGHGH(h_{n+2}, g_{n+1}, h_{1}, w, p, a) identified along the corresponding images of the subgroup < g_{n+1}, h_{1}, w, p, a >. With the obtained group we naturally associate a finite collection of maximal limit groups. With each such maximal limit group we associate its canonical multi-graded Makanin-Razborov diagram with respect to the non-QH, nonabelian vertex groups and edge groups in the multi-graded decomposition of the quotient limit group Q^{n}(t, y, h_{1}, w, p, a) we have started this step with, call each of the resolutions in this diagram an auxiliary resolution, and each terminal limit group an auxiliary limit group. Each of the obtained auxiliary limit groups is set to be an anvil, Anv(MGQRes). With the developing resolution we associate the the family of formal solutions x_{α}(u, y, h_{3}, g_{2}, h_{1}, w, p, a) defined over the developing resolution and parameterized by the graded modular groups associated with the graded auxiliary resolution and the graded formal resolution GF Res(x, u, y, g_{n+1}, h_{1}, w, p, a). Since the associated family of formal solutions is defined over the developing resolution, and the developing resolution is canonically mapped into the anvil, the canonical family of formal solutions associated with the developing resolution is naturally defined over the anvil as well.

(5) By part (1) we may assume that Q^{n}(y, h_{1}, w, p, a) is isomorphic to the limit group Q^{n−1}(y, h_{1}, w, p, a) associated with the anvil, Anv(MGQ^{n−1}Res)(t, y, w, p, a), and by parts (2)-(3) we may assume that Q^{n}(t_{s(n)−1}, y, w, p, a) is isomorphic to the subgroup Q^{n−1}(t_{s(n)−1}, y, w, p, a) associated with the anvil. Part

(3) treats all the cases in which the n-th multi-graded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) is not of maximal complexity. Hence, the only case left in presenting the general step of our procedure for validation of a sentence, is the case of an n-th multi-graded quotient resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) of maximal possible complexity, i.e., a multi-graded quotient resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

consists of a single level, and the abelian decomposition associated with this level is identical to the abelian decomposition associated with the top level of the multi-graded quotient resolution

MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a)

we have started the n-th step of the procedure with. We treat this case as we treated it in the minimal rank case in the previous section and in the general step of the procedure for validation of a sentence presented in section 4 of [Se4]. In case the abelian decompositions associated with MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) and the top

level of MGQ^{n−1}Res(t, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a) are identical, we use the modular groups associated with the abelian decomposition associated with MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) to map the subgroup Q^{n}(t, y, w, p, a) into the limit group associated with the (appropriate) next level of the corresponding part of the anvil, Anv(MGQ^{n−1}Res)(t, y, w, p, a), i.e., into the subgroup associated with the next level of the developing resolution associated with that anvil. By the construction of the developing resolution associated with the anvil, Anv(MGQ^{n−1}Res)(t, y, w, p, a), this image of the appropriate subgroup of Q^{n}(t, y, w, p, a) is a proper quotient of itself, and the relevant part of the developing resolution associated with the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a) is composed of smaller number of levels. We now analyze the remaining part of the anvil in the same way we have analyzed n-th multi-graded quotient resolution, according to parts (1)-(5). As long as the abelian decompositions associated with the constructed multi-graded quotient resolutions are of maximal possible complexity we continue analyzing multi-graded quotient resolutions associated with various parts of the anvil we have started with, until one of the parts (1)-(4) applies to the corresponding multi-graded quotient resolution. If there exists a level for which one of the parts (1)-(4) applies, we set a developing resolution and an anvil with the limit group associated with this level according to the part (1)-(4) that applies to it. With that anvil we associate a canonical finite collection of auxiliary resolutions associated with that level. With each auxiliary resolution we set its terminal (rigid or solid) limit group to be the associated auxiliary limit group, and the modular groups associated with the auxiliary resolution to be the auxiliary modular groups (see part (4) of the general step). Given an auxiliary resolution we set the developing resolution to be the resolution induced by the subgroup < y, h_{1}, w, p, a > from the parts of the resolution above that level followed by the image of the developing resolution associated with that level in the associated auxiliary limit group. We set the anvil, Anv(MGQ^{n}Res)(t, y, w, p, a), to be the subgroup generated by the the auxiliary limit group associated with the auxiliary resolution and the subgroup Q^{n}(t, y, w, p, a), amalgamated along the copies of the limit group associated with the level to which one of the steps (1)-(4) was applied. We further associate with the anvil a finite collection of auxiliary resolutions and limit groups. With the developing resolution and its associated anvil, we naturally associate a family of formal solutions, x_{α}(t, y, w, p, a), parameterized by the elements of the auxiliary modular groups.

If all the abelian decompositions associated with the multi-graded resolutions used for the construction of the developing resolution are of maximal complexity, i.e., if none of the parts (1)-(4) applies to any of these multigraded resolutions, we examine the structure of the developing resolution. The developing resolution is built from a sequence of induced resolutions. Each of the induced resolutions is a resolution induced by the (image of the) subgroup < y, h_{1}, w, p, a >, and with each level of the induced resolution there is associated an (induced) abelian decomposition (see section 3 of [Se4] for the construction of the induced resolution).

Proposition 2.7. Suppose that all the abelian decompositions associated with the multi-graded resolutions used for the construction of the developing resolution are of maximal possible complexity. Let < v, y, h_{1}, w, p, a > be the subgroup generated by the closure of the developing resolution in the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a). From each of the multi-graded resolutions used to construct the developing resolution, there is a resolution induced by the (image of the) subgroup < v, y, h_{1}, w, p, a >.

Then either the structure of the resolution composed from the resolutions induced by the subgroup < v, y, h_{1}, w, p, a > from the various multi-graded resolutions used to construct the developing resolution, is identical to the structure of the developing resolution, or there exists some level j, so that the structure of the abelian decompositions associated with the resolutions induced by the subgroup < v, y, h_{1}, w, p, a > above level j are identical to the structure of the abelian decompositions associated with the corresponding resolution induced by the subgroup < y, h_{1}, w, p, a >, and in level j, either the number of factors in the free decomposition associated with the abelian decomposition associated with the resolution induced by < v, y, h_{1}, w, p, a > is strictly smaller than the number of factors in the corresponding free decomposition associated with the resolution induced by the subgroup < y, h_{1}, w, p, a >, and in case of equality in the number of factors, the complexity of the abelian decomposition associated with the resolution induced by < v, y, h_{1}, w, p, a > is strictly smaller than the complexity of the abelian decomposition associated with the resolution induced by < y, h_{1}, w, p, a >.

Proof: Identical to the proof of proposition 4.8 of [Se4].

□

If the structure of the resolution composed from the resolutions induced by the subgroup < v, y, h_{1}, w, p, a > from the various multi-graded resolutions used to construct the developing resolution, is identical to the structure of the developing resolution, we reached a terminal point of our branching procedure. With the multi-graded resolution

MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

we associate the terminal rigid or solid limit group of the resolution composed from the resolutions induced by the subgroup < v, y, h_{1}, w, p, a >, which we denote W P (HG)^{n+1}. If the structure of the resolution composed from the resolutions induced by the subgroup < v, y, h_{1}, w, p, a > from the various multi-graded resolutions used to construct the developing resolution, is not identical to the structure of the developing resolution, proposition 2.7 implies that there exists some level j for which the structure of the abelian decompositions associated with the resolutions induced by the subgroup < v, y, h_{1}, w, p, a > above level j are identical to the structure of the abelian decompositions associated with the corresponding resolution induced by the subgroup < y, h_{1}, w, p, a >, and in level j, either the number of factors in the free decomposition associated with the abelian decomposition associated with the resolution induced by < v, y, h_{1}, w, p, a > is strictly smaller than the number of factors in the corresponding free decomposition associated with the resolution induced by the subgroup < y, h_{1}, w, p, a >, and in case of equality in the number of factors, the complexity of the abelian decomposition associated with the resolution induced by < v, y, h_{1}, w, p, a > is strictly smaller than the complexity of the abelian decomposition associated with the resolution induced by < y, h_{1}, w, p, a >. In this case we do the following.

Let CRes(u, y, h_{1}, w, p, a) be the graded resolution that is composed from the resolutions induced by the subgroup < v, y, h_{1}, w, p, a >. With the resolution CRes(u, y, h_{1}, w, p, a) we associate its terminal rigid or solid limit group (with respect to the parameter subgroup < h_{1}, w, p >, which we denote: W P (HG)^{n+1}(g_{n+1}, h_{n+1}, g_{n}, h_{1}, w, p, a). We collect all the formal solutions defined over ungraded resolutions covered by the graded resolution CRes(u, y, h_{1}, w, p, a). This collection of formal solutions factors through a canonical collection of graded formal limit groups. With each graded formal limit group we associate its graded formal Makanin-Razborov diagram as we did in section 3 of [Se2]. We continue with each of the graded formal resolutions that appear in these diagrams in parallel. Let GF Res(x, u, y, g_{n+1}, h_{n+1}, g_{n}, h_{1}, w, p, a) be a graded formal resolution in one of these diagrams, and let W P (HG)^{n+1}H(h_{n+2}, g_{n+1}, h_{n+1}, g_{n}, h_{1}, w, p, a) be its terminating rigid or solid (not formal!) limit group. With the graded formal resolution GF Res(x, u, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a) we associate the resolution GRes(û, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a), which is the graded (not formal!) resolution associated with the terminal formal limit group of the graded formal resolution GF Res(x, u, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a). With a graded formal resolution we naturally associate a family of formal solutions parameterized by elements of the graded formal modular groups associated with the graded formal resolution. We continue with each of the graded formal resolutions separately. If the subgroup generated by < y, h_{1}, w, p, a > in a graded formal resolution GF Res(x, u, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a) is a proper quotient of the limit group Q^{n}(y, h_{1}, w, p, a) we have started the analysis of the multi-graded resolution MGQ^{n}Res(t, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) with, we replace this graded formal resolution by starting the preliminary step of the procedure with the limit group generated by < y, h_{1}, w, p, a > in the limit group associated with the graded formal resolution GF Res(x, u, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a). Hence, for the continuation we may assume that these two limit groups generated by < y, h_{1}, w, p, a > are isomorphic. In this case we set the developing resolution to be the graded resolution GRes(û, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a). We set the anvil, which we denote Anv(MGQ^{n}Res)(t, y, w, p, a), to be the group generated by the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a), to which we add roots according the roots added to the graded resolution

GRes(û, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a)

and the terminal limit group of the graded resolution

GRes(û, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a)

W P (HG)^{n+1}H(h_{n+2}, g_{n+1}, h_{n+1}, g_{n}, h_{1}, w, p, a), by adding relations that force the corresponding copies of the subgroups generated by < g_{n+1}, h_{n+1}, g_{n}, h_{1}, w, p, a > to be equal. In this case the developing resolution is not necessarily embedded into the anvil, but it is clearly canonically mapped into it. With the anvil, Anv(MGQ^{n}Res)(t, y, w, p, a) we associate the graded auxiliary resolution, and graded auxiliary modular groups associated with the anvil Anv(MGQ^{n−1}Res)(t, y, w, p, a) it was constructed from, and the graded formal auxiliary resolution associated with the graded formal resolution: GF Res(x, u, y, h_{n+2}, g_{n+1}, h_{n+1}, h_{1}, w, p, a). With both the developing resolution and the anvil, Anv(MGQ^{n}Res)(t, y, w, p, a) we associate the family of formal solutions x_{α}(t, y, w, p, a), parameterized by the graded auxiliary modular groups and the graded formal modular groups associated with the anvil.

Given the developing resolution, the anvil, and the family of formal solutions defined over them, we construct the staggered data-structure according to definition 2.5. With the anvil constructed in the general step of the iterative procedure, we have associated a family of formal solutions {x_{α}(t, y, w, p, a)}. We start step n + 1 of the iterative procedure, by imposing on the specializations that factor through and are taut with respect to the resolutions associated with the anvil, the (finitely many) systems of equations: ψ_{j}(x_{α}(t, y, w, p, a), y, w, p, a) = 1, a system for each fixed index j, and the entire family of formal solutions {x_{α}(t, y, w, p, a)}.

III: Termination of the first iterative procedure

Defining the preliminary, initial and general steps of our iterative procedure for analyzing the set EAE(p), we are required to prove its termination. To prove termination of our iterative procedure in the minimal rank case (section 1), we used the strict decrease in the complexity of the resolutions associated with successive steps of the procedure, a strict decrease that forces termination. Unlike our procedure in the minimal rank case, in the general procedure we do not obtain a strict decrease in the complexity of the resolutions associated with successive steps of the procedure. To obtain termination in the general case, we need to look at limit groups (or alternatively Zariski closures) and complexities of various resolutions and decompositions associated with the staggered resolutions and anvils constructed along the steps of the procedure. Our ultimate goal in proving the termination of the procedure, is to show that after finitely many steps of it, the iterative procedure is applied not to specializations of the limit QRlim(y, h_{1}, w, p, a) but rather to specializations of a proper quotient of it. Our entire argument is similar to the one used to prove the termination of the procedure for validation of a sentence ([Se4],4.12).

Theorem 2.8. The iterative procedure for the analysis of the set EAE(p) terminates after finitely many steps.

Proof: Identical to the proof of theorem 4.12 of [Se4].

§3. The Sieve Method in Few Special Cases

As in the procedure for the analysis of an EAE set in the minimal (graded) rank case, presented in the first section, the outcome of the ”trial and error” procedure gives us a tree of stratified sets which encodes all the (finitely many) possible sequences of families of formal solutions that are needed in order to validate that a certain specialization p_{0} of the defining parameters p is indeed in the set EAE(p). This stratification is the basis for our analysis of the structure of the set EAE(p) in the general case as well.

Let EAE(p) be the set defined by the predicate:

EAE(p) = ∃w ∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1.

Recall (definition 1.19) that a specialization w_{0} of the variables w, is said to be a witness for a specialization p_{0} of the defining parameters p if the following sentence:

∀y ∃x Σ(x, y, w_{0}, p_{0}, a) = 1 ∧ Ψ(x, y, w_{0}, p_{0}, a) ≠ 1

is a truth sentence. Clearly, if there exists a witness for a specialization p_{0} then p_{0} ∈ EAE(p), and every p_{0} ∈ EAE(p) has a witness.

In order to show that a specialization p_{0} of the defining parameters p is in the set EAE(p), we need to find a witness w_{0} for the specialization p_{0}. As in the minimal rank case, the construction of the tree of stratified sets, guarantees that a witness w_{0} for a specialization p_{0}, proves that p_{0} ∈ EAE(p) using a certain ”proof system” which is built from a finite sequence of (families of) formal solutions, which correspond to boundedly many paths along the tree of stratified sets. Given p_{0} ∈ EAE(p), we are not able to say much about a possible witness for p_{0} using the information we have collected so far. However, as in the minimal rank case, with each ”proof system”, i.e., with each path of (families of) formal solutions that goes along the tree of stratified sets, one can associate a certain Diophantine set of possible witnesses. As demonstrated in the first section, the bound on the form and number of all possible ”proof systems” associated with all possible witnesses suggested by the tree of stratified sets, forces every possible witness for p_{0} to belong to one of the finitely many Diophantine sets associated with the (finite) collection of all proof systems. In parallel with the minimal rank case, in the sequel we will show that if a specialization p_{0} ∈ EAE(p), and it can be shown that p_{0} ∈ EAE(p) using a witness w_{0} with a corresponding proof system, then every ”generic” specialization of w that belongs to either (a closure of) some modular block or some Diophantine set associated with the Diophantine set which is associated with the specific proof system is a witness for p_{0} using the same proof system as w_{0}. This will reduce the analysis of the set EAE(p) to the analysis of the Diophantine sets and the modular blocks and further Diophantine sets associated with each ”proof system”, and eventually will enable us to show that the set EAE(p) is in the Boolean algebra of AE sets, which finally concludes the proof of theorem 1.4.

In parallel with the minimal rank case (definition 1.20), if p_{0} ∈ EAE(p) is a specialization of the defining parameters p, and w_{0} is a witness for p_{0}, then the construction of the tree of stratified sets, implies that one can associate a proof system with the couple (w_{0}, p_{0}), which corresponds to a (finite) collection of paths in the tree of stratified sets. Note that there may be several proof systems associated with a given couple (w_{0}, p_{0}), but the construction of the tree of stratified sets guarantees that the number of proof systems associated with the couple (w_{0}, p_{0}) is globally bounded. As in the first section we will say that a given proof system associated with the couple (w_{0}, p_{0}) is of depth d, if all the paths associated with the proof system terminate after d steps (levels) of the tree of stratified sets.

As we did in the minimal rank case, we will start by demonstrating our approach for the analysis of the set EAE(p), by analyzing those specializations of the defining parameters p that have witnesses with proof systems of ”depth 0”, i.e., those that have witnesses with proof systems that terminate after the preliminary step of the construction of the tree of stratified sets. We will continue by analyzing the specializations of the defining parameters p for which there are witnesses with proof systems of depth at most 1, i.e., those that have witnesses with proof systems that terminate after the initial step of the construction of the tree of stratified sets, and then present the analysis of the entire set EAE(p).

Lemma 3.1. Let T_{0}(p) ⊂ EAE(p) be the subset of all specializations p_{0} ∈ EAE(p) of the defining parameters p, that have a witness with a proof system that terminates after the preliminary step of the construction of the tree of stratified sets. Then T_{0}(p) is an EA set.

Proof: The preliminary step of the iterative procedure for the construction of the tree of stratified sets in the general case, is identical to the first step of the iterative procedure for the construction of the tree of stratified sets in the minimal rank case. Hence, the proof of lemma 3.1 is identical with the proof of lemma 1.21.

□

Lemma 3.1 proves that the set of specializations p_{0} of the defining parameters p that have a witness with a proof system that terminates after the preliminary step of the iterative procedure for the construction of the tree of stratified sets, is an EA set. Before analyzing the entire set EAE(p) we analyze the set of specializations p_{0} that have witnesses with a proof system that terminates after the initial step of the iterative procedure for the construction of the tree of stratified sets. The analysis of specializations p_{0} ∈ EAE(p) having witnesses with such proof systems is much more complicated than the analysis of witnesses with proof systems that terminate after the preliminary step, and it is based, though it is somewhat different, on the analysis of the set of specializations p_{0} having proof systems of depth 2, T_{2}(p), in the minimal rank case (theorem 1.22).

Theorem 3.2. Let T_{1}(p) ⊂ EAE(p) be the subset of all specializations p_{0} ∈ EAE(p) of the defining parameters p, that have witnesses with a proof system that terminates after the initial step of the procedure for the construction of the tree of stratified sets. Then T_{1}(p) is in the Boolean algebra of AE sets.

Proof: To start the proof of theorem 3.2 recall that a valid P S statement, presented in definition 1.23, is a statement that satisfies a list of properties required by the corresponding proof system (see definition 1.23).

As in the minimal rank case, the tree of stratified sets guarantees that there are finitely many proof systems of depth 1 (i.e., those that terminate after the initial step of the procedure for the construction of the tree of stratified sets). Once we fix a proof system of depth 1, we have fixed the rigid or solid limit group W P H(h, w, p, a), the number of rigid or strictly solid families of specializations of each of the limit group W P HG(g_{1}, h_{1}, w, p, a), and the number of rigid or solid specializations of each of the limit groups W P HGH(h_{2}, g_{1}, h_{1}, w, p, a). As in the first section, we start the analysis of the set T_{1}(p) enumerating all the possible proof systems, and for each proof system we collect all possible (configurations of) valid P S statements: ((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) (the integer ν(ps) depends on the fixed proof system), which by the standard arguments presented in section 5 of [Se1], factor through a (canonical) collection of maximal limit groups P SHGH_{1}, …, P SHGH_{m}, which we call P S (proof system) limit groups.

By construction, for each p_{0} ∈ T_{1}(p) there exists some witness w_{0} and a corresponding proof system, so that a specialization of the form:

((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that is associated with the specialization p_{0}, the witness w_{0} and that proof system, is a valid P S statement (i.e., it satisfies conditions (i)-(iv) of definition 1.23), and factors through a limit group P SHGH_{j}. As in the minimal rank case, our main goal will be to show that these valid P S statements are ”generic” in some modular blocks associated in the sequel with each of the P S limit groups P SHGH. The ”sieve” procedure for the analysis of the valid P S statements that factor through a given P S limit group combines the procedure presented in the first section for the minimal ranks case, with tools used in the construction of the tree of stratified sets in the general case, and with the notion of a core resolution presented in the next section. As in the first section (proposition 1.24), we start by analyzing those P S limit groups which are rigid or solid with respect to the parameter subgroup P =< p >.

Proposition 3.3. Suppose that a P S limit group P SHGH is rigid or solid with respect to the parameter subgroup P =< p >, and if it is solid suppose that the subgroup W P =< w, p > is a subgroup of the distinguished vertex group in the graded JSJ decomposition of P SHGH (i.e., the vertex stabilized by the subgroup AP =< p, a >). The set of specializations p_{0} that have a witness w_{0}, and a rigid or solid specialization ((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) of P SHGH which is a valid P S statement, which we denote P S(p), is in the Boolean algebra of AE sets.

Proof: Identical to the proof of proposition 1.24.

□

Proposition 3.3 proves theorem 3.2 in case the P S limit groups P SHGH are rigid or solid with no flexible quotients, and the subgroup W P =< w, p > is a subgroup of the distinguished vertex group in the graded JSJ decomposition of P SHGH. In this special case, the number of possible witnesses w_{0} associated with each specialization p_{0} of the defining parameter p is finite and globally bounded. In the general case, the number of possible witnesses associated with each specialization p_{0} of the defining parameters p is infinite. Our goal in the analysis of the set T_{1}(p) in the general case, is to find finitely many modular blocks associated with each of the P S limit groups P SHGH, modular blocks in which a generic specialization of the variables w is a witness for the corresponding specialization p_{0} of the defining parameters P. To achieve this goal, i.e., to find the finitely many modular blocks associated with each of the P S limit groups P SHGH, we present a ”trial and error” procedure, based on the one used to construct the tree of stratified sets (in the general case). As in the minimal rank case, the output of the iterative ”trial and error” procedure, i.e., the finitely many modular blocks associated with each P S limit group P SHGH, are later used to derive a predicate in the Boolean algebra of AE predicates that describes the set T_{1}(p).

Let P =< p > be the group of defining parameters. With each of the limit groups P SHGH_{i} we associate its canonical taut graded Makanin-Razborov diagram (with respect to the parameter subgroup P), which contains finitely many graded resolutions which we denote P SHGHRes_{j}, and each graded resolution P SHGHRes_{j} is defined over the rigid or solid limit group P B_{j}(b, p, a). We will treat the limit groups P SHGH_{i} and their graded resolutions P SHGHRes_{j} and rigid or solid limit groups P B_{j}(b, p, a) in parallel, hence, we will omit the indices of the limit group and its graded resolution. In the sequel, we will treat each strata in the singular locus of the graded resolutions P SHGHRes separately, and do it in parallel.

Precisely as we did in the minimal rank case, we start our analysis of the set of valid P S statements by associating with (the completion of) the graded P S resolution P SHGHRes a (canonical) finite collection of Non-Rigid and Non-Solid P S limit groups (see definition 1.25). We denote the non-rigid P S limit groups associated with the P S resolution P SHGHRes, NRgdP S_{1}, …, NRgdP S_{q}, and the nonsolid P S limit groups associated with P SHGHRes, NSldP S_{1}, …, NSldP S_{r}.

Recall (definition 1.25) that the graded formal closures associated with the collection of non-rigid and non-solid PS limit groups, determine those ”generic” specializations that factor through and are taut with respect to the various P S resolutions P SHGHRes but fail to be valid P S statements with respect to the (fixed) proof system because certain specializations that are required to be rigid or strictly solid actually factor through flexible quotients of the corresponding rigid or solid limit groups. As in the minimal rank case, ”generic” specializations that factor through the P S resolutions P SHGHRes can fail to be valid P S statements in other ways as well. In particular, we need to collect all the the test sequences that factor through the P S resolutions P SHGHRes, and for which for at least one of the tuples: (h_{j}^{2}(n), g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}, a) there exist some specialization g_{j}^{2}(n) so that the (combined) specialization: (g_{j}^{2}(n), h_{j}^{2}(n), g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}, a) factors through (at least) one of the limit groups W P (HG)^{2}, or there exists a specialization g_{1}(n) so that the (combined) specialization: (g_{1}(n), h_{1}(n), w_{n}, p_{n}, a) factors through (at least) one of the limit groups W P HG, which is not associated with one of the paths associated with our fixed proof system (cf. definition 1.31).

Definition 3.4. Let Comp(P SHGHRes)(v, (h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{1}, w, p, a) be the completion of the graded resolution P SHGHRes. we look at the entire collection of all test sequences that factor through the graded completion Comp(P SHGHRes):

{v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{ν(ps)}^{2}(n), g_{ν(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{n}, a)}.

We look at those graded test sequences for which for some index j there exists some specialization g_{j}^{2}(n) so that the combined specialization: (g_{j}^{2}(n), h_{j}^{2}(n), g_{j}^{1}(n), h_{1}(n), w_{n}, p_{n}) which is not supposed to factor through any of the limit groups W P (HG)^{2} according to our fixed proof system (which has depth 1), factors through at least one of these limit groups, or for which there exists some specialization g_{1}(n) so that the combined specialization: (g_{1}(n), h_{1}(n), w_{n}, p_{n}) factors through a limit group W P HG which is not associated with a path that is associated with our fixed proof system. The collection of all these (graded) test sequences factor through a (canonical) collection of maximal left P S limit groups LeftP S_{1}, …, LeftP S_{m}. The analysis of graded formal limit groups presented in section 3 of [Se2] associates (canonically) with each Left PS limit group LeftP S_{i} a graded formal Makanin-Razborov diagram, and each such graded formal resolution is in fact a one level graded resolution, which is a graded formal closure of the graded resolution P SHGHRes, GF Cl(P SHGHRes). Clearly, no specialization ((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) that factors through the resolution P SHGH, and which is a valid P S statement with respect to our fixed proof system, factors through one of the LeftP S limit groups LeftP S_{1}, …, LeftP S_{m}, and their associated resolutions.

”Generic” specializations that factor through the P S resolutions P SHGHRes can fail to be valid P S statements also if there exist additional rigid or strictly solid specializations of the limit groups P SHG(g_{1}, h_{1}, w, p, a) that are not specified by the given specializations. As in the minimal rank case, the ”generic” specializations for which there exists ”surplus” in rigid or strictly solid specializations are collected in extra P S (graded) limit groups and graded resolutions (definition 1.26). We denote the extra P S limit groups associated with the graded P S resolution P SHGHRes, ExtraP S_{1}, …, ExtraP S_{ℓ}.

The extra P S limit groups and their associated graded formal closures collect all the ”generic” specializations (i.e., all the test sequences) of the P S limit groups P SHGH for which there exist rigid or strictly solid families in addition to those specified by the generic specializations. For a general specialization of the P S limit groups P SHGH, i.e., a specialization which is not necessarily ”generic”, it may as well be that the additional rigid or strictly solid specializations, collected by the extra P S limit groups and their associated graded formal closures, do become flexible or do coincide with the rigid or strictly solid families of the various specializations (g_{j}^{1}, h_{0}^{1}, w_{0}, p_{0}). In the minimal rank case, we collected all the specializations that factor through one of the extra P S graded formal closures associated with one of the extra P S limit groups ExtraP S, in which such a ”collapse” of the additional rigid and strictly solid families occurs, in a canonical collection of collapsed extra P S (graded) limit groups and their associated (collapsed extra P S) graded resolutions (definition 1.35). In the general case, the analysis of the specializations that factor through one of the extra P S graded formal closures, in which such a ”collapse” of the additional rigid and strictly solid families occurs, and the construction of the associated collapsed extra P S (graded) limit groups and their collapsed extra P S graded resolutions is more involved, and so is the iterative procedure for the analysis of the set T_{1}(p). Hence, mostly for presentation purposes, in this section we present the construction of the collapsed P S limit groups and their associated graded resolutions, and the iterative procedure for the analysis of the set T_{1}(p) in the special case in which all the limit groups W P HG are rigid. The approach we use in the rigid case can be somewhat generalized, however, the general case of general rigid and solid W P HG seems to be considerably more involved, and it is treated in the next sections.

In this part we assume that all the limit groups W P HG associated with one of the paths associated with our fixed proof systems are rigid limit groups. In this case the analysis of the set T_{1}(p) is conceptually similar to the procedure presented in the minimal rank case, combined with the iterative procedure for the construction of the tree of stratified sets. Let ExtraP SRes be one of the extra P S graded resolutions associated with one of the extra P S graded limit groups ExtraP S_{i}. Note that by construction, ExtraP SRes is a graded formal closure of the P S resolution P SHGHRes we have started the analysis with. In the case the groups W P HG are all rigid, we will say that a specialization that factors through and is taut with respect to the extra P S graded resolution ExtraP SRes is collapsed if the variables added for each of the additional rigid specializations (i.e., the ones that were not specified by the proof system) satisfy one of the following:

(1) A specialization of the variables added for one of the additional rigid specializations becomes trivial or flexible.

(2) A specialization of the variables added for one of the additional rigid specializations becomes equal to one of rigid specializations specified by the proof system, i.e., with one of the specializations g_{j}^{1} in the specialization:

(u, v, (h_{1}^{2}, r_{1}^{1}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, r_{ν(ps)}^{1}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}).

Defining the finitely many possibilities for collapse forms, we collect all the specializations that are collapsed specializations in finitely many closures of the resolution P SHGHRes we have started with, which we call formal collapsed Extra P S (graded) resolutions, and finitely many graded limit groups, which we call collapsed Extra P S (graded) limit groups. We start by collecting all test sequences that factor through an extra P S resolution and satisfy the conditions of one of the possible collapsed forms, in finitely many formal collapsed extra P S limit groups and resolutions, and then collect the set of all collapsed specializations in a finite collection of collapsed extra P S limit groups. The definition of the formal collapsed extra P S resolutions (definition 3.5) is general and does not depend on the graded limit groups W P HG being rigid or solid.

Definition 3.5. Let ExtraP SRes be one of the extra P S graded resolutions associated with one of the extra P S graded limit groups ExtraP S_{i}. Note that by construction, ExtraP SRes is a graded formal closure of the P S resolution P SHGHRes we have started the analysis with. With the Extra P S resolution we associate all its possible collapse forms.

Let ExtraP SRes(v, , (h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{1}, w, p, a) be an extra P S resolution. We look at the entire collection of all test sequences that factor through the extra P S resolution. We look at those graded test sequences for which each specialization of the sequence satisfies the conditions of one of the (finitely many) possible collapse forms associated with the extra P S resolution. The collection of all these (graded collapsed) test sequences factor through a (canonical) collection of maximal formal collapsed extra P S limit groups F CollapseExtraP S_{1}, …, F CollapseExtraP S_{t}. The analysis of graded formal limit groups presented in section 3 of [Se2] associates (canonically) with each formal collapse extra P S limit group F CollapseExtraP S a graded formal Makanin-Razborov diagram, and each such graded formal resolution is in fact a one level graded resolution, which is a graded formal closure of the graded resolution P SHGHRes, GF Cl(P SHGHRes). We call each such graded formal closure, a formal collapsed extra P S (graded) resolution.

Definition 3.6. Let ExtraP SRes be one of the extra P S graded resolutions associated with one of the extra P S graded limit groups ExtraP S_{i}. With the Extra P S resolution we associate all its possible collapse forms. ExtraP SRes is a graded formal closure of the P S resolution P SHGHRes we have started the analysis with, so with the resolution ExtraP SRes we can associate its canonical collection of graded auxiliary resolutions. Note that if a specialization that factors through a graded auxiliary resolution associated with the extra P S resolution, ExtraP SRes, satisfies the conditions of one of the collapsed forms associated with it, the specialization has to satisfy a (fixed) system of equations associated with the specific collapsed form.

We go over all the graded auxiliary resolutions and all the collapsed forms associated with the extra P S resolution, ExtraP SRes. Given a graded auxiliary resolution and a collapsed form, we look at all the specializations that factor through the given graded auxiliary resolution and satisfy the system of equations associated with the specified collapse form associated with the resolution ExtraP SRes. By our standard method presented in section 5 of [Se1], the collection of specializations that factor through a given auxiliary resolution and satisfy the system of equations associated with a specified collapsed form, factor through a canonical (finite) collection of maximal limit groups, which we call collapsed extra P S limit groups, and denote: CollapseExtraP S_{1}, …, CollpaseExtraP S_{d}. Note that by construction, each collapsed extra P S limit group is a proper quotient of the limit group associated with the extra P S resolution, ExtraP SRes we have started with.

The P S limit groups P SHGH and their associated resolutions, together with the non-rigid and left P S limit groups and their associated graded resolutions, the extra P S resolutions, the formal collapsed extra P S resolutions, and the collapsed extra P S limit groups, enable us to generalize the main principle that was used in the procedure for quantifier elimination under the minimal (graded) rank assumption, to the case in which the graded limit group W P HG are all rigid. As in the minimal rank case, we show that if a valid P S statement factors through one of the resolutions P SHGHRes then either there exists a valid P S statement that factors through one of the collapsed extra P S limit groups and their associated graded auxiliary resolutions, or there exists a ”generic” valid P S statement, i.e., a test sequence of valid P S statements that factor through the P S resolution P SHGHRes. This principle does not depend on the graded limit groups W P HG being rigid.

Theorem 3.7. Let P SHGHRes be one of the P S graded resolutions, and suppose there exists a valid P S statement: ((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) that factors through the completion of the resolution P SHGHRes with respect to our fixed proof system. Then one of the following holds:

(1) there exists a test sequence of valid P S statements (with respect to our fixed proof system):

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{ν(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

that factor through the completion of the P S resolution P SHGHRes (note that the specialization p_{0} of the defining parameters p are fixed along the entire test sequence).

(2) the valid P S statement factors through one of the graded auxiliary resolutions associated with the P S resolution P SHGHRes and one of its associated collapsed extra P S limit groups.

Proof: Similar to the proof of theorem 1.32.

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As in the minimal rank case, theorem 3.7 reduces the analysis of the set T_{1}(p) to the set of specializations of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through the various resolutions P SHGHRes. Proposition 3.8 below does not depend on the graded limit groups W P HG being rigid.

Proposition 3.8. Let P SHGHRes be one of the (non-trivial) P S graded resolutions. Let T SP S(p) be the set of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements:

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{d(ps)}^{2}(n), g_{ν(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

that factor through the completion of the P S resolution P SHGHRes, Comp(P SHGHRes). Then T SP S(p) is in the Boolean algebra of AE sets.

Proof: Identical to the proof of proposition 1.33.

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As in the minimal (graded) rank case, at this stage we have all the tools needed for showing that the set T_{1}(p) is in the Boolean algebra of AE sets. By construction, if p_{0} ∈ T_{1}(p) then there must exist a valid P S statement of the form: ((h_{1}^{2}, r_{1}^{1}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, r_{ν(ps)}^{1}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) that factors through one the of the P S resolutions P SHGHRes constructed with respect to all proof systems of depth 1.

By proposition 3.8, the sets T SP S(p) associated with the various P S resolutions P SHGHRes, i.e., the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of the P S resolutions P SHGHRes, is in the Boolean algebra of AE sets. By theorem 3.7 if there exists a valid P S statement that factors through a P S resolution P SHGHRes, then either there exists a test sequence of valid P S statements that factors through that P S resolution, or there must exist a specialization that factors through one of the graded auxiliary resolutions associated with one of the P S resolutions P SHGHRes, and one of its associated collapsed extra P S limit groups.

We continue with each of the graded auxiliary resolutions, and each of their associated collapsed extra P S limit groups associated with the various P S resolutions P SHGHRes. Given a graded auxiliary resolution and an associated collapsed extra P S limit group, we apply the procedure of the general step of the procedure for the construction of the tree of stratified sets. The output of this general step is a finite collection of anvils and their associated developing resolutions. Exactly as we did with each of the P S resolutions P SHGHRes, we associate (canonically) with each developing resolution its Non-Rigid and Left P S resolutions, extra P S resolutions, and formal collapsed extra P S resolutions. with each graded auxiliary resolution associated with an extra P S resolution, we further associate a collapsed extra P S limit group. By proposition 3.8 applied, the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of the developing resolutions, are in the Boolean algebra of AE sets. By theorem 3.7 applied to the various developing resolutions, if there exists a valid P S statement that factors through a given developing resolution, then either there exists a test sequence of valid P S statements that factors through that developing resolution, or there must exist a specialization that factors through one of the associated graded auxiliary resolutions and through its associated collapsed extra P S limit group.

Continuing iteratively with a procedure similar to the one used to construct the tree of stratified sets, we obtain a terminating iterative procedure by the proof of theorem 2.8 which is identical to the proof of theorem 4.12 of [Se4]. The iterative procedure we have constructed has to terminate with either a rigid limit group with respect to the defining parameters P =< p >, or a solid limit group in which the subgroup W P =< w, p > is contained in the distinguished vertex group in the associated graded abelian JSJ decomposition.

By iteratively applying proposition 3.8 to the various developing resolutions constructed along the iterative procedure, the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of these developing resolutions are in the Boolean algebra of AE sets. By proposition 3.3 the set of specializations p_{0} of the defining parameters P =< p > for which there exists a valid P S statement that factors through one of the terminating rigid or solid P S limit groups is in the Boolean algebra of AE sets. The entire set T_{1}(p) is precisely the set of specializations p_{0} of the defining parameters P =< p > for which there exists a valid P S statement, hence, the set T_{2}(p) is in the Boolean algebra of AE sets, and the proof of theorem 3.2 in case all the graded limit groups W P HG are rigid is completed.

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§4. Core Resolutions

In the first section we have shown that an EAE set is in the Boolean algebra of AE sets, assuming the limit groups defining it are of minimal rank. The procedure used to analyze the structure of an EAE set presented in the first section, is combined from two iterative procedures. The first iterative procedure constructs the tree of stratified sets, from which a finite collection of proof systems is obtained, and the second is a sieve procedure for finding those specializations of the defining parameters for which there exists a valid proof statement which is in the form of one of the proof systems derived from the tree of stratified sets. In the second section, we combined concepts and techniques used in the minimal rank case, with the iterative procedure for validation of an AE sentence presented in section 4 of [Se4], to generalize the iterative procedure for the construction of the tree of stratified sets to a general EAE sets. In the third section we considered few special cases of families of EAE sets, for which the techniques used in the procedure for validation of an AE sentence can be used in order to get a sieve procedure for these sets. However, the techniques used in the procedure for validation of a sentence presented in [Se4] are not sufficient in order to construct a sieve procedure for a general EAE set. In order to construct a general sieve procedure, we will need several additional tools and notions. In this section we present the needed tools and in the next one we use these tools to construct the sieve procedure for a general EAE set. To construct the iterative procedure for validation of an AE sentence, we needed to present geometric and induced resolutions (see section 3 of [Se4] for these notions). We start this section with the definition of a core resolution of a resolution Res(t, v, a) and its complexity.

Definition 4.1. Let Res(t, v, a) be a a well-separated resolution, and let Comp(Res)(u, t, v, a) be its completion. Let CRes(r, v, a) be a geometric subresolution of the completed resolution Res(u, t, v, a). We say that the resolution CRes(r, v, a) is a core resolution of the subgroup < v, a > in the completion Comp(Res)(u, t, v, a), if the resolution CRes(r, v, a) is a firm subresolution of the subgroup < r, v, a > ([Se4], definition 3.9). i.e., it has the following properties:

(i) the rank of the resolution CRes(r, v, a), rk(CRes(r, v, a)), is equal to the rank of the subgroup < r, v, a > with respect to the completed resolution Comp(Res)(u, t, v, a).

(ii) there exists a firm test sequence for the subgroup < r, v, a >.

(iii) Let A_{1}, …, A_{m} be all the non-cyclic pegged abelian groups that appear along the completed resolution Comp(Res)(t, v, a), let peg_{1}, …, peg_{m} be the pegs of the abelian groups A_{1}, …, A_{m}, and let {peg_{i}, q_{1}^{i}, …, q_{j_{i}}^{i}}_{i=1}^{m} be an arbitrary basis for the collection of the subgroups A_{1}, …, A_{m}. Then for any set of integers {(s_{j}^{i}, n_{j}^{i})}, where n_{j}^{i} ≥ 2 and 0 ≤ s_{j}^{i} ≤ n_{j}^{i}, there exist a firm test sequence {t_{n}, v_{n}, a} of the subgroup < v, a >, so that for every index n, the specialization of each of the pegs peg_{i} is an element h_{i} with no non-trivial

roots, and the specialization of each of the basis elements q_{j}^{i} is h_{i}^{r_{j}^{i}} where r_{j}^{i} = u_{j}^{i} · n_{j}^{i} + s_{j}^{i} for some positive integer u_{j}^{i}.

We denote a core resolution, Core(< v, a >, Res(t, v, a))(r, v, a). In exactly the same way we define a graded core resolution, and a multi-graded core resolution. Since a core resolution is in particular a geometric subresolution, we set the complexity of a core resolution to be its complexity as a geometric subresolution.

A (graded, multi-graded) core resolution is generally not unique, but given a well-separated resolution Res(t, v, a), and a subgroup < v, a > of its corresponding limit group, there always exists a core resolution of the subgroup < v, a >, simply the completion Comp(Res)(u, t, v, a) itself. However, in general, picking the completion of the ambient resolution as a core resolution for a given subgroup is not ”economical”, i.e., its complexity is going to be much larger than the complexity of other core resolutions that can be associated with the same subgroup. Hence, we continue by modifying the procedure for the construction of the induced resolution to get a procedure for the construction of a core resolution. The construction of a core resolution is composed from 2 iterative procedures. The procedure used for the first part is essentially identical to the iterative procedure used to construct the induced resolution. In the second part we use an iterative procedure that either reduces the rank of the resolution constructed in the first part, or alternatively, shows that the resolution constructed in the first part is a firm subresolution, hence, it is a core resolution. For presentation purposes, we present the second step of the construction in the case in which none of the abelian decompositions associated with the various levels of the resolution Res(t, v, a) we have started with, contain QH vertex groups, and then generalize the construction to an arbitrary resolution.

Let Comp(Res)(t, v, a) be a completed well-separated resolution, and let < v, a > < Rlim(t, v, a) be a subgroup of the limit group Rlim(t, v, a). With each level of the completed resolution Comp(Res)(t, v, a) there is associated a (possibly trivial) free decomposition and principal abelian decompositions of each of the factors. We denote the decompositions associated with the various levels of the completed resolution Comp(Res)(t, v, a): Λ_{1}, …, Λ_{q}, and the canonical epimorphisms between consecutive levels we denote: η_{1}, …, η_{q−1}. Furthermore, each vertex group in these decompositions which is neither quadratically hanging nor abelian is embedded into the next level by the canonical epimorphisms between consecutive levels. For the first part of the construction of a core resolution of the subgroup < v, a > in the completed resolution Comp(Res)(t, v, a) we use a finite iterative procedure, which is essentially identical to the first step in the construction of the induced resolution presented in section 3 of [Se4]. We start by describing its first step.

(i) Using standard Bass-Serre theory, the subgroup < v, a > < Rlim(t, v, a) inherits a decomposition Δ_{1} with abelian and trivial edge groups from the decomposition Λ_{1}. Note that if < v, a > intersects a conjugate of a QH vertex group in the decomposition Λ_{1} in a subgroup of finite index, then the intersection appears as a QH vertex group in the inherited decomposition Δ_{1}. If < v, a > intersects a conjugate of a QH vertex group in the decomposition Λ_{1} in a non-trivial subgroup of infinite index, then by ([Se4],1.5) the intersection gives rise to a free factorization (and a possible free factor) in the decomposition Δ_{1} of the group < v, a >.

(ii) Suppose the free decomposition inherited by the subgroup < v, a > from the decomposition Δ_{1} is:

< v, a >= F_{1}∗ < v_{1}, a > ∗ < v_{2} > ∗ … ∗ < v_{b} >

where F_{1} is a free group which is the free product of free factors contributed by subgroups of infinite index in QH vertex groups in Λ_{1} and Bass-Serre generators of loops with trivial stabilizer in Δ_{1}. < v_{1}, a > is the connected component that contains the vertex stabilized by F_{k} =< a_{1}, …, a_{k} > itself.

We continue with each of the factors < v_{1}, a >, < v_{2} >, …, < v_{b} > separately. We will denote each of these factors by V_{i}. Each factor V_{i} inherits an abelian splitting Δ_{1}^{i} from the decomposition Λ_{1}. Each edge e in Δ_{1}^{i} that connects two non-QH, non-abelian vertex groups, is composed from a couple of edges e_{1} and e_{2} that are adjacent and are both in the orbit of the same edge e′ in the Bass-Serre tree corresponding to the graph of groups Λ_{1} of Comp(Res)(t, v, a). Furthermore, e′ connects a non-abelian vertex group to an abelian vertex group in the decomposition Λ_{1}. Let A be the abelian vertex group that stabilizes the common vertex v of e_{1} and e_{2} in the Bass-Serre tree corresponding to the decomposition Λ_{1}. There exists an element a ∈ A that is mapped to the identity element by η_{1}, and conjugates the vertex adjacent to v in e_{2} to the vertex adjacent to v in e_{1}. We modify the factor V_{i} by adding to its generators the element a. We act in the same way on the factor V_{i}, in case an abelian vertex group in Δ_{1}^{i} is connected to two non-QH, non-abelian vertex groups, which are necessarily in the same orbit of a non-QH, non-abelian vertex group in the graph of groups Λ_{1}. Repeating this operation for all the edges connecting two non-QH, nonabelian vertex groups in the decomposition Δ_{1}^{i} of the factor V_{i}, and for all couples of edges connecting an abelian vertex group to two non-QH nonabelian vertex groups in Δ_{1}^{i}, we get a subgroup V̂_{i} < Comp(Res)(t, v, a), so that V_{i} < V̂_{i}, η_{1}(V_{i}) = η_{1}(V̂_{i}), and in the decomposition Δ̂_{1}^{i} inherited by V̂_{i} from the decomposition Λ_{1}, which is a ”folding” of the decomposition Δ_{1}^{i} of V_{i}, a non-QH, non-abelian vertex group is connected only to QH vertex groups and to abelian vertex groups and not to any other non-QH, non-abelian vertex groups. Furthermore, an abelian vertex group in Δ̂_{1}^{i} is connected to at most one non-QH, non-abelian vertex group as it is in the decomposition Λ_{1}.

(iii) As we did with a general well-separated resolution (definition 2.2 of [Se4]), we associate a decomposition Θ̂_{1}^{i} with the decomposition Δ̂_{1}^{i}, by cutting each of the punctured surfaces that correspond to QH vertex groups in Δ̂_{1}^{i} along a maximal collection of disjoint, non-homotopic (non-boundary parallel) s.c.c. which are mapped to the identity element by the homomorphism η_{1}. As we did in completing a general well-structured resolution ([Se2],1.12), we modify the factor V̂_{i} by adding Bass-Serre generators so that each connected punctured subsurface in the decomposition Θ̂_{1}^{i} that is connected to a non-QH vertex group in Θ̂_{1}^{i}, will be connected to a unique non-QH vertex group, and get a subgroup Ṽ_{i} with corresponding graphs of groups Δ̃_{1}^{i} and Θ̃_{1}^{i}. As in the construction of the completed resolution of a well-separated resolution, every connected component in the decomposition Θ̃_{1}^{i} that contains a non-QH vertex group, contains a unique non-QH, nonabelian vertex group, and (possibly) few abelian vertex groups all connected (only) to the (unique) non-QH, non-abelian vertex group in their connected component. All the QH vertex groups in a connected component in Θ̃_{1}^{i} are also connected only to the (unique) non-QH, non-abelian vertex group in that connected component. Since Comp(Res)(t, v, a) is a completed and well-separated resolution, all the Bass-Serre generators we have added to the factor V_{i} are elements of the completed resolution Rlim(t, v, a), and the subgroup Ṽ_{i} obtained after adding the Bass-Serre generators is embedded in the completed limit group Rlim(t, v, a). Furthermore, η_{1}(Ṽ_{i}) = η_{1}(V_{i}).

item”(iv)” With each factor Ṽ_{i} and each connected component in the decomposition Θ̃_{1}^{i} that contains a non-QH vertex group, we associate a subgroup J_{s}^{i} and continue to the second level of the completed resolution Comp(Res)(t, v, a) with each of the subgroups J_{s}^{i} separately. For each index i, and each connected component (indexed by s) in T̃heta_{1}^{i}, We set J_{s}^{i} to be the image under the canonical epimorphism η_{1} of the fundamental group of the corresponding connected component. Note that the Bass-Serre elements, and the elements from abelian vertex groups that were added to the various factors V_{i}, do not change its image under the epimorphism η_{1}.

(v) As we did in part (i), for each of the subgroups J_{s}^{i} we use standard Bass-Serre theory to get a decomposition Δ_{2}^{(i,s)} inherited by the subgroup J_{s}^{i} from the decomposition Λ_{2} associated with the second level of the completed resolution Comp(Res)(t, v, a). From the decomposition Δ_{2}^{(i,s)}, the subgroup J_{s}^{i} inherits a free decomposition and an abelian decomposition of each of the factors. We add elements for each edge connecting two non-QH, nonabelian vertex groups in the abelian decompositions of the different factors as we did in part (ii), and conjugating elements corresponding to boundary components of QH vertex groups in one of the abelian decompositions of the different factors using the corresponding decomposition Θ_{2}^{(i,s)} associated with Δ_{2}^{(i,s)} as we did in part (iii). With each factor of J_{s}^{i} we associate finitely many subgroups M_{(i,s,j)} corresponding to the different connected components in the decomposition Θ_{2}^{(i,s)} exactly in the same way we associated the subgroups J_{s}^{i} with the factors Ṽ_{i} in part (iv). We continue to the third level of the completed resolution Comp(Res)(t, v, a) with the subgroups M_{(i,s,j)} associated with each of the connected components of the various graphs Θ_{2}^{(i,s)} separately, analyze the decompositions inherited by each of the subgroups M_{(i,s,j)} from the Λ_{3} associated with the third level of the completed resolution Comp(Res)(t, v, a), add additional elements according to parts (ii) and (iii), and subsequently continue to the next levels of the completed resolution Comp(Res)(t, v, a).

The first step in the iterative procedure used in the first part of the construction of the core resolution constructs a resolution Res(u, v, a) of the subgroup < v, a > < Rlim(t, v, a) by going through the levels of the completed resolution Comp(Res)(t, v, a) and applying steps (i)-(v) above. To start the second step in the iterative procedure used for the first part of the construction of the core resolution, we set IRes_{1}(u, v, a) to be the subgroup of the completed resolution Comp(Res)(t, v, a) generated by the different factors Ṽ_{i} and their images in the lower levels of the completed resolution Comp(Res)(t, v, a) obtained by steps (i)-(v) above. Defining the subgroup IRes_{1}(u, v, a) < Rlim(t, v, a), we start going through the levels of the completed resolution Comp(Res)(t, v, a), starting with the subgroup IRes_{1}(u, v, a) instead of the subgroup < v, a > we have started with in the first step. IRes_{1}(u, v, a) being a subgroup of Rlim(t, v, a) inherits a decomposition from the abelian decomposition Λ_{1} associated with the first level of the completed resolution Comp(Res)(t, v, a). Let this decomposition be IΔ_{1}. Since the subgroups < v, a > and IRes_{1}(u, v, a) differs only in the stabilizers of the unique non-abelian, non-QH vertex group in each connected component of the various decompositions Θ̃_{1}^{i} associated with the decompositions Δ̃_{1}^{i} of the factors Ṽ_{i}, if the graph of groups IΔ_{1} is not combinatorially similar to the graph of groups Δ̃_{1} then one of the following occurs:

(1) In the free decomposition IΔ_{1} inherited by the subgroup IRes_{1}(u, v, a) from the graph of groups Λ_{1} either the number of factors is dropping, or the rank of the free factor corresponding to Bass-Serre generators of loops with trivial edge stabilizers and free factors contributed by infinite index subgroups of QH vertex groups in Θ_{1} is dropping. In this case each of the factors V_{i} of < v, a > and the subgroups Ṽ_{i} associated with it are subgroups of a factor in the free decomposition of IRes_{1}(u, v, a) inherited from Λ_{1}.

(2) The number of factors and the rank of the free group corresponding to Bass-Serre generators of loops with trivial edge stabilizers and free factors contributed by infinite index subgroups of QH vertex groups in the free decomposition inherited by IRes_{1}(u, v, a) from Λ_{1} stays identical to their values in the free decomposition inherited by the subgroup < v, a > from the decomposition Δ̃_{1}. In this case each of the factors V_{i} of < v, a > is a subgroup of a unique factor in the free decomposition of IΔ_{1} inherited by IRes_{1}(u, v, a) from Λ_{1}. The combinatorics of the graph of groups IΔ_{1} is strictly smaller than the combinatorics of the graphs of groups Δ̃_{1}^{i} in correspondence. i.e., for at least one of the factors in the decomposition IΔ_{1}, the combinatorics of its corresponding graph of groups is strictly smaller than the combinatorics of the corresponding graph of groups in Δ̃_{1}^{i}. i.e., either the number of edges and vertices is smaller or the genus or the (absolute value of the) Euler characteristic of some of the QH vertex groups is smaller.

Applying steps (i)-(v) to the subgroup IRes_{1}(u, v, a), we set IRes_{2}(u, v, a) to be the subgroup of the completed limit group Rlim(t, v, a) generated by the different factors of the subgroup IRes_{1}(u, v, a) and their images in the lower levels of the completed resolution Comp(Res)(t, v, a) obtained in the second step of the iterative procedure, i.e. obtained by performing steps (i)-(v) above starting with the subgroup IRes_{1}(u, v, a) < Rlim(t, v, a). Defining the subgroup IRes_{2}(u, v, a) < Rlim(t, v, a), we say that the procedure for the construction of the induced resolution terminates if IRes_{2}(u, v, a) = IRes_{1}(u, v, a). Otherwise, we perform the third step of the iterative procedure for constructing the induced resolution by going through the levels of the completed resolution Comp(Res)(t, v, a) starting with the subgroup IRes_{2}(u, v, a) instead of the subgroups < v, a > and IRes_{1}(u, v, a) as we did in the first two steps. In the general step, we say that the iterative procedure used in the first part of the construction of the core resolution terminates if IRes_{n}(u, v, a) = IRes_{n−1}(u, v, a). Otherwise, we continue to the next step by applying steps (i)-(v) starting with the subgroup IRes_{n}(u, v, a) < Rlim(t, v, a).

Lemma 4.2.

(i) The various ranks of the resolutions constructed along the iterative procedure used in the first part of the construction of the core resolution is a nonincreasing sequence.

(ii) The iterative procedure used in the first part of the construction of the core resolution terminates after finitely many steps.

Proof: Part (ii) is identical to lemma 3.3 in [Se4]. To prove part (i), note that by construction, at each step j of the iterative procedure, the terminal free group of the resolution constructed in the j-th step of the procedure, IRes_{j}(u, v, a), is an image of the terminal free group of the resolution constructed in step j − 1 of the procedure, IRes_{j−1}(u, v, a).

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A core resolution is a geometric subresolution, hence, the output of the first part of the procedure needs to be a complete resolution.

Lemma 4.3. Let IRes_{f}(u, v, a) be the terminal resolution obtained by the above procedure. Then IRes_{f}(u, v, a) is a geometric subresolution of the completed resolution Comp(Res)(t, v, a).

Proof: Identical to lemma 3.4 of [Se4].

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The first part of the construction of the core resolution is essentially identical to the construction of the induced resolution. The resolution obtained by the iterative procedure used in this part, IRes_{f}(u, v, a), is a geometric subresolution of the complete well-separated resolution Res(t, v, a) we have started with, hence, if IRes_{f}(u, v, a) is a resolution of minimal rank it is a core resolution. However, in general it is not a firm subresolution, hence, can not serve as a core resolution. To get a firm subresolution, we usually need to modify the resolution IRes_{f}(u, v, a) by sequentially reducing its rank. We do that by sequentially ”fill-in” various abelian and QH vertex groups associated with the resolution Res(t, v, a) and related to the resolution IRes_{f}(u, v, a), in a way that does not increase the ranks of the obtained resolutions, and so that if the entire sequence of ”fill-in” operations does not manage to eventually reduce the rank of the obtained resolution, then it is guaranteed that the resolution IRes_{f}(u, v, a), we have started the second step with, is a firm subresolution, so it can be taken as a core resolution.

Before presenting the second part of the construction of a core resolution, we present two preliminary iterative procedures. The first preliminary procedure fills in inefficient QH vertex groups, and the second preliminary procedure adds pegs to the abelian vertex groups in the various abelian decompositions associated with the various levels of a given induced resolution. As we will see in the sequel, the iterative procedure we use for the second part of the construction of a core resolution, takes care of these two operations, so the preliminary procedures are not really necessary for the construction of the core, but they may be of importance for other purposes.

Definition 4.4. Let IRes_{f}(u, v, a) be the terminal resolution obtained by the iterative procedure used in the first part of the construction of the core resolution. Let Q be a QH vertex group in the abelian decomposition associated with one of the levels of the resolution IRes_{f}(u, v, a). By construction, Q is a finite index subgroup of a QH vertex group Q̂ in the complete resolution Res(t, v, a) we have started with. With the QH vertex group Q there is an associated surface S, and with the QH vertex group Q̂ there is an associated surface Ŝ which is finitely covered by S. The resolutions IRes_{f}(u, v, a) and Res(t, v, a) are well-separated, so with S and Ŝ there is an associated collection of non-homotopic disjoint s.c.c. that are mapped to the identity in the next levels of the corresponding resolutions. We say that the QH vertex group Q (or the corresponding surface S) of IRes_{f}(u, v, a) is inefficient if it is not of minimal rank and one of the following conditions hold:

(i) the rank of the free group dropped from the QH vertex group Q is strictly bigger than the rank of the QH vertex group dropped from the QH vertex group Q̂.

(ii) the number of connected components in the surface S after cutting it along its associated collection of s.c.c. is strictly bigger than the number of connected components of the surface Ŝ after cutting it along its associated collection of s.c.c.

We start the second part of the construction of the core resolution by sequentially ”fill” inefficient surfaces (QH vertex groups). By the way inefficient surfaces are defined, each time we fill an inefficient surface, the rank of the associated (geometric) resolution strictly decreases. We start the second part of the construction of the core resolution with the resolution obtained by the first part of the construction, IRes_{f}(u, v, a). If the resolution IRes_{f}(u, v, a) contains no inefficient surfaces (QH vertex groups) we do not change the resolution IRes_{f}(u, v, a) and continue with it to the next iterative procedure used in the second part of the construction of the core resolution. Suppose Q is an inefficient QH vertex group in one of the levels of the resolution IRes_{f}(u, v, a), and let Q̂ be the QH vertex group that contain Q in the resolution Res(t, v, a) we have started the procedure with. With the (complete) resolution IRes_{f}(u, v, a) we naturally associate the subgroup < u, v, a > which is generated by the limit groups associated with the various levels of IRes_{f}(u, v, a). By the construction of the resolution IRes_{f}(u, v, a), the subgroup < u, v, a > is naturally a subgroup of the limit group Rlim(t, v, a), associated with the (complete) resolution Res(t, v, a).

Let < q, u, v, a > be the subgroup generated by the subgroups < u, v, a > and Q̂ in the limit group Rlim(t, v, a). At this point we apply the procedure used in the first part of the construction to the limit group < q, u, v, a > and the resolution Res(t, v, a) we have started with. Let the obtained (geometric) resolution be F iRes(s, v, a).

Proposition 4.5. The rank of the resolution F iRes(s, v, a), obtained by filling the QH subgroup Q, is strictly smaller than the rank of the resolution IRes_{f}(u, v, a) we have started the second step with.

Proof: The terminal free group of the resolution IRes_{f}(u, v, a) is naturally mapped into the terminal free group of the resolution F iRes(s, v, a). Since the QH vertex group Q̂ is obtained from the QH vertex group Q by adding finitely many elements which are roots of elements in Q, the terminal free group of the resolution F iRes(s, v, a) is generated by the image of the terminal free group of the resolution IRes_{f}(u, v, a) in addition to finitely many roots of elements in that image. Hence, the rank of the resolution F iRes(s, v, a) is bounded by the rank of the resolution IRes_{f}(u, v, a).

The QH vertex group Q is inefficient. If the rank of the free factor dropped from it in the resolution IRes_{f}(u, v, a) is strictly smaller than the rank of the free factor dropped from the QH vertex group Q̂ in the resolution Res(t, v, a), then by the above structure of the terminal free group of F iRes(s, v, a), the rank of that terminal free group is strictly smaller than the rank of the terminal free group of IRes_{f}(u, v, a).

Let S be the surface associated with the QH vertex group Q, and Ŝ be the surface associated with the QH vertex group Q̂. S finitely covers Ŝ, and with each of them there is an associated collection of s.c.c. that are mapped to the identity in the next levels of the resolutions IRes_{f}(u, v, a) and Res(t, v, a) in correspondence. Suppose the ranks of the free factors dropped from Q and Q̂ are identical. In that case the number of connected components of the surface S after we cut it along its associated collection of s.c.c. is strictly bigger than the number of components of Ŝ after we cut it along its associated collection of s.c.c. Let C_{1}, …, C_{ℓ} be the subgroups associated with the various components of the surface S. Let q_{1}, …, q_{d} ∈ Q̂ be elements that conjugate distinct couples of components of the surface S in the QH subgroup Q̂. For each i, let C_{i} and C_{j(i)} be the subgroups of < u, v, a > associated with the two components conjugated by the element q_{i} ∈ Q̂. Since the group C_{i} intersect q_{i}C_{j(i)}q_{i}^{−1} non-trivially, the rank of the image of the subgroup < C_{i}, q_{i}C_{j(i)}q_{i}^{−1} > in the terminal free group of the resolution F iRes(s, v, a) is strictly smaller than the sum of the image of the ranks of C_{i} and C_{j(i)} in the terminal free group of the resolution IRes_{f}(u, v, a) we have started the second step with.

Let ν be the map from the subgroup H =< C_{1}, …, C_{ℓ}, Q̂ > into the terminal free group of the resolution F iRes(s, v, a), and suppose that the ranks of the terminal free groups of the resolutions IRes_{f}(u, v, a) and F iRes(s, v, a) are identical. Then the rank of the free group ν(H) is equal to the rank of the image of the subgroup < C_{1}, …, C_{ℓ} > in the terminal free group of the resolution IRes_{f}(u, v, a), which is the sum of the ranks of the images of the various subgroups C_{i} in that terminal free group. Since, in addition, the rank of the subgroup ν(< C_{i}, q_{i}C_{j(i)}q_{i}^{−1} >) is strictly smaller than the sum of the ranks of the images of C_{i} and C_{j(i)} in the terminal free group of IRes_{f}(u, v, a), ν(H) admits a free decomposition: ν(H) = ν(q_{1}) ∗ … ∗ ν(q_{d}) ∗ ν(H_{1}), and since for some integer n, q_{1}^{n} ∈ Q, ν(q_{1})^{n} ∈ ν(< q_{2}, …, q_{d}, H_{1} >, q_{1}^{−1} < q_{2}, …, q_{d}, H_{1} > q_{1}). But clearly, if G =< a > ∗B, and the element a is of infinite order, then < B, a^{−1}Ba >= B ∗ a^{−1}Ba, hence, for every n > 1, a^{n} ∉< B, a^{−1}Ba >, a contradiction.

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If the resolution F iRes(s, v, a) contains no inefficient QH vertex groups we replace the resolution IRes_{f}(u, v, a) by the resolution F iRes(u, v, a) and proceed to the next iterative procedure used in the second step of the construction of the core resolution. If the resolution F iRes(s, v, a) contains an inefficient QH vertex group Q, we ”fill” this incomplete surface precisely as we filled an incomplete surface in the resolution IRes_{f}(u, v, a) obtained by the first part of the construction. We continue iteratively by filling inefficient QH vertex groups in the previously obtained resolutions. Since, by proposition 4.5, each filling of an inefficient QH vertex group strictly decreases the rank of the obtained resolution, this (filling) iterative procedure terminates after finitely many steps, which enable us to replace the resolution IRes_{f}(u, v, a) by the obtained resolution, and continue to the next iterative procedure used in the second step of the construction of the core resolution.

Let IRes_{e}(u, v, a) be the resolution obtained by iteratively filling inefficient QH vertex groups. The resolution IRes_{e}(u, v, a) is a geometric subresolution of Res(t, v, a) by construction, but in general it may not be a firm subresolution. The second preliminary procedure adds pegs to abelian vertex groups in the abelian decompositions associated with the various levels of the resolution IRes_{e}(u, v, a). Suppose that the resolution IRes_{e}(u, v, a), obtained by filling inefficient QH vertex groups in the resolution IRes_{f}(u, v, a), constructed in the first part of the construction of the core resolution, is not of minimal rank. In the second preliminary procedure we add pegs of abelian subgroups intersected non-trivially by the subgroup associated with the resolution IRes_{e}(u, v, a) iteratively.

(1) We start by adding the pegs of abelian subgroups associated with the top level of the resolution Res(t, v, a) that are intersected non-trivially by the subgroup associated with the resolution IRes_{e}(u, v, a).

(2) We construct the resolution induced from the resolution Res(t, v, a) by the subgroup generated by the subgroup associated by the resolution IRes_{e}(u, v, a) and the additional pegs, and denote the obtained resolution IRes_{1}(u, v, a). Since the pegs we added are either roots of elements that belong to the subgroup associated with IRes_{e}(u, v, a), or they commute with (non-trivial) elements in the subgroup associated with IRes_{e}(u, v, a), the rank of the obtained resolution IRes_{1}(u, v, a) is bounded by the rank of the resolution IRes_{e}(u, v, a):

rk(IRes_{1}(u, v, a)) ≤ rk(IRes_{e}(u, v, a)).

(3) If rk(IRes_{1}(u, v, a)) < rk(IRes_{e}(u, v, a)) we start the second step of the procedure with the resolution IRes_{1}(u, v, a), so we may suppose that rk(IRes_{1}(u, v, a)) = rk(IRes_{e}(u, v, a)). In this case, if a peg that was added to the subgroup associated with the resolution IRes_{e}(u, v, a) is not a root of an element of this subgroup, then the additional peg has to be mapped into a free generator in the terminal free subgroup of the obtained (induced) resolution IRes_{1}(u, v, a), and the subgroup generated by the collection of those additional pegs is mapped into a free factor (of the same rank) in the terminal free subgroup of IRes_{1}(u, v, a).

Let Λ_{1} be the abelian decomposition associated with the top level of the resolution Res(t, v, a). In the free decomposition IΔ_{1} inherited by the subgroup IRes_{1}(u, v, a) from Λ_{1} either the number of factors is dropping, or the rank of the free factor corresponding to Bass-Serre generators of loops with trivial edge stabilizers and free factors contributed by infinite index subgroups of QH vertex groups is dropping, or the number of free factors and the rank of the free factor corresponding to Bass-Serre generators with trivial edge stabilizers and free factors contributed by infinite index subgroups of QH vertex groups are identical to the corresponding number and rank in the abelian decomposition inherited by the subgroup (associated with) IRes_{e}(u, v, a) from Λ_{1}.

In case the number of free factors and the rank of the above free factor in IΔ_{1} are identical to the corresponding number of factors and rank in the abelian decomposition inherited by the subgroup IRes_{e}(u, v, a) from the abelian decomposition Λ_{1}, the complexity of the graph of groups IΔ_{1}, inherited by the subgroup IRes_{1}(u, v, a) from Λ_{1}, is bounded by complexity of the graph of groups inherited by IRes_{e}(u, v, a) from Λ_{1} in addition to the number of abelian groups for which we have added a peg and this peg is not a root of an element in the subgroup associated with IRes_{e}(u, v, a). Note that the number of those abelian groups is bounded by the rank of the resolution IRes_{e}(u, v, a), rk(IRes_{e}(u, v, a)).

(4) We continue as in steps (1) and (2), by adding all the pegs of abelian subgroups associated with the top level of the resolution Res(t, v, a) that are intersected non-trivially by the subgroup associated with the resolution IRes_{1}(u, v, a), and constructing the resolution induced from the resolution Res(t, v, a) by the subgroup generated by the subgroup associated by the resolution IRes_{1}(u, v, a) and the additional pegs, and denote the obtained resolution IRes_{2}(u, v, a), and continue iteratively. By construction, the ranks of all the obtained resolutions, which we denote IRes_{n}(u, v, a), satisfy:

rk(IRes_{n}(u, v, a)) ≤ rk(IRes_{n−1}(u, v, a)) ≤ rk(IRes_{e}(u, v, a)).

If at some step n, rk(IRes_{n}(u, v, a)) < rk(IRes_{n−1}(u, v, a)) we start the second step of the procedure with the resolution IRes_{n}(u, v, a), hence, we may assume that the ranks of the obtained resolutions IRes_{n}(u, v, a) does not decrease along the iterative procedure. The subgroup generated by the pegs that were added to the subgroups associated with the resolutions IRes_{e}(u, v, a), IRes_{1}(u, v, a), …, IRes_{n−1}(u, v, a), that are not a root of an element in the subgroup associated with the preceding resolution, are mapped isomorphically onto a free factor (of the same rank) in the terminal free subgroup of the obtained (induced) resolution IRes_{n}(u, v, a). In the free decomposition inherited by the subgroup IRes_{n}(u, v, a) from Λ_{1}, the abelian decomposition associated with the top level of Res(t, v, a), either the number of factors is dropping, or the rank of the free factor corresponding to Bass-Serre generators of loops with trivial edge stabilizers and free factors contributed by infinite index subgroups of QH vertex groups is dropping, or the number of free factors and the rank of the free factor corresponding to Bass-Serre generators with trivial edge stabilizers and free factors contributed by infinite index subgroups of QH vertex groups are identical to the corresponding number and rank in the abelian decomposition inherited by the subgroup IRes_{n−1}(u, v, a) from Λ_{1}. In case the number of free factors and the rank of the above free factor in the abelian decomposition inherited by IRes_{n}(u, v, a) from Λ_{1} are identical to the corresponding number of factors and rank in the abelian decomposition inherited by the subgroup IRes_{n−1}(u, v, a) from the abelian decomposition Λ_{1}, the complexity of the graph of groups inherited by the subgroup IRes_{n}(u, v, a) from Λ_{1}, is bounded by complexity of the graph of groups inherited by IRes_{n−1}(u, v, a) from Λ_{1} in addition to the number of abelian groups for which we have added a peg and this peg is not a root of an element in the subgroup associated with IRes_{n−1}(u, v, a). Since the total number of those abelian groups that are added to the entire sequence of resolutions IRes_{e}(u, v, a), IRes_{1}(u, v, a), …, IRes_{n}(u, v, a) is bounded by the rank of the resolution IRes_{e}(u, v, a), the sequence terminates after a finite time., i.e., we obtain a resolution IRes_{n}(u, v, a) for which if the subgroup associated with it intersects a (conjugate of an) abelian vertex group in the graded abelian decomposition associated with the top level of Res(t, v, a) then this intersection contains the peg of the abelian subgroup.

(5) At this point we continue by adding pegs to abelian subgroups associated with the second level of the resolution Res(t, v, a) that are intersected nontrivially by the subgroup associated with the resolution IRes_{n}(u, v, a), and continue iteratively as we did in part (4). Note that after adding pegs of abelian subgroups associated with the second level of Res(t, v, a), we may need to add pegs of abelian subgroups associated with the top level of Res(t, v, a) as well. Since eventually, either the rank of an obtained resolution decreases or the complexity of the abelian decompositions inherited by the obtained resolutions from the abelian decomposition associated with either the top level or the second level of the resolution Res(t, v, a) decreases, the iterative procedure for adding pegs to abelian subgroups associated with the top two levels of the resolution Res(t, v, a) terminates after finitely many steps. Continuing to the next levels of the resolution Res(t, v, a), we finally obtain a resolution, which we denote IRes_{p}(t, v, a), for which if a conjugate of an abelian vertex group associated with one of the levels of Res(t, v, a) intersects non-trivially the subgroup associated with the obtained resolution IRes_{p}(u, v, a), then this intersection contains the peg of the abelian subgroup.

The iterative procedure for the addition of pegs terminates after a finite number of steps with a resolution, IRes_{p}(u, v, a). To the obtained resolution IRes_{p}(u, v, a) we apply the procedure for filling inefficient QH vertex groups, and if an inefficient vertex group was indeed filled, we apply the procedure for adding pegs once again, until we obtain a resolution, which we still denote IRes_{p}(u, v, a), that does not contain any inefficient QH vertex groups, and for which if a conjugate of an abelian vertex group associated with one of the levels of Res(t, v, a) intersects nontrivially the subgroup associated with the obtained resolution IRes_{p}(u, v, a), then this intersection contains the peg of the abelian subgroup. However, the obtained resolution IRes_{p}(u, v, a), which is a geometric subresolution by construction, is not necessarily a firm subresolution. To modify the resolution IRes_{p}(u, v, a) in order to obtain a firm subresolution, we use an iterative procedure, that is aimed towards sequentially reducing the rank of the obtained resolutions. The iterative procedure we present either reduces the rank after finitely many steps, or guarantees (after finitely many steps) that the resolution IRes_{p}(u, v, a) is indeed a firm subresolution. Unlike the procedure for the addition of pegs, the procedure used in the second part of the second step of the construction of the core resolution, starts from the bottom level of the resolution Res(t, v, a) and iteratively climbs towards its top level.

(1) The terminal level (which we denote by ℓ) of the resolution Res(t, v, a) is a free group F_{k} ∗ F, hence, we start the procedure from the level above the terminal one (level ℓ − 1). According to the construction of an induced resolution, the subgroup associated with the resolution IRes_{p}(u, v, a) inherits a (possibly trivial) free decomposition from each of the levels of the resolution IRes_{p}(u, v, a), and it is mapped to a subgroup associated with each of the levels. In particular, in accordance with the free decomposition inherited by the subgroup associated with the resolution IRes_{p}(u, v, a) from the levels that lie above the ℓ−1 level, the image of the subgroup associated with IRes_{p}(u, v, a), G^{ℓ−1}, admits a free decomposition when mapped into the ℓ − 1 level, G^{ℓ−1} = H_{1}^{ℓ−1} ∗ … ∗ H_{s(ℓ−1)}^{ℓ−1}. We treat the factors H_{i}^{ℓ−1} in parallel.

(2) We fix a system of generators of H_{i}^{ℓ−1}, H_{i}^{ℓ−1} =< h_{1}^{ℓ−1}, …, h_{r(ℓ−1,i)}^{ℓ−1} >. If no (non-trivial) subgroup of the factor H_{i}^{ℓ−1} fixes a vertex in the Bass-Serre tree associated with the ℓ − 1 level of the resolution Res(t, v, a), we have started with, we leave the factor H_{i}^{ℓ−1} unchanged. Suppose that a (non-trivial) subgroup of H_{i}^{ℓ−1} fixes a vertex in the abelian decomposition associated with the ℓ − 1 level of the resolution Res(t, v, a).

Let T_{ℓ−1} be the Bass-Serre tree corresponding to the abelian decomposition associated with the ℓ − 1 level of the ambient resolution Res(t, v, a), and let Λ_{H_{i}^{ℓ−1}} be the graph of groups inherited by H_{i}^{ℓ−1} from its action on the Bass-Serre tree T_{ℓ−1}. Suppose that the abelian decomposition Λ_{H_{i}^{ℓ−1}},

inherited by H_{i}^{ℓ−1} from its action on the Bass-Serre tree T_{ℓ−1}, contains a couple of QH vertex groups, Q_{1}, Q_{2}, that satisfy the following conditions:

(i) Q_{1} is a finite index subgroup in a subgroup Q′_{1} that is conjugate to a QH vertex group in the abelian decomposition associated with the ℓ − 1 level of the ambient resolution Res(t, v, a), and Q_{2} is a finite index subgroup in a subgroup Q′_{2} that is conjugate to a QH vertex group Q′_{2} in the abelian decomposition associated with the ℓ − 1 level of the ambient resolution Res(t, v, a), and Q′_{1} is conjugate to Q′_{2} in the limit group associated with the ℓ − 1 level of the ambient resolution Res(t, v, a).

(ii) the QH vertex group Q′_{1} (hence, also Q′_{2}) is not of minimal rank, i.e., there exists a s.c.c. on S′_{1} (the surface associated with Q′_{1}) that is mapped to the trivial element in the next level of the resolution Res(t, v, a). In this case we say that the abelian decomposition Λ_{H_{i}^{ℓ−1}} contains a reducing QH couple.

If the abelian decomposition Λ_{H_{i}^{ℓ−1}} contains a reducing QH couple, Q_{1}

and Q_{2}, we set the subgroup Ĥ_{i} to be the subgroup generated by H_{i}^{ℓ−1}, the QH vertex groups Q′_{1} and Q′_{2} that contain Q_{1} and Q_{2} as subgroups of finite index, and an element in the limit group associated with the ℓ − 1 level of the ambient resolution Res(t, v, a) that conjugates Q′_{1} to Q′_{2}. We set H′_{i} to be the limit group associated with the resolution induced by the subgroup Ĥ_{i} from the ambient resolution Res(t, v, a). Since the QH vertex groups Q′_{1} and Q′_{2} are not of minimal rank, and since Q_{1} and Q_{2} are not conjugate in the subgroup H_{i}^{ℓ−1} we have started with, and Q′_{1} and Q′_{2} are conjugate in H′_{i}:

rk(H′_{i}) < rk(H_{i}^{ℓ−1})

in case there exists a reducing QH couple in Λ_{H_{i}^{ℓ−1}}. In this case we replace

the factor H_{i}^{ℓ−1} by H′_{i}, and repeat part (2) of the construction with the newly obtained subgroup H_{i}^{ℓ−1}.

Suppose that the abelian decomposition Λ_{H_{i}^{ℓ−1}} does not contain a reducing QH couple. Let η_{ℓ−1} be the map from the limit group associated with the ℓ − 1 level of Res(t, v, a) to the (free) limit group associated with the (terminal) ℓ level of Res(t, v, a). Let t_{0} ∈ T_{ℓ−1} be its base point, and let T′_{ℓ−1} be the finite subtree of T_{ℓ−1} spanned by the points t_{0}, h_{1}^{ℓ−1}(t_{0}), …, h_{r(ℓ−1,i)}^{ℓ−1}(t_{0}). To continue our treatment of the factor H_{i}^{ℓ−1} we need the notions of floating and absorbed surfaces.

Definition 4.6. Let Q be a QH vertex group in the finite tree T′_{ℓ−1}, and let S be its associated surface. Since the ambient resolution Res(t, v, a) is well-separated, the image of the QH vertex group Q in the next level of the resolution Res(t, v, a) is non-abelian. We say that Q is a floating QH vertex group (S is a floating surface) with respect to the geometric subresolution IRes(u, v, a), if Q does not intersect the subgroup H_{i}^{ℓ−1} in a subgroup of finite index, and one of the three conditions hold:

(i) the QH vertex group Q is not of minimal rank, i.e., there exists a s.c.c. on the surface Ŝ that is mapped to the trivial element in the limit group associated with the next level of the ambient resolution Res(t, v, a).

(ii) none of the vertex groups that are adjacent to Q in T′_{ℓ−1} intersect non-

trivially the subgroup H_{i}^{ℓ−1}. (iii) the QH vertex group Q is of minimal rank, and for every vertex group V

in T′_{ℓ−1} that is adjacent to Q in T′_{ℓ−1} and intersects H_{i}^{ℓ−1} non-trivially:

rk(η_{ℓ−1}(< H_{i}^{ℓ−1} ∩ V, Q >)) > rk(η_{ℓ−1}(H_{i}^{ℓ−1} ∩ V)).

We say that Q is an absorbed vertex group (S is an absorbed surface) if it is not floating.

In the case in which there is no reducing QH couple, we set Ĥ_{i} to be the subgroup generated by the factor H_{i}^{ℓ−1} and one of the following if it exists:

(i) an abelian vertex group in the finite tree T′_{ℓ−1} that is intersected nontrivially by the factor H_{i}^{ℓ−1}, but is not contained in it.

(ii) an absorbed QH vertex group in the finite tree T′_{ℓ−1} (with respect to the

subgroup H_{i}^{ℓ−1}), that is not contained in H_{i}^{ℓ−1}. (iii) an (abelian) edge group E in the finite tree T′_{ℓ−1}, that is not contained in

H_{i}^{ℓ−1}, and is adjacent to a non-abelian, non-QH vertex group V in T′_{ℓ−1} that is intersected non-trivially by H_{i}^{ℓ−1}, for which:

rk(η_{ℓ−1}(< H_{i}^{ℓ−1} ∩ V, E >)) ≤ rk(η_{ℓ−1}(H_{i}^{ℓ−1} ∩ V)).

(iv) an element v_{0} ∈ V, v_{0} ∉ H_{i}^{ℓ−1}, where V is a non-abelian, non-QH vertex group in the finite tree T′_{ℓ−1} that is intersected non-trivially by the subgroup H_{i}^{ℓ−1}, for which there exist two edge groups E_{1}, E_{2} that are adjacent to V in the finite tree T′_{ℓ−1} and are not conjugate in the subgroup H_{i}^{ℓ−1}, so that v conjugates E_{1} to E_{2} in the limit group associated with the ℓ − 1 level of the ambient resolution Res(t, v, a), and:

rk(η_{ℓ−1}(< H_{i}^{ℓ−1} ∩ V, v_{0} >)) ≤ rk(η_{ℓ−1}(H_{i}^{ℓ−1} ∩ V)).

We further set H′_{i} to be the subgroup associated with the resolution induced by the subgroup Ĥ_{i}. Since Ĥ_{i} is generated by H_{i}^{ℓ−1}, the absorbed QH vertex groups in T′_{ℓ−1} with respect to H_{i}^{ℓ−1}, abelian vertex groups that are intersected non-trivially by H_{i}^{ℓ−1}, and additional elements that do not increase the ranks of the corresponding vertex groups, the rank of H′_{i} is bounded by the rank of H_{i}^{ℓ−1}, rk(H′_{i}) ≤ rk(H_{i}^{ℓ−1}). Furthermore, both groups H′_{i} and H_{i}^{ℓ−1} inherit free decompositions from the abelian decompositions associated with their corresponding actions on the Bass-Serre tree T_{ℓ−1}, and both the number of factors as well as the rank of the additional free group in the abelian decomposition associated with H′_{i} (the Kurosh rank) are bounded by the number of factors and the corresponding rank in the abelian decomposition associated with H_{i}^{ℓ−1}. If H′_{i} is identical to H_{i}^{ℓ−1} we have concluded our treatment of the factor H_{i}^{ℓ−1}. Otherwise, we replace the factor H_{i}^{ℓ−1} by H′_{i}, and repeat the construction of a subgroup Ĥ_{i} associated with the newly obtained subgroup H_{i}^{ℓ−1} (without changing the finite tree T′_{ℓ−1}), and the construction of the subgroup H′_{i} associated with the resolution induced by Ĥ_{i} from the ambient resolution Res(t, v, a), and repeat our treatment of the subgroup H′_{i}. Since in each step we either reduce the rank of the obtained subgroup, or we add a new edge group, an abelian vertex group or absorbed QH vertex group from the finite tree T′_{ℓ−1} to the subgroup associated with H_{i}^{ℓ−1}, or we add an element that conjugates two edge groups in T′_{ℓ−1} that were not conjugated previously, we conclude our treatment of the factor H_{i}^{ℓ−1} after finitely many steps. If the ranks of at least one of the factors H_{i}^{ℓ−1} strictly decreased by the iterative procedure, we replace the resolution IRes_{p}(u, v, a) with the resolution induced by the subgroup generated by the subgroup associated with IRes_{p}(u, v, a) and the newly obtained factors H_{1}^{ℓ−1}, …, H_{s(ℓ−1)}^{ℓ−1}, and denote the obtained resolution IRes_{ℓ−1}(u, v, a). In this case, rk(IRes_{ℓ−1}(u, v, a)) < rk(IRes_{p}(u, v, a)), and we continue by starting the second step of the construction of the core resolution with the resolution IRes_{ℓ−1}(u, v, a) (instead of IRes_{f}(u, v, a)). If none of the ranks of the various factors H_{i}^{ℓ−1} decreases, we continue by analyzing the next (ℓ − 2) level of the resolution IRes_{p}(u, v, a).

(3) The iterative procedure presented in part (2) concludes our treatment of the various factors H_{i}^{ℓ−1} of the subgroup G^{ℓ−1} = H_{1}^{ℓ−1}∗…∗H_{s(ℓ−1)}^{ℓ−1}, which is the image of the subgroup associated with the resolution IRes_{p}(u, v, a) in the subgroup associated with the ℓ−1 level of the ambient resolution Res(t, v, a). We continue by iteratively increasing the index m, and analyzing the various factors of the image of the subgroup G^{ℓ−m} = H_{1}^{ℓ−m} ∗ … ∗ H_{s(ℓ−m)}^{ℓ−m}, which is the image of the subgroup associated with the resolution IRes_{p}(u, v, a) in the subgroup associated with the ℓ − m level of the ambient resolution Res(t, v, a), according to the procedure presented in part (2).

Suppose that we analyzed all the images of the subgroup associated with the resolution IRes_{p}(u, v, a) in the subgroups associated with all levels below the ℓ−m level of the ambient resolution Res(t, v, a): G^{ℓ−1}, …, G^{ℓ−m+1}, and suppose that the rank of the obtained resolution did not decrease along these levels. In accordance with the free decomposition inherited by the subgroup associated with the resolution IRes_{p}(u, v, a) from the levels that lie above the ℓ − m level, the image of the subgroup associated with IRes_{p}(u, v, a), G^{ℓ−m}, admits a free decomposition when mapped into the ℓ − m level, G^{ℓ−m} = H_{1}^{ℓ−m} ∗ … ∗ H_{s(ℓ−m)}^{ℓ−m}. We treat the factors H_{i}^{ℓ−m} according to the procedure presented in part (2). We start by fixing a system of generators of H_{i}^{ℓ−m}, H_{i}^{ℓ−m} =< h_{1}^{ℓ−m}, …, h_{r(ℓ−m,i)}^{ℓ−m} >. If no (non-trivial) subgroup of the factor H_{i}^{ℓ−m} fixes a vertex in the Bass-Serre tree associated with the ℓ − m level of the resolution Res(t, v, a), we have started with, we leave the factor H_{i}^{ℓ−m} unchanged. Suppose that a (non-trivial) subgroup of H_{i}^{ℓ−m} fixes a vertex in the abelian decomposition associated with the ℓ − m level of the resolution Res(t, v, a). Let T_{ℓ−m} be the Bass-Serre tree corresponding to the abelian decomposition associated with the ℓ − m level of the ambient resolution Res(t, v, a), and let Λ_{H_{i}^{ℓ−m}} be the graph of groups inherited by H_{i}^{ℓ−m} from its action on the Bass-Serre tree T_{ℓ−m}. If the abelian decomposition Λ_{H_{i}^{ℓ−m}} contains a

reducing QH couple, Q_{1} and Q_{2}, we set the subgroup Ĥ_{i} to be the subgroup generated by H_{i}^{ℓ−m}, the QH vertex groups Q′_{1} and Q′_{2} that contain Q_{1} and Q_{2} as subgroups of finite index, and an element in the limit groups associated with the ℓ − m level of the ambient resolution Res(t, v, a) that conjugates Q′_{1} to Q′_{2}. We set H′_{i} to be the limit group associated with the resolution induced by the subgroup Ĥ_{i} from the ambient resolution Res(t, v, a). Since the QH vertex groups Q′_{1} and Q′_{2} are not of minimal rank, and since Q_{1} and Q_{2} are not conjugate in the subgroup H_{i}^{ℓ−m} we have started with, and Q′_{1} and Q′_{2} are conjugate in H′_{i}:

rk(H′_{i}) < rk(H_{i}^{ℓ−m})

in case there exists a reducing QH couple in Λ_{H_{i}^{ℓ−m}}. In this case we replace

the factor H_{i}^{ℓ−m} by H′_{i}, and start the second part of the construction of the core resolution, with the resolution induced from the ambient resolution Res(t, v, a), by the subgroup generated by the newly obtained subgroup H_{i}^{ℓ−m}, and the subgroup associated with the resolution IRes_{p}(u, v, a). Note that the rank of the obtained induced resolution is strictly smaller than the rank of the resolution IRes_{p}(u, v, a) we have started the second step with.

Suppose that the abelian decomposition Λ_{H_{i}^{ℓ−1}} does not contain a reducing QH couple. Let η_{ℓ−m} be the map from the limit group associated with the ℓ − m level of Res(t, v, a) to the (free) limit group associated with the (terminal) ℓ level of Res(t, v, a). Let t_{0} ∈ T_{ℓ−m} be its base point, and let T′_{ℓ−m} be the finite subtree of T_{ℓ−m} spanned by the points t_{0}, h_{1}^{ℓ−m}(t_{0}), …, h_{r(ℓ−m,i)}^{ℓ−m}(t_{0}). In this case, we set Ĥ_{i} to be the subgroup generated by the factor H_{i}^{ℓ−m} and one of the following if it exists:

(i) an abelian vertex group in the finite tree T′_{ℓ−m} that is intersected nontrivially by the factor H_{i}^{ℓ−m}, but is not contained in it.

(ii) an absorbed QH vertex group in the finite tree T′_{ℓ−m} (with respect to the

subgroup H_{i}^{ℓ−m}), that is not contained in H_{i}^{ℓ−m}. (iii) an (abelian) edge group E in the finite tree T′_{ℓ−m}, that is not contained in

H_{i}^{ℓ−m}, and is adjacent to a non-abelian, non-QH vertex group V in T′_{ℓ−m} that is intersected non-trivially by H_{i}^{ℓ−m}, for which:

rk(η_{ℓ−m}(< H_{i}^{ℓ−m} ∩ V, E >)) ≤ rk(η_{ℓ−m}(H_{i}^{ℓ−m} ∩ V)).

(iv) an element v_{0} ∈ V, v_{0} ∉ H_{i}^{ℓ−m}, where V is a non-abelian, non-QH vertex group in the finite tree T′_{ℓ−m} that is intersected non-trivially by the subgroup H_{i}^{ℓ−m}, for which there exist two edge groups E_{1}, E_{2} that are adjacent to V in the finite tree T′_{ℓ−m}, and are not conjugate in the subgroup H_{i}^{ℓ−m}, so that v conjugates E_{1} to E_{2} in the limit group associated with the ℓ − m level of the ambient resolution Res(t, v, a), and:

rk(η_{ℓ−m}(< H_{i}^{ℓ−m} ∩ V, v_{0} >)) ≤ rk(η_{ℓ−m}(H_{i}^{ℓ−m} ∩ V)).

We further set H′_{i} to be the subgroup associated with the resolution induced by the subgroup Ĥ_{i}. Since Ĥ_{i} is generated by H_{i}^{ℓ−m}, the absorbed QH vertex groups in T′_{ℓ−m} with respect to H_{i}^{ℓ−m}, abelian vertex groups that are intersected non-trivially by H_{i}^{ℓ−m}, and additional elements that do not increase the ranks of the corresponding vertex groups, the rank of H′_{i} is bounded by the rank of H_{i}^{ℓ−m}, rk(H′_{i}) ≤ rk(H_{i}^{ℓ−m}). Furthermore, both groups H′_{i} and H_{i}^{ℓ−m} inherit free decompositions from the abelian decompositions associated with their corresponding actions on the Bass-Serre tree T_{ℓ−m}, and both the number of factors as well as the rank of the additional free group in the abelian decomposition associated with H′_{i} (the Kurosh rank) are bounded by the number of factors and the corresponding rank in the abelian decomposition associated with H_{i}^{ℓ−m}. If H′_{i} is identical to H_{i}^{ℓ−m} we have concluded our treatment of the factor H_{i}^{ℓ−m}. Otherwise, we replace the factor H_{i}^{ℓ−m} by H′_{i}, and repeat the analysis of the resolution induced by the (newly obtained) subgroup H_{i}^{ℓ−m} from the ambient resolution Res(t, v, a), according to parts (2) and (3), without changing the finite tree T′_{ℓ−m}. Since in each step we either reduce the rank of the obtained subgroup, or we add a new edge group, an abelian vertex group or absorbed QH vertex group from the finite tree T′_{ℓ−m} to the subgroup associated with H_{i}^{ℓ−m}, or we add an element that conjugates two edge groups in T′_{ℓ−m} that were not conjugated previously, we conclude our treatment of the factor H_{i}^{ℓ−m} after finitely many steps. If the ranks of at least one of the factors H_{i}^{ℓ−m} strictly decreased by the iterative procedure, we replace the resolution IRes_{p}(u, v, a) with the resolution induced by the subgroup generated by the subgroup associated with IRes_{p}(u, v, a) and the newly obtained factors H_{1}^{ℓ−m}, …, H_{s(ℓ−m)}^{ℓ−m}, and denote the obtained resolution IRes_{ℓ−m}(u, v, a). In this case, rk(IRes_{ℓ−m}(u, v, a)) < rk(IRes_{p}(u, v, a)), and we continue by starting the second step of the construction of the core resolution with the resolution IRes_{ℓ−m}(u, v, a) (instead of IRes_{f}(u, v, a)). If none of the ranks of the various factors H_{i}^{ℓ−1} decreases, we continue by analyzing the next (ℓ − (m + 1)) level of the resolution IRes_{p}(u, v, a).

The iterative procedure used for the second part of the construction of the core resolution terminates after finitely many steps. Using it we obtain a geometric subresolution of the ambient resolution Res(t, v, a), which we denote IRes_{s}(u, v, a), which is set to be either the resolution IRes_{f}(u, v, a) obtained by the first part of the construction, in case the procedure used for the second part of the construction of the core resolution has not reduced the rank of the resolution it has constructed, or it is the resolution constructed by the procedure used in the second part of the construction of the core resolution, in case this resolution is of strictly smaller rank than the resolution IRes_{f}(u, v, a), constructed by the procedure used in the first part of the construction. The obtained resolution IRes_{s}(u, v, a) is a geometric subresolution of Res(t, v, a) by construction, in addition it is guaranteed to be a firm subresolution by the following theorem, hence, it may serve as a core resolution, Core(< v, a >, Res(t, v, a)).

Theorem 4.7. The resolution IRes_{s}(u, v, a), obtained by the procedure for the construction of a core resolution, is a firm subresolution of the resolution Res(t, v, a).

Proof: From the two parts of the construction of the core resolution, we obtain a geometric subresolution of the ambient resolution, Res(t, v, a), that we denote IRes_{sec}(u, v, a). We set the resolution IRes_{s}(u, v, a) to be the resolution IRes_{sec}(u, v, a) in case rk(IRes_{sec}) < rk(IRes_{f}), and IRes_{f}(u, v, a) in case rk(IRes_{sec}) = rk(IRes_{f}). Furthermore, if rk(IRes_{sec}) = rk(IRes_{f}) then for any specialization of the ambient resolution Res(t, v, a), the specialization of the terminal free group of the resolution IRes_{sec}(u, v, a) is obtained from the specialization of the terminal free group of the resolution IRes_{f}(u, v, a) by successively adding elements that do not increase the rank. Hence, if rk(IRes_{sec}) = rk(IRes_{f}), and IRes_{sec}(u, v, a) is a firm subresolution, so is the resolution IRes_{f}(u, v, a) which is set to be the resolution IRes_{s}(u, v, a) in this case. Therefore, to prove theorem 4.7, it is enough to prove that the resolution obtained by the procedure for the construction of a core resolution, IRes_{sec}(u, v, a) is a firm subresolution.

We prove that IRes_{sec}(u, v, a) is a firm subresolution according to the order of the second part of the construction of the resolution IRes_{s}(u, v, a). We start by proving it in case the ambient resolution, Res(t, v, a), is a one level resolution, and the associated abelian decomposition contains no QH vertex groups, we continue by proving it for ambient resolutions Res(t, v, a) for which the abelian decompositions associated with their various levels contain no QH vertex groups, and then we generalize the proof to arbitrary ambient resolutions.

Lemma 4.8. Suppose that the ambient resolution, Res(t, v, a), contains a single level, and the abelian decomposition associated with that level contains no QH vertex groups. Then the resolution IRes_{sec}(u, v, a), obtained by the procedure for the construction of a core resolution, is a firm subresolution of the resolution Res(t, v, a).

Proof: Let H^{1} be the subgroup associated with the resolution IRes_{sec}(u, v, a), and let T be the Bass-Serre tree corresponding to the abelian decomposition associated with the single level of the ambient resolution Res(t, v, a). In case the ambient resolution contains a single level, only part (1) and (2) of the procedure used for the second part of the construction of a core resolution are applied along the iterative procedure.

Suppose that no (non-trivial) subgroup of H^{1} fixes a vertex in the Bass-Serre tree T. Then there exists a test sequence of the ambient resolution Res(t, v, a), so that the corresponding sequence of specializations of the subgroup H^{1} are bi-Lipschitz equivalent to the action of H^{1} on the Bass-Serre tree T. Hence, the rank of each specialization of H^{1} is equivalent to the rank of H^{1}, that implies that IRes_{sec}(u, v, a) is a firm subresolution of Res(t, v, a).

Suppose that there are non-trivial subgroups of H^{1} that fix vertices in the Bass-Serre tree T. In this case H^{1} inherits a free decomposition from its action on the Bass-Serre tree T: H^{1} = B_{1} ∗ … ∗ B_{m} ∗ F_{s}, where each of the factors B_{j} inherits a (possibly trivial) abelian decomposition with non-trivial abelian edge groups from its action on the action on the tree T. Furthermore, the canonical map from the limit group associated with the top level of the ambient resolution, Res(t, v, a), to the limit group associated with its terminal (second) level, maps each of the factors B_{j} onto a free group D_{j}, which is a subgroup of the terminal free group of the ambient resolution, Res(t, v, a). Since IRes_{sec}(u, v, a) is a completed resolution, H^{1} naturally contains a subgroup isomorphic to M = D_{1} ∗ … ∗ D_{m} ∗ F_{s}. Then there exists a test sequence of the ambient resolution Res(t, v, a), so that the corresponding sequence of specializations of the subgroup M < H^{1} are isomorphic to M. Hence, the rank of each of the specializations of H^{1} is equivalent to the rank of the subgroup M, which is equivalent to the rank of IRes_{sec}(u, v, a), that implies that IRes_{sec}(u, v, a) is a firm subresolution of Res(t, v, a).

□

An argument similar to the one used to prove lemma 4.8 allows us to prove theorem 4.7 for ambient resolutions for which their associated abelian decompositions contain no QH vertex groups.

Lemma 4.9. Suppose that the abelian decompositions associated with the various levels of the ambient resolution, Res(t, v, a), contain no QH vertex groups. Then the resolution IRes_{sec}(u, v, a), obtained by the procedure for the construction of a core resolution, is a firm subresolution of the resolution Res(t, v, a).

Proof: Let H = H^{1} be the subgroup associated with the obtained resolution IRes_{sec}(u, v, a), and let T_{1} be the Bass-Serre tree corresponding to the abelian decomposition associated with the top level of the ambient resolution, Res(t, v, a). The subgroup H^{1} inherits an abelian decomposition from its action on the Bass-Serre tree T_{1}, an abelian decomposition that gives rise to a free decomposition: H^{1} = B_{1}^{1} ∗ … ∗ B_{m_{1}}^{1} ∗ F_{s_{1}}. Each of the factors B_{j}^{1} is mapped into the limit group associated with the second level of the ambient resolution Res(t, v, a), and we denote its image H_{j}^{2}. Setting T_{2} to be the Bass-Serre tree corresponding to the abelian decomposition associated with the second level of the ambient resolution Res(t, v, a), each of the subgroups H_{j}^{2} inherits an abelian decomposition from its action on the Bass-Serre tree T_{2}, an abelian decomposition that gives rise to a free decomposition of the subgroup H_{j}^{2}. Each of the factors B_{j}^{2} in the free decomposition of H_{j}^{2} is naturally mapped into the limit group associated with the third level of the ambient resolution Res(t, v, a), and we denote its image H_{j}^{3}. Continuing inductively, by going down through the levels of the ambient resolution Res(t, v, a), we obtain the set of subgroups H_{j}^{ℓ−m} of the limit group associated with the ℓ − m level of the ambient resolution Res(t, v, a), m = 0, …, ℓ − 1, j = 1, …, q_{ℓ−m}, and the factors in the free decompositions the subgroups H_{j}^{ℓ−m} inherit from their action on the Bass-Serre tree T_{ℓ−m}, that we denote B_{j}^{ℓ−m}.

We prove that the obtained resolution IRes_{sec}(u, v, a) is a firm subresolution, by going through the levels of the ambient resolution Res(t, v, a) from bottom to top, and showing inductively, that the resolutions induced by the subgroups H_{j}^{ℓ−m} associated with the corresponding level, are indeed firm subresolutions of the ambient resolution Res(t, v, a). The subgroups associated with the bottom level H_{j}^{ℓ} are subgroups of the terminal free group of the ambient resolution Res(t, v, a), and the resolutions induced by them from the ambient resolution Res(t, v, a), are trivial, so these are indeed firm subresolutions of Res(t, v, a). According to lemma 4.8, the resolutions induced by the various subgroups associated with the ℓ − 1 level, H_{j}^{ℓ−1}, from Res(t, v, a), are firm subresolutions of Res(t, v, a).

Suppose that the resolutions induced by the various subgroups H_{j}^{ℓ−m+1} from the ambient resolution Res(t, v, a) are firm subresolutions, and let H_{j}^{ℓ−m} be one of the factors in the free decomposition inherited by the image of the subgroup associated with the resolution IRes_{sec}(u, v, a) in the limit group associated with the ℓ − m level of the ambient resolution Res(t, v, a). Suppose that no (non-trivial) subgroup of H_{j}^{ℓ−m} fixes a vertex in the Bass-Serre tree T_{ℓ−m}. Then there exists a test sequence of the ambient resolution Res(t, v, a), so that the corresponding sequence of specializations of the subgroup H_{j}^{ℓ−m} are bi-Lipschitz equivalent to the action of H_{j}^{ℓ−m} on the Bass-Serre tree T_{ℓ−m}. Hence, the rank of each specialization of H_{j}^{ℓ−m} is equivalent to the rank of H_{j}^{ℓ−m}, that implies that IRes_{sec}(u, v, a) is a firm subresolution of Res(t, v, a).

Suppose that there are non-trivial subgroups of H_{j}^{ℓ−m} that fix vertices in the Bass-Serre tree T_{ℓ−m}. In this case H_{j}^{ℓ−m} inherits a free decomposition from its action on the Bass-Serre tree T_{ℓ−m}: H_{j}^{ℓ−m} = B_{1}^{ℓ−m} ∗ … ∗ B_{r}^{ℓ−m} ∗ F_{s}, where each of the factors B_{k}^{ℓ−m} inherits a (possibly trivial) abelian decomposition with non-trivial abelian edge groups from its action on the action on the tree T_{ℓ−m}. Furthermore, the canonical map from the limit group associated with the ℓ − m level of the ambient resolution, Res(t, v, a), to the limit group associated with its ℓ − m + 1 level, maps each of the factors B_{k}^{ℓ−m} onto a subgroup H_{k}^{ℓ−m+1}. Since by our inductive assumption the resolution induced by each of the factors H_{k}^{ℓ−m+1} is a firm subresolution, each of the factors H_{k}^{ℓ−m+1} is mapped onto the terminal free group of the resolution induced by the various factors H_{k}^{ℓ−m+1}, that we denote D_{k}^{ℓ−m+1}. Since IRes_{sec}(u, v, a) is a completed resolution, H_{j}^{ℓ−m} naturally contains a subgroup isomorphic to M = D_{1} ∗ … ∗ D_{r} ∗ F_{s}. Then there exists a test sequence of the ambient resolution Res(t, v, a), so that the corresponding sequence of specializations of the subgroup M < H_{j}^{ℓ−m} are isomorphic to M. Hence, the rank of each of the specializations of H_{j}^{ℓ−m} is equivalent to the rank of the subgroup M, which is equivalent to the rank of the resolution induced by the subgroup H_{j}^{ℓ−m} from the ambient resolution Res(t, v, a), that implies that the resolution induced by the subgroup H_{j}^{ℓ−m} is a firm subresolution of Res(t, v, a). By induction on the various levels of the ambient resolution Res(t, v, a), the resolution IRes_{sec}(u, v, a) is a firm subresolution of the ambient resolution Res(t, v, a).

□

In lemmas 4.8 and 4.9 we proved theorem 4.7 in case the abelian decompositions associated with the various levels of the ambient resolution Res(t, v, a) contain no QH vertex groups. To prove theorem 4.7 in the general case we use the combinatorial properties of (quadratic) test sequences together with the arguments used to prove lemmas 4.8 and 4.9. As in analyzing the resolution IRes_{sec}(u, v, a) in case the ambient resolution, Res(t, v, a), contains no QH vertex groups, we analyze the resolution IRes_{sec}(u, v, a) inductively from bottom to top, starting with the factors associated with the ℓ − 1 level of the ambient resolution Res(t, v, a).

Let H_{i}^{ℓ−1} be a factor in the free decomposition inherited by the subgroup associated with the resolution IRes_{sec}(u, v, a), from the abelian decompositions associated with all the levels of the ambient resolution Res(t, v, a) that lie above the ℓ − 1 level. let T_{ℓ−1} be the Bass-Serre tree corresponding to the abelian decomposition associated with the single level of the ambient resolution Res(t, v, a). Let H_{i}^{ℓ−1} =< h_{1}^{ℓ−1}, …, h_{r(ℓ−1,i)}^{ℓ−1} > be a fixed generating set for H_{i}^{ℓ−1}, let t_{0} ∈ T_{ℓ−1} be its base point, and let T′_{ℓ−1} be the finite tree spanned by the vertices: t_{0}, h_{1}^{ℓ−1}(t_{0}), …, h_{r(ℓ−1,i)}^{ℓ−1}(t_{0}) in the Bass-Serre tree T_{ℓ−1}. By lemma 1.5 of [Se4], from the abelian decomposition associated with the ℓ−1 level of the ambient resolution Res(t, v, a), the factor H_{i}^{ℓ−1} inherits an abelian decomposition that we denote Λ_{H_{i}^{ℓ−1}}, in which all edge groups are edge groups in the Bass-Serre tree T_{ℓ−1}, and the vertex groups are either abelian or QH vertex groups in T_{ℓ−1}, or they are contained in non-QH, non-abelian vertex groups in T_{ℓ−1}.

If the finite subtree T′_{ℓ−1} contains no floating QH vertex groups, the resolution induced by the subgroup H_{i}^{ℓ−1} is firm by the argument used to prove lemma 4.8. Suppose that the finite subtree T′_{ℓ−1} contains a floating QH vertex group. Let Q be a QH vertex group in the abelian decomposition associated with the ℓ−1 level of the ambient resolution Res(t, v, a), let Θ_{Q} be the abelian decomposition obtained from the abelian decomposition associated with the ℓ − 1 level of the ambient resolution Res(t, v, a) by collapsing all the edges except those that are connected to the QH vertex group Q, and suppose that the free decomposition inherited by the factor H_{i}^{ℓ−1} from Θ_{Q} according to lemma 1.5 of [Se4] is a non-trivial free decomposition of H_{i}^{ℓ−1}, that implies that some conjugates of the QH vertex group Q are floating QH vertex groups in the finite tree T′_{ℓ−1}.

Recall, that given a test sequence of the resolution Res(t, v, a), a quadratic test sequence is associated with the QH vertex Q. A quadratic test sequence is presented in definition 1.5 of [Se2]. According to this definition, with the surface S associated with the QH vertex group Q, we associate a collection of non-homotopic, non-boundary parallel s.c.c. b_{1}, …, b_{q} so that S \∪{b_{1}, …, b_{q}} is a disjoint union of three-punctured spheres and once-punctured Mobius bands, and another collection of non-homotopic, non-boundary parallel s.c.c. d_{1}, …, d_{t} so that each of the curves d_{i} intersects non-trivially at least one of the curves b_{j}, and the collection of curves b_{1}, …, b_{q}, d_{1}, …, d_{t} fill the surface S. Let φ_{1}, …, φ_{q} be the automorphisms of Q that correspond to Dehn twists along the s.c.c. b_{1}, …, b_{q}, and ψ_{1}, …, ψ_{t} be the automorphisms of Q that correspond to Dehn twists along the s.c.c. d_{1}, …, d_{t}. To construct a quadratic test sequence, we define the following sequences of automorphisms of the QH vertex group Q, {ν_{n}, τ_{n}}, iteratively. We set τ_{1} = id., and ν_{1} to be:

ν_{1} = ψ_{1}^{ℓ_{1}^{1}} ∘ ψ_{2}^{ℓ_{2}^{1}} ∘ … ∘ ψ_{t}^{ℓ_{t}^{1}}.

For every index n > 1 we define τ_{n} to be:

τ_{n} = φ_{1}^{m_{1}^{n}} ∘ φ_{2}^{m_{2}^{n}} ∘ … ∘ φ_{q}^{m_{q}^{n}} ∘ ν_{n−1}

and:

ν_{n} = ψ_{1}^{ℓ_{1}^{n}} ∘ ψ_{2}^{ℓ_{2}^{n}} ∘ … ∘ ψ_{t}^{ℓ_{t}^{n}} ∘ τ_{n}.

The automorphisms {ν_{n}, τ_{n}} naturally extend to automorphisms of the limit group Rlim_{ℓ−1}(t, v, a) associated with the ℓ−1 level of the ambient resolution Res(t, v, a), and we set the sequence of homomorphisms λ_{n} : Rlim_{ℓ−1}(t, v, a) → F_{k}, to be a sequence of homomorphisms of the form:

λ_{n} = μ_{n} ∘ φ_{1}^{e_{n}} ∘ φ_{2}^{e_{n}} ∘ … ∘ φ_{q}^{e_{n}} ∘ ν_{n}

where μ_{n} is a homomorphism obtained from a composition of the automorphisms associated with the other parts of the abelian decomposition associated with the ℓ − 1 level of the ambient resolution Res(t, v, a), and the (canonical) epimorphism η_{ℓ−1} from the group associated with the ℓ − 1 level of Res(t, v, a) to the free group associated with the (terminal) ℓ level.

The decomposition of the QH vertex group Q along the collection of s.c.c. b_{1}, …, b_{q}, can naturally be extended to a decomposition of the limit group Rlim_{ℓ−1}(t, v, a), associated with the ℓ − 1 level of the ambient resolution Res(t, v, a). Let G be the Bass-Serre tree corresponding to this decomposition of Rlim_{ℓ−1}(t, v, a). Before analyzing the image under the homomorphism λ_{n} of the factor H_{i}^{ℓ−1}, λ_{n}(H_{i}^{ℓ−1}), in the terminal free group of the ambient resolution Res(t, v, a), we analyze the action on the tree G, of the image of the factor H_{i}^{ℓ−1} under the automorphism ν_{n} of Rlim_{ℓ−1}(t, v, a).

Proposition 4.10. Let Γ_{n} be the graph of groups inherited by the subgroup ν_{n}(H_{i}^{ℓ−1}) from its action on the Bass-Serre tree G, and let Γ′_{n} be the free decomposition of ν_{n}(H_{i}^{ℓ−1}) obtained from the graph of groups Γ_{n} by collapsing all the edges with non-trivial stabilizers. The free decomposition Γ′_{n} of ν_{n}(H_{i}^{ℓ−1}) naturally transfers to a free decomposition (still denoted Γ′_{n}) of H_{i}^{ℓ−1}. Then there exists a test sequence with corresponding sequence of automorphisms, {(τ_{n}, ν_{n})}, for which there exists an index n_{0}, so that for every index n > n_{0}, the free decompositions Γ′_{n} are identical to the free decomposition inherited by H_{i}^{ℓ−1} from the decomposition Θ_{Q} according to lemma 1.5 of [Se4].

Proof: Let Δ_{H_{i}^{ℓ−1}} be the free decomposition inherited by H_{i}^{ℓ−1} from the decomposition Θ_{Q} according to lemma 1.5 of [Se4], i.e., the free decomposition obtained from the abelian decomposition Γ_{H_{i}^{ℓ−1}}. By our assumptions, Δ_{H_{i}^{ℓ−1}} is a non-trivial

free decomposition of H_{i}^{ℓ−1}. The collection of s.c.c. b_{1}, …, b_{q} and d_{1}, …, d_{t} are chosen to fill the surface S associated with the QH vertex group Q, and by possibly enlarging the collection d_{1}, …, d_{t} we may assume that the subgroup generated by the curves b_{1}, …, b_{q}, d_{1}, …, d_{t} and their roots in Q (cores of corresponding Mobius bands), is Q itself.

By the way a test sequence is defined, larger and larger (finite) sets of elements in the subgroup H_{i}^{ℓ−1} that are either hyperbolic with respect to the abelian decomposition Θ_{Q}, or they are conjugate to non-boundary parallel elements in the QH subgroup Q, are guaranteed to be mapped by the automorphisms ν_{n}, to elements that act hyperbolicly on the Bass-Serre tree G that corresponds to the abelian decomposition obtained from Θ_{Q}, by decomposing the QH subgroup Q along the disjoint, non-homotopic collection of s.c.c. b_{1}, …, b_{q}. Furthermore, the axis of the image of such elements under the automorphisms ν_{n} when acting on G, is composed from high powers of elements that are conjugate to the s.c.c. b_{1}, …, b_{q}, d_{1}, …, d_{t}.

Since we have assumed that the elements b_{1}, …, b_{q}, d_{1}, …, d_{t} and their roots in Q generate the subgroup Q, and since the free decomposition Δ_{H_{i}^{ℓ−1}} is non-trivial, there must exist some index n_{1} for which the free decomposition Γ′_{n_{1}} corresponding to the abelian decomposition Γ_{n_{1}} inherited by the subgroup ν_{n_{1}}(H_{i}^{ℓ−1}) from its action on the Bass-Serre tree G is non-trivial. Clearly, the free decomposition Γ′_{n_{1}} is compatible with the free decomposition Δ_{H_{i}^{ℓ−1}}, i.e., each of the factors in Δ_{H_{i}^{ℓ−1}} can be conjugated into a factor in Γ′_{n}.

Suppose that the free decomposition Δ_{H_{i}^{ℓ−1}} is a proper refinement of the free decomposition Γ′_{n}. By properly choosing the powers that define the sequence of automorphisms {(τ_{n}, ν_{n})}, we can guarantee that all the next free decompositions Γ′_{n} corresponding to the abelian decompositions Γ_{n} that are inherited by the subgroup ν_{n}(H_{i}^{ℓ−1}) from its action on the Bass-Serre tree G, are compatible with the free decomposition Γ′_{n_{1}}. Since there exists a factor in Γ′_{n_{1}} that can not be conjugated into a factor in Δ_{H_{i}^{ℓ−1}}, the same argument that guarantees the existence of a non-trivial free decomposition Γ′_{n_{1}}, guarantees the existence of and index n_{2}, for which the free decomposition Γ′_{n_{2}} that is compatible with the free decomposition Γ′_{n_{1}}, is a proper refinement of the free decomposition Γ′_{n_{1}}. Continuing inductively, we obtain an index n_{0}, for which the free decomposition Γ′_{n_{0}} is identical to the free decomposition Δ_{H_{i}^{ℓ−1}}. Each factor in the automorphism Γ′_{n_{0}} is elliptic in the abelian decomposition Θ_{Q}, so if we start with the automorphism ν_{n_{0}}, and choose the powers that define the next automorphisms {(τ_{n}, μ_{n})} in our test sequence to be large enough, we guarantee that all the next free decompositions Γ′_{n} are identical to the free decomposition Γ_{n_{0}}, and the proposition follows.

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Suppose that the (non-trivial) free decomposition inherited by the subgroup H_{i}^{ℓ−1} from the abelian decomposition Θ_{Q} according to lemma 1.5 of [Se4] is: H_{i}^{ℓ−1} = B_{1} ∗ … ∗ B_{m} ∗ F_{s}, where the factors B_{1}, …, B_{m} can be conjugated into the non-QH vertex groups in the abelian decomposition Θ_{Q}. Since by proposition 4.10 there exists a sequence of automorphisms {(τ_{n}, μ_{n})}, for which there exists some index n_{0}, so that for every index n > n_{0} the free decomposition Γ′_{n} of H_{i}^{ℓ−1} is identical to the free decomposition H_{i}^{ℓ−1} = B_{1} ∗ … ∗ B_{m} ∗ F_{s}, then the sequence of automorphisms {(τ_{n}, μ_{n})} can be extended to a test sequence of the two bottom levels of the ambient resolution Res(t, v, a), so that for every n > n_{0}:

λ_{n}(H_{i}^{ℓ−1}) = λ_{n}(B_{1}) ∗ … ∗ λ_{n}(B_{m}) ∗ λ_{n}(F_{s})

where λ_{n} maps F_{s} isomorphically onto λ_{n}(F_{s}), which clearly implies that the given test sequence is firm with respect to the entire subgroup H_{i}^{ℓ−1}. By iteratively adding the QH vertex groups in the abelian decomposition associated with the ℓ − 1 level of the ambient resolution Res(t, v, a), we get that the resolutions induced by the various subgroups H_{i}^{ℓ−1} are indeed firm subresolutions of Res(t, v, a).

Analyzing the resolutions induced by the various subgroups H_{i}^{ℓ−1} from the ambient resolution Res(t, v, a), to complete the proof of theorem 4.7, i.e., to show that the entire resolution IRes_{sec}(u, v, a) is a firm subresolution, we continue the analysis of the resolution IRes_{sec}(u, v, a) by climbing through the levels of the ambient resolution Res(t, v, a) as we did in proving lemma 4.9, and for each level we use the same argument used in the analysis of the factors H_{i}^{ℓ−1}.

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Given a well-separated complete resolution Res(t, v, a), and a subgroup < v, a > of the limit group Rlim(t, v, a) associated with this resolution, the procedure for the construction of a core resolution, enables us to start with the resolution induced by the subgroup < v, a > from the resolution Res(t, v, a), and modify it sequentially by reducing the ranks of the obtained resolution, to get a firm geometric subresolution of the resolution Res(t, v, a), which can be taken to be a core resolution, Core(< v, a >, Res(t, v, a)(r, v, a). Naturally, the procedure used for the construction of a core resolution generalizes to the construction of a (graded) core of a (well-separated) graded resolution, which we denote GCore(< v, a > , GRes(t, v, p, a))(r, v, a), and to the construction of a (multi-graded) core of a multigraded resolution, which we denote MGCore(< v, a >, MGRes)(r, v, a). Note that in the graded and multi-graded cases, the second part of the construction of the core resolution either reduces the sum of the rank and the number of the terminating factors of the resolution IRes_{f}(u, v, a) obtained in the first part, or it proves that it is indeed a firm subresolution, hence, may serve as a (graded, multi-graded) core resolution.

In the procedure for the construction of the tree of stratified sets, we were able to bound the complexity of multi-graded resolutions by the complexity of multigraded resolutions produced in previous steps of the iterative procedure. In the general sieve method, presented in the next section, we will need similar bounds on the cores of the constructed multi-graded resolutions. To state these bounds, we first need to present the complexity of a (graded, multi-graded) core resolution, which is a slight modification of our definition of the complexity of a resolution.

Definition 4.11. Let Res(t, v, a) be an (ungraded) well-separated complete resolution, and let Core(< v, a >, Res(t, v, a)) be a core resolution. We set the complexity of the core, Core(< v, a >, Res(t, v, a)), Cmplx(Core(< v, a >, Res(t, v, a))), to be the complexity of the core, Core(< v, a >, Res(t, v, a)), viewed as an induced resolution ([Se4], definition 3.2). Unlike the ungraded case, in case the ambient resolution is multi-graded (or graded), we need to slightly modify the complexity of the core.

Let MGRes(t, v, P, R_{1}, …, R_{d}, a) be a well-separated complete multi-graded resolution, and let MGCore(< v, a >, MGRes(t, v, P, R_{1}, …, R_{d}, a)) be a (multigraded) core resolution. Let Q_{1}, …, Q_{m} be the QH subgroups that appear in the core MGCore(< v, a >, MGRes(t, v, P, R_{1}, …, R_{d}, a)). Each QH vertex group Q_{j} is a subgroup of finite index in a QH vertex group in the ambient (multi-graded) resolution, so with each QH we associate the (punctured) surface S_{j} that is associated with the QH vertex group of the ambient resolution that contains Q_{j}. With each (punctured) surface S_{j} we may associate an ordered couple (genus(S_{j}), |χ(S_{j})|).

We will assume that the QH subgroups Q_{1}, …, Q_{m} are ordered according to the lexicographical (decreasing) order of the ordered couples associated with their corresponding surfaces. Let rk(MGCore) be the rank of the core, MGCore(< v, a > , MGRes(t, v, P, R_{1}, …, R_{d}, a)), and let factor(MGCore) be the number of factors it is terminating with (i.e., the number of factors in the free decomposition inherited by the terminal subgroup of the multi-graded core from the free decomposition of the terminal subgroup of the ambient multi-graded resolution). Let Abrk(Res(t, a)) be the abelian rank of the resolution (see definition 1.15 of [Se4]). We set the complexity of the multi-graded core:

MGCore(< v, a >, MGRes(t, v, P, R_{1}, …, R_{d}, a))

denoted Cmplx(MGCore), to be:

Cmplx(MGCore) = (rk(MGCore) + factor(MGCore),
(genus(S_{1}), |χ(S_{1})|), …, (genus(S_{m}), |χ(S_{m})|), Abrk(Res(t, a))).

On the set of core resolutions we can define a partial order. Let MGCore_{1} and MGCore_{2} be two (multi-graded) core resolutions. We say that Cmplx(MGCore_{1}) = Cmplx(MGCore_{2}) if the tuples defining the two complexities are identical. We say that Cmplx(MGCore_{1}) < Cmplx(MGCore_{2}) if:

(1) rk(MGCore_{1})+factor(MGCore_{1}) is smaller than rk(MGCore_{2})+factor(MGCore_{2}).

(2) the above numbers are equal and the tuple:

((genus(S_{1}^{1}), |χ(S_{1}^{1})|), …, (genus(S_{m_{1}}^{1}), |χ(S_{m_{1}}^{1}|))

is smaller in the lexicographical order than the tuple:

((genus(S_{1}^{2}), |χ(S_{1}^{2})|), …, (genus(S_{m_{2}}^{2}), |χ(S_{m_{2}}^{2}|)).

(3) the above numbers and tuples are equal and Abrk(MGCore_{1}) < Abrk(MGCore_{2}).

To get bounds on the complexity of core resolutions of the multi-graded resolutions constructed along our quantifier elimination iterative procedure, we need to study some basic properties of the core resolution.

Theorem 4.12. Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) be a well-separated complete multi-graded resolution containing a single level (i.e., the multi-graded limit group associated with its second level is either rigid or solid), and let Λ_{1} be the graph of groups with fundamental group Rlim(v, P, a) associated with the single level of MGRes_{1}, where R_{1}, …, R_{m}, P are the non-QH, non-abelian vertex groups in the graph of groups Λ_{1}. Let MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) be a well-separated complete resolution of a limit group Rlim(t, v, P, a), so that there is an embedding ν : Rlim(v, P, a) → Rlim(t, v, P, a), that maps each of the subgroups R_{1}, …, R_{m} into a conjugate of one of the subgroups L_{j}.

Since the multi-graded resolution MGRes_{1}(v, R_{1}, …, R_{m}, P, a) is well-separated, with each QH vertex groups in its associated multi-graded abelian decomposition, Λ_{1}, there is an associated collection of s.c.c. that are mapped to the trivial element in the terminal level of the resolution MGRes_{1}(v, R_{1}, …, R_{m}, P, a).

Each QH vertex group in Λ_{1}, naturally inherits a sequence of abelian decompositions from the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a). Suppose that for every such QH vertex group Q, this sequence of multi-graded abelian decompositions is compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the terminal level of the multi-graded resolution MGRes_{1}(v, R_{1}, …, R_{m}, P, a).

Then the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) can be modified to a well-separated multi-graded resolution MGRes_{3}(u, t, v, L_{1}, …, L_{n}, P, a), so that every specialization of the subgroup Rlim(t, v, P, a) that factors through the multi-graded resolution MGRes_{3} factors through the multi-graded resolution MGRes_{2} and vice versa, and for which there exists a core, MGCore(< v, P, a > , MGRes_{3}), produced by modifying the above procedure for the construction of a (multi-graded) core resolution, that has the following properties:

(i) the complexity of the multi-graded core is bounded by the complexity of the multi-graded resolution MGRes_{1}:

Cmplx(MGCore(< v, P, a >, MGRes_{3})) ≤ Cmplx(MGRes_{1}).

(ii) if these complexities are equal, then the structure of the core MGCore(< v, P, a >, MGRes_{3}) is identical to the structure of the multi-graded resolution MGRes_{1}.

Proof: Since the multi-graded resolution MGRes_{2} is ”compatible” with the multigraded resolution MGRes_{1}, every specialization of the subgroup < v, P, a > that is the restriction of a specialization of the subgroup < t, v, P, a > that is taut with respect to the multi-graded resolution MGRes_{2}, is taut with respect to the multigraded abelian decomposition Λ_{1} of MGRes_{1}. Hence, the rank of the induced resolution, IndRes(< v, P, a >, MGRes_{2}), is bounded by the rank of the multigraded resolution MGRes_{1}, and:

rk(IndRes(< v, P, a >, MGRes_{2})) + factor(IndRes(< v, P, a >, MGRes_{2})) ≤

≤ rk(MGRes_{1}) + factor(MGRes_{1}).

Applying the iterative procedure for the construction of the core resolution, to the subgroup < v, P, a > and the multi-graded resolution MGRes_{2}, we obtain a multigraded core, MGCore(< v, P, a >, MGRes_{2}). By construction, the sum of the rank and the number of factors of this core is bounded by the sum of the rank and the number of factors of the induced resolution, IndRes(< v, P, a >, MGRes_{2}), and if the two sums are equal, then the second part of the construction was not applied, i.e., the multi-graded core and the induced resolution are identical. Hence:

rk(MGCore(< v, P, a >, MGRes_{2})) + factor(MGCore(< v, P, a >, MGRes_{2})) ≤

≤ rk(MGRes_{1}) + factor(MGRes_{1}).

If:

rk(MGCore(< v, P, a >, MGRes_{2})) + factor(MGCore(< v, P, a >, MGRes_{2})) <

< rk(MGRes_{1}) + factor(MGRes_{1})

then we set the core to be the output of the procedure, MGCore(< v, P, a > , MGRes_{2}), and set the multi-graded resolution MGRes_{3} to be the the multigraded resolution MGRes_{2}, and the theorem follows. Hence, we may assume:

rk(MGCore(< v, P, a >, MGRes_{2})) + factor(MGCore(< v, P, a >, MGRes_{2})) =

= rk(MGRes_{1}) + factor(MGRes_{1}).

In this case, the multi-graded core constructed by our iterative procedure, MGCore(< v, P, a >, MGRes_{2})), is identical to the induced resolution, IndRes(< v, P, a > , MGRes_{2}). If the multi-graded abelian decomposition Λ_{1}, associated with the multi-graded resolution MGRes_{1}, contains no surviving QH vertex groups (see section 1 of [Se4] for the notion of a surviving surface), the theorem follows from our analysis of taut homomorphisms of maximal rank, presented in section 2 of [Se4]. Suppose that the multi-graded abelian decomposition Λ_{1} of MGRes_{1}, contains a surviving QH vertex group. Let Q be a surviving vertex group in Λ_{1}, and let Q̂ be a QH vertex group in an abelian decomposition associated with one of the levels of MGRes_{2}, so that Q is mapped isomorphically onto Q̂, and Q is not mapped isomorphically onto a QH vertex group in an abelian decomposition associated with a level that lies above the level to which Q̂ belongs in the multi-graded resolution MGRes_{2}. We vary the multi-graded resolution by ”pushing down” a QH vertex isomorphic to Q̂ which is set to be the image of the QH vertex Q:

i.e., we push down a QH vertex group which is set to be the image of Q, leave the levels below the QH vertex group Q̂ unchanged, and change the order of the QH vertex groups that lie above Q̂ into which the QH vertex group Q is mapped. We repeat this ”pushing down” operation for all surviving QH vertex groups Q_{j} in the abelian decomposition Λ_{1}, that are mapped isomorphically onto QH vertex groups Q̂_{j} that are not conjugate in the limit group associated with the resolution MGRes_{2}. We denote the obtained resolution MGRes_{3}(u, t, v, L_{1}, …, L_{n}, P, a). With the multi-graded resolution MGRes_{3} we associate a core, MGCore(< v, P, a > , MGRes_{3}), by applying our algorithm for the construction of a core resolution. If:

rk(MGCore(< v, P, a >, MGRes_{3})) + factor(MGCore(< v, P, a >, MGRes_{3})) <

< rk(MGRes_{1}) + factor(MGRes_{1})

the theorem follows. Suppose:

rk(MGCore(< v, P, a >, MGRes_{3})) + factor(MGCore(< v, P, a >, MGRes_{3})) =

= rk(MGRes_{1}) + factor(MGRes_{1})

If there are no two surviving QH vertex groups in Λ_{1} that are mapped onto conjugate QH vertex groups in MGRes_{3}, or more generally, if every two surviving QH vertex groups in Λ_{1} that are mapped onto conjugate vertex groups in MGRes_{3} belong to the same factor in the abelian decomposition Λ_{1} of MGRes_{1}, the theorem follows by our analysis of taut homomorphisms of maximal rank ([Se4], section 2). Otherwise, every two surviving QH vertex groups Q_{1} and Q_{2} in Λ_{1} that do not belong to the same factor in Λ_{1}, and are mapped isomorphically onto conjugate QH vertex groups in MGRes_{3}, must belong to different factors in the free decomposition associated with the terminal level of the core MGCore(< v, P, a >, MGRes_{3}). Let Q_{1} and Q_{2} be such surfaces. In this case we replace the subgroup < v, P, a > by the subgroup G_{1} generated by < v, P, a > and the element that conjugates Q̂_{1} to Q̂_{2} in the subgroup associated with the resolution MGRes_{3}. We repeat this operation for all such couples of surviving surfaces until we obtain a group G_{s}. Note that by adding such a conjugating element, we may increase the rank of the core of the corresponding subgroup by at most 1, but we necessarily reduce the number of terminating factors by at least 1. At this point we look at the core, MGCore(G_{s}, MGRes_{3}), obtained by our procedure for the construction of a core resolution. By construction:

rk(MGCore(G_{s}, MGRes_{3})) + factor(MGCore(G_{s}, MGRes_{3})) ≤
≤ rk(MGRes_{1}) + factor(MGRes_{1})

and if the inequality is strict the theorem follows. If:

rk(MGCore(G_{s}, MGRes_{3})) + factor(MGCore(G_{s}, MGRes_{3})) =
= rk(MGRes_{1}) + factor(MGRes_{1})

then, the core, MGCore(G_{s}, MGRes_{3}), is just the induced resolution, IndRes(G_{s}, MGRes_{3}), so by our analysis of taut homomorphisms of maximal rank:

Cmplx(MGCore(G_{s}, MGRes_{3})) < Cmplx(MGCore(G_{s}, MGRes_{3}))

and the theorem follows.

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Theorem 4.12 bounds the complexity of the core of a multi-graded resolution, by the complexity of the abelian decomposition of a one level multi-graded resolution that is ”compatible” with it. In a similar way it is possible to bound the complexity of the core of a multi-graded resolution, by the complexity of the core of an abelian decomposition of a one level multi-graded resolution that is ”compatible” with it.

Theorem 4.13. Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) and MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) be multi-graded resolutions that satisfy the assumptions of theorem 4.12, and let < y, a > be a subgroup of the limit group Rlim(v, P, a). Then the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) can be modified to a well-separated multi-graded resolution MGRes_{3}(u, t, v, L_{1}, …, L_{n}, P, a), so that every specialization of the subgroup Rlim(t, v, P, a) that factors through the multigraded resolution MGRes_{3} factors through the multi-graded resolution MGRes_{2} and vice versa, and for which there exists a core, MGCore(< y, a >, MGRes_{3}), that has the following properties:

(i) the complexity of the multi-graded core is bounded by the complexity of the corresponding core of the multi-graded resolution MGRes_{1}:

Cmplx(MGCore(< y, a >, MGRes_{3})) ≤ Cmplx(MGCore(< y, a >, MGRes_{1}))

(ii) if these complexities are equal, then the structure of the core MGCore(< y, a >, MGRes_{3}) is identical to the structure of the corresponding core of the multi-graded resolution MGRes_{1}, MGCore(< y, a >, MGRes_{1}).

Proof: Let G_{1} be the image of the subgroup corresponding the core, MGCore(< v, a >, MGRes_{1}), in the group < t, v, P, a >. The argument used to prove theorem 4.12 applied to the core, MGCore(G_{1}, MGRes_{2}), proves theorem 4.13.

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Theorem 4.13 bounds the complexity of the core of a multi-graded resolution, by the complexity of the core of a one level multi-graded resolution that is ”compatible” with it. In addition to the bound on the complexity of the core of the entire multigraded resolution, we will need bounds on the core resolutions of multi-graded resolutions composed from some successive levels of it.

Theorem 4.14. Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) and MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) be well-separated complete multi-graded resolutions that satisfy the assumptions of theorem 4.12, and let < y, a > < Rlim(v, P, a). Suppose that the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) has ℓ levels, let 1 ≤ i_{1} ≤ i_{2} ≤ ℓ, and let MGRes_{3}(t, v, L_{1}, …, L_{n}, P, a) be the multi-graded resolution composed from levels i_{1}, …, i_{2} of the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a).

Then the multi-graded resolution MGRes_{3}(t, v, L_{1}, …, L_{n}, P, a) can be modified to a well-separated multi-graded resolution MGRes_{4}(u, t, v, L_{1}, …, L_{n}, P, a), so that every specialization of the subgroup Rlim(t, v, P, a) that factors through the multi-graded resolution MGRes_{4} factors through the multi-graded resolution MGRes_{3} and vice versa, and for which there exists a core, MGCore(< y, a > , MGRes_{4}), that has the following properties:

(i) the complexity of the multi-graded core is bounded by the complexity of the corresponding core of the multi-graded resolution MGRes_{1}:

Cmplx(MGCore(< y, a >, MGRes_{4})) ≤ Cmplx(MGCore(< y, a >, MGRes_{1})).

(ii) if these complexities are equal, then the structure of the core, MGCore(< y, a >, MGRes_{4}), is identical to the structure of the corresponding core of the multi-graded resolution MGRes_{1}, MGCore(< y, a >, MGRes_{1}).

(iii) If the complexities are equal, then the multi-graded resolution:

MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a)

can be modified to a well-separated multi-graded resolution:

MGRes_{5}(u, t, v, L_{1}, …, L_{n}, P, a)

so that every specialization of the subgroup Rlim(t, v, P, a) that factors through the multi-graded resolution MGRes_{2} factors through the multi-graded resolution MGRes_{5} and vice versa, and for which there exists a core, MGCore(< y, a >, MGRes_{5}), that has the same structure as the corresponding core of the multi-graded resolution MGRes_{1}, MGCore(< y, a >, MGRes_{1}).

Proof: Parts (i) and (ii) follow by the argument used to prove theorem 4.13. Suppose that the complexities, Cmplx(MGCore(< y, a >, MGRes_{4})) and Cmplx(MGCore(< y, a >, MGRes_{1})) are equal. To prove part (iii), suppose that using the construction used in the proofs of theorems 4.12 and 4.13, the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) can be modified to a well-separated multi-graded resolution MGRes_{5}(u, t, v, L_{1}, …, L_{n}, P, a), so that every specialization of the subgroup Rlim(t, v, P, a) that factors through the multi-graded resolution MGRes_{2} factors through the multi-graded resolution MGRes_{5} and vice versa, and:

Cmplx(MGCore(< y, a >, MGRes_{5})) < Cmplx(MGCore(< y, a >, MGRes_{1})).

If:

rk(MGCore(< y, a >, MGRes_{5})) + factor(MGCore(< y, a >, MGRes_{5})) <

< rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1}))

then by construction, the multi-graded resolution MGRes_{5} is identical to the multigraded resolution MGRes_{2}, so:

rk(MGCore(< y, a >, MGRes_{3})) + factor(MGCore(< y, a >, MGRes_{3})) ≤

≤ rk(MGCore(< y, a >, MGRes_{2})) + factor(MGCore(< y, a >, MGRes_{2})) <

< rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1}))

and part (iii) follows in this case. Hence, for the rest of the argument we may assume:

rk(MGCore(< y, a >, MGRes_{5})) + factor(MGCore(< y, a >, MGRes_{5})) =

= rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1})).

Since MGRes_{3} is composed from a ”block” of consecutive levels of the multi-graded resolution MGRes_{2}, every surviving QH vertex group in the multi-graded abelian decomposition associated with the core, MGCore(< y, a >, MGRes_{1}) with respect to the multi-graded resolution MGRes_{3}, is also a surviving QH vertex group with respect to the multi-graded resolution MGRes_{2}. Therefore:

Cmplx(MGCore(< y, a >, MGRes_{4})) ≤ Cmplx(MGCore(< y, a >, MGRes_{5})) <

< Cmplx(MGCore(< y, a >, MGRes_{1}))

and part (iii) of the theorem follows.

□

If we restrict theorem 4.14 to a single level of the multi-graded resolution MGRes_{2}, we are able to bound the complexity of the core of each of the multi-graded abelian decompositions associated with the various levels of a multi-graded resolution in terms of the complexity of the core of MGRes_{1}.

Corollary 4.15. Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) and MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) be well-separated complete multi-graded resolutions that satisfy the assumptions of theorem 4.12, and let < y, a > < Rlim(v, P, a). Let Θ_{1}, …, Θ_{ℓ} be the multi-graded abelian decompositions associated with the various levels of the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a). Then either:

(i) The multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) can be modified to a well-separated multi-graded resolution MGRes_{3}(u, t, v, L_{1}, …, L_{n}, P, a), so that every specialization of the subgroup Rlim(t, v, P, a) that factors through the multi-graded resolution MGRes_{3} factors through the multi-graded resolution MGRes_{2} and vice versa, and for which there exists a core, MGCore(< y, a >, MGRes_{3}), that has the same structure as the core of the multi-graded resolution MGRes_{1}, MGCore(< y, a >, MGRes_{1}).

(ii) For every abelian decomposition, Θ_{i}, associated with a level of the multigraded resolution, MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a), the complexity of the core associated with that level, MGCore(< y, a >, Θ_{i}), satisfies:

Cmplx(MGCore(< y, a >, Θ_{i})) < Cmplx(MGCore(< y, a >, MGRes_{1})).

As in the iterative procedure used for the construction of the tree of stratified sets, resolutions of maximal complexity play an essential role in the general sieve procedure.

Definition 4.16. Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) and MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) be well-separated complete multi-graded resolutions that satisfy the assumptions of theorem 4.12, and let < y, a > < Rlim(v, P, a). We say that the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) is a resolution of maximal complexity, if the multi-graded resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) can be modified to a wellseparated multi-graded resolution MGRes_{3}(u, t, v, L_{1}, …, L_{n}, P, a), so that every specialization of the subgroup Rlim(t, v, P, a) that factors through the multi-graded resolution MGRes_{3} factors through the multi-graded resolution MGRes_{2} and vice versa, and for which there exists a core, MGCore(< y, a >, MGRes_{3}), that has the same structure as core of the multi-graded resolution MGRes_{1}, MGCore(< y, a > , MGRes_{1}). In case the core, MGCore(< y, a >, MGRes_{3}), has the same structure as the core MGCore(< y, a >, MGRes_{1}), we call the part of the core that includes the edge groups and the abelian and QH vertex groups, the formed part of the (maximal complexity) core resolution.

Maximal core resolutions and their properties play an essential role in the sieve procedure presented in the next section. One of their main properties is the fact that only maximal core resolutions can ”cover” maximal core resolutions.

Theorem 4.17. Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) be a well-separated complete multi-graded resolution containing a single level with corresponding limit group Rlim_{1}(v, P, a), so that the subgroups R_{1}, …, R_{m}, P are the non-abelian, non-QH vertex groups in the multi-graded abelian decomposition associated with

MGRes_{1}(v, R_{1}, …, R_{m}, P, a)

and let < y, a > < Rlim_{1}(v, P, a). Let Rlim_{2}(t, v, M_{1}, …, M_{d}, P, a) and

Rlim_{3}(w, t, v, L_{1}, …, L_{n}, P, a)

be limit groups for which:

(1) there exists an embedding μ : Rlim_{1}(v, P, a) → Rlim_{2}(t, v, P, a), that maps each of the subgroups R_{1}, …, R_{m} into a conjugate of one of the subgroups M_{j}.

(2) there exists a homomorphism: ν : Rlim_{2}(t, v, P, a) → Rlim_{3}(w, t, v, P, a), that embeds naturally the subgroup < v, P, a >, and maps each of the subgroups M_{1}, …, M_{d} into a conjugate of one of the subgroups L_{j}.

Let MGRes_{3}(w, t, v, L_{1}, …, L_{n}, P, a) be a multi-graded resolution of the limit group Rlim_{3}(w, t, v, L_{1}, …, L_{n}, P, a), for which the resolutions MGRes_{1}(v, R_{1}, …, R_{m}, P, a) and MGRes_{3}(w, t, v, L_{1}, …, L_{n}, P, a) satisfy the assumptions of theorem 4.12. Then either:

(i) the resolution MGRes_{3}(w, t, v, L_{1}, …, L_{n}, P, a) can be modified to a resolution MGRes_{4}(u, w, t, v, L_{1}, …, L_{n}, P, a) so that every specialization of the subgroup Rlim_{3}(w, t, v, P, a) that factors through the resolution

MGRes_{3}(w, t, v, L_{1}, …, L_{n}, P, a)

also factors through the resolution MGRes_{4}(u, w, t, v, L_{1}, …, L_{n}, P, a), and vice versa, and there exists a core, MGCore(< y, a >, MGRes_{4}) that has smaller complexity than the core MGCore(< y, a >, MGRes_{1}).

(ii) the multi-graded resolution MGRes_{3} is of maximal complexity, and there exist resolutions in the taut multi-graded Makanin-Razborov diagram of

Rlim_{2}(t, v, M_{1}, …, M_{d}, P, a) :

MGRes_{1}^{2}(t, v, M_{1}, …, M_{d}, P, a), …, MGRes_{e}^{2}(t, v, M_{1}, …, M_{d}, P, a)

that are all compatible with the resolution MGRes_{1}(v, R_{1}, …, R_{m}, P, a) (i.e., satisfy the assumptions of theorem 4.13), all are of maximal complexity, and so that every specialization of the subgroup < y, a > that (can be completed to a specialization) that is taut with respect to the completion of the resolution MGRes_{3}(w, t, v, L_{1}, …, L_{n}, P, a), can be completed to a specialization that is taut with respect to at least one of the maximal complexity resolutions MGRes_{i}^{2}(t, v, M_{1}, …, M_{d}, P, a).

Proof: If MGRes_{3} is not a resolution of maximal complexity part (i) of the theorem holds, hence, we may assume that MGRes_{3} is a resolution of maximal complexity. MGRes_{3}(w, t, v, L_{1}, …, L_{n}, P, a) is a multi-graded resolution of the limit group Rlim_{3}(w, t, v, P, a), and there is a homomorphism ν : Rlim_{2}(t, v, P, a) → Rlim_{3}(w, t, v, P, a) that maps the subgroups M_{1}, …, M_{d} into conjugates of the subgroups L_{1}, …, L_{n}, hence, every specialization of the subgroup Rlim_{2}(t, v, P, a) that can be extended to a specialization that factors through the multi-graded resolution MGRes_{3}, and for which the corresponding specialization of the subgroup < v, P, a > factors and is taut with respect to the multi-graded resolution MGRes_{1}, factors and is taut with respect to at least one of the resolutions in the taut multigraded Makanin-Razborov diagram of the limit group Rlim_{2}(w, t, v, P, a) that is compatible with the resolution MGRes_{1}(v, R_{1}, …, R_{m}, P, a).

We look at the collection of all test sequences that factor through the resolution MGRes_{3}, and for which the corresponding specializations of the subgroup < v, P, a > factor and are taut with respect to the resolution MGRes_{1}. The specializations of the subgroup Rlim_{2}(t, v, P, a) corresponding to the specializations of each such test sequence factor and are taut with respect to at least one of the resolutions in the multi-graded taut Makanin-Razborov diagram of Rlim_{2}(t, v, M_{1}, …, M_{d}, P, a), hence, by passing to appropriate subsequences, we can assume that the specializations of the subgroup Rlim_{2}(t, v, P, a) corresponding to a given test sequence of Rlim_{3}(w, t, v, P, a) factor and are taut with respect to a fixed resolution in the multi-graded taut Makanin-Razborov diagram of Rlim_{2}(t, v, M_{1}, …, M_{d}, P, a). With each test sequence of MGRes_{3}, for which the corresponding specializations of Rlim_{2} factor and are taut with respect to a (fixed) multi-graded resolution MGRes_{i}^{2}, which is a resolution in the taut multi-graded Makanin-Razborov diagram of Rlim_{2}(t, v, M_{1}, …, M_{d}, P, a), we apply the techniques used for the construction of formal solutions (section 1 of [Se2]), and associate (canonically) a finite collection of closures of the multi-graded resolution MGRes_{3}, and with each such closure we further associate a map from the completion of MGRes_{i}^{2} into this closure. We further apply the techniques used for the construction of graded and multi-graded formal limit groups, and associate with the entire collection of such test sequences of MGRes_{3}, a finite collection of closures of the multi-graded resolution MGRes_{3}, and with each such closure we further associate a map from the completion of the corresponding resolution MGRes_{i}^{2} in the taut multi-graded Makanin-Razborov diagram of Rlim_{2}(t, v, M_{1}, …, M_{d}, P, a) into this closure.

If all the resolutions MGRes_{i}^{2} that are mapped into the closures of MGRes_{3} are of maximal complexity, part (ii) of the theorem follows. Hence, we may assume that at least one of the multi-graded resolutions MGRes_{i} is not of maximal complexity, i.e.:

Cmplx(MGCore(< y, a >, MGRes_{i}^{2})) < Cmplx(MGCore(< y, a >, MGRes_{1})).

Suppose:

rk(MGCore(< y, a >, MGRes_{i}^{2})) + factor(MGCore(< y, a >, MGRes_{i}^{2})) =

= rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1})).

Since MGRes_{3} is of maximal complexity, every QH vertex group in the abelian decomposition associated with the core, MGCore(< y, a >, MGRes_{i}^{2}), is a surviving surface with respect to the resolution MGRes_{3}, hence, it must be a surviving surface with respect to the resolution MGRes_{i}^{2}. So the multi-graded resolution MGRes_{i}^{2} has to be of maximal complexity as well, a contradiction. Therefore, we may assume:

rk(MGCore(< y, a >, MGRes_{i}^{2})) + factor(MGCore(< y, a >, MGRes_{i}^{2})) <

< rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1})).

Let G_{i} be the image of the subgroup associated with the core of MGRes_{i}^{2} in a corresponding closure of MGRes_{3}. At this point we apply our iterative procedure for the construction of the core resolution, to construct the multi-graded core, MGCore(G_{i}, Cl(MGRes_{3})), where Cl(MGRes_{3}) is the corresponding (multigraded) closure of MGRes_{3}. If:

rk(MGCore(G_{i}, Cl(MGRes_{3}))) + factor(MGCore(G_{i}, Cl(MGRes_{3}))) <

< rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1}))

then since < y, a >< G_{i}, we may associate with the core, MGCore(G_{i}, Cl(MGRes_{3})), a corresponding core MGCore(< y, a >, MGRes_{3}), for which:

rk(MGCore(< y, a >, MGRes_{3})) + factor(MGCore(< y, a >, MGRes_{3})) <

< rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1}))

and part (i) of the theorem follows. Hence, for the rest of the argument we may assume that:

rk(MGCore(G_{i}, Cl(MGRes_{3}))) + factor(MGCore(G_{i}, Cl(MGRes_{3}))) ≥

≥ rk(MGCore(< y, a >, MGRes_{1})) + factor(MGCore(< y, a >, MGRes_{1})).

In this case for every test sequence that factor through the given closure Cl(MGRes_{3}), and is firm with respect to the subgroup G_{i}, the corresponding specializations of the limit group Rlim_{2}(t, v, P, a), can not factor through the multi-graded resolution MGRes_{i}^{2}, hence, they must factor through another resolution in the taut multi-graded Makanin-Razborov diagram of Rlim_{2}(t, v, P, a) that is compatible with the multi-graded resolution MGRes_{1}. Therefore, we may omit the resolution MGRes_{i}^{2} and the corresponding map from its completion into the closure Cl(MGRes_{3}), from our list of such closures and maps. Repeating these alterations for all the multi-graded resolutions MGRes_{i}^{2} in the taut multi-graded Makanin-Razborov diagram of Rlim_{2}(t, v, P, a) that appear in our list, we either construct a core, MGCore(< y, a >, MGRes_{3}) that is not of maximal complexity, and get part (i) of the theorem, or we are left with resolutions MGRes_{i}^{2} which are all of maximal complexity, so part (ii) of the theorem holds.

□

In addition to the natural ”covering” property of maximal core resolutions, presented in theorem 4.17, we need a correspondence between maximal core resolutions, and maximal core resolutions containing two parts, the top obtained by enlarging the parameter subgroups to include the formed part of the core resolution, and the bottom obtained using the original parameter subgroups.

Theorem 4.18. Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) and MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) be multi-graded resolutions that satisfy the assumptions of theorem 4.12, let < y, a > < Rlim_{1}(v, P, a), and suppose MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) is of maximal complexity. Then either there exists a multi-graded core, MGCore(< y, a >, MGRes_{2}), that has strictly smaller complexity than the core, MGCore(< y, a >, MGRes_{1}), or there exists a collection of multi-graded resolutions:

MGRes_{1}^{2}(t, v, L_{1}, …, L_{n}, P, a), …, MGRes_{e}^{2}(t, v, L_{1}, …, L_{n}, P, a)

of the limit group Rlim_{2}(t, v, L_{1}, …, L_{n}, P, a), that satisfy the following properties:

(1) each resolution MGRes_{j}^{2} is compatible with the resolution MGRes_{1} (i.e., the two resolutions satisfy the assumptions of theorem 4.12).

(2) each resolution MGRes_{j}^{2} is of maximal complexity.

(3) each resolution MGRes_{j}^{2} is composed from two parts. The top, T MGRes_{j}^{2}, being a resolution in the taut multi-graded Makanin-Razborov diagram of the limit group Rlim_{2}(t, v, P, a) with respect to the parameter subgroups L_{1}, …, L_{n}, P and the formed part of the core resolution, MGCore(< y, a > , MGRes_{1}), and the bottom part being a one level resolution of the terminal limit group of the top T MGRes_{j}^{2}, that has the same structure as the formed part of the core, MGCore(< y, a >, MGRes_{1}).

(4) every specialization of the subgroup < y, a > that can be extended to a specializations that is taut with respect to the maximal complexity resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a), can be extended to a specialization that is taut with respect to at least one of the (maximal complexity) resolutions MGRes_{j}^{2}.

Proof: We start with the collection of multi-graded resolutions of the limit group Rlim_{2}(t, v, L_{1}, …, L_{n}, P, a) that are composed from two parts. The top being a resolution in the multi-graded taut Makanin-Razborov diagram of Rlim_{2}(t, v, P, a) with respect to the subgroups L_{1}, …, L_{n}, P and the formed part of the maximal complexity core, MGCore(< y, a >, MGRes_{1}), that we denote T MGRes, and the bottom being a resolution in the taut multi-graded Makanin-Razborov diagram of the terminal rigid or solid (multi-graded) limit group of T MGRes with respect to the subgroups L_{1}, …, L_{n}, P, that we denote BMGRes.

By theorem 4.17 either there exists a core, MGCore(< y, a >, MGRes_{2}), that has strictly smaller complexity than the core, MGCore(< y, a >, MGRes_{1}), or there exists a collection of multi-graded resolutions:

MGRes_{1}^{2}(t, v, L_{1}, …, L_{n}, P, a), …, MGRes_{e}^{2}(t, v, L_{1}, …, L_{n}, P, a)

from the list described above, that are all of maximal complexity, compatible with the resolution MGRes_{1} (i.e., the resolutions satisfy the assumptions of theorem 4.12), and every specialization of the subgroup < y, a > that can be extended to a specialization that is taut with respect to the maximal complexity resolution MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a), can be extended to a specialization that is taut with respect to at least one of the (maximal complexity) resolutions MGRes_{j}^{2}. Since each of the multi-graded resolutions, MGRes_{j}^{2}, are of maximal complexity, we can use the modular groups associated with the bottom level of the top part of the multi-graded resolution MGRes_{j}^{2}, T MGRes, as we did in analyzing maximal complexity resolutions along the proof of theorem 4.12, to modify the bottom level of MGRes_{j}^{2}, BMGRes, to be a single level (multi-graded) resolution that has the same structure as the formed part of the core, MGCore(< y, a >, MGRes_{1}), and the theorem follows.

□

Given a well-separated resolution Res(t, v, a), and a subgroup < v, a > of its corresponding limit group Rlim(t, v, a), we presented an iterative procedure for the construction of a core resolution, Core(< v, a >, Res(t, v, a)), and used the algorithm to prove some basic properties of core resolutions, which play an essential role in the general sieve procedure presented in the next section. The core resolution, Core(< v, a >, Res(t, v, a)), is embedded in the (complete) ambient resolution Res(t, v, a), however, its terminal free group is not embedded ”discretely” into the resolution Res(t, v, a). i.e., the image of the terminal free group of a core resolution can intersect a QH vertex group in the ambient resolution, in a subgroup of finite index, or it can inherit a cyclic decomposition from an abelian vertex group in the ambient resolution. To guarantee the termination of the sieve procedure, we need the core resolutions we study, to be ”comparable” metrically with their embeddings in the ambient resolutions, hence, we need to prevent ”indiscrete” embeddings of the terminal free group of the core resolutions we study. To achieve that we present penetrated core resolutions.

Definition 4.19. Let Res(t, v, a) be a well-separated (complete) resolution, suppose we fix an ordered basis for each of the abelian and pegged abelian vertex groups in the resolution Res(t, v, a), and an order on the various QH and abelian vertex groups in each of the levels of the resolution Res(t, v, a). Fixing these orders, we can naturally associate with the resolution Res(t, v, a) a finite sequence of actions of subgroups of Rlim(t, v, a) on corresponding real trees. Let < v, a > be a subgroup of Rlim(t, v, a), the limit group associated with the resolution Res(t, v, a). We say that a core resolution, Core(< v, a >, Res(t, v, a)), is a penetrated core resolution, if the abelian decompositions inherited by the core from its embedding into the ambient resolution, are identical to the abelian decompositions associated with it. i.e., the embedding of the core resolution into the ambient resolution does not contain a finite index subgroup of a QH vertex group, which is not a QH vertex group in the core itself, and every abelian decomposition inherited by the core from its embedding into the ambient resolution, is the natural image of (parts) of abelian decompositions associated with the core itself.

The algorithm for the construction of a core resolution presented in the beginning of this section, actually constructs penetrated core resolutions, since it is based on the iterative procedure for the construction of the induced resolution. However, its final output and the modification of this algorithm used for the construction of core resolutions that satisfy the inequalities of theorems 4.12-4.18, does not give penetrated core resolutions in general. Along the sieve procedure, we use penetrated core resolutions, only in case the (multi-graded) resolutions in question are of maximal complexity. Hence, for the purposes of the sieve procedure, we present a process that extends a core resolution to a penetrated core resolution, in case the resolution in question is of maximal complexity (definition 4.16).

Let MGRes_{1}(v, R_{1}, …, R_{m}, P, a) be a well-separated complete multi-graded resolution containing a single level, let Λ_{1} be the graph of groups with fundamental group Rlim(v, R_{1}, …, R_{m}, P, a) associated with the single level of MGRes_{1}, where R_{1}, …, R_{m}, P are the non-QH, non-abelian vertex groups in the graph of groups Λ_{1}. Let < y, a > < Rlim(v, R_{1}, …, R_{m}, P, a). Let MGRes_{2}(t, v, L_{1}, …, L_{n}, P, a) be the completion of a resolution of a limit group Rlim(t, v, L_{1}, …, L_{n}, P, a), so that there is an embedding ν : Rlim(v, P, a) → Rlim(t, v, L_{1}, …, L_{n}, P, a), that maps each of the subgroups R_{1}, …, R_{m} into a conjugate of one of the subgroups L_{j}. Suppose that the resolutions MGRes_{1} and MGRes_{2} satisfy the assumptions of theorem 4.12, that the resolution MGRes_{2} is of maximal complexity (definition 4.16), and that the resolution MGRes_{2} is composed from two parts, the top, T MGRes_{2}, being a multi-graded resolution in the taut (multi-graded) Makanin-Razborov diagram of the limit group Rlim(t, v, L_{1}, …, L_{n}, P, a) with respect to the parameter subgroups L_{1}, …, L_{n}, P and the formed part of the core MGCore(< y, a >, MGRes_{1}), and the bottom being a one level resolution that has the same structure as the formed part of the core, MGCore(< y, a >, MGCore_{1}).

Proposition 4.20. Let P enMGCore(< y, a >, MGRes_{2}) be the resolution induced by the subgroup associated with the core, MGCore(< y, a >, MGRes_{1}), from the multi-graded resolution MGRes_{2}. Then:

(i) P enMGCore(< y, a >, MGRes_{2}) is a penetrated core resolution. (ii) every QH vertex group that appears in P enMGCore(< y, a >, MGRes_{2}),

and not in the core, MGCore(< y, a >, MGRes_{2}), is of minimal rank. (iii) Let J_{i}(y, a) be the image of the subgroup < y, a > in the limit group W_{i},

which is the natural free product of the limit group associated with the ith level of P enMGCore(< y, a >, MGRes_{2}), and the free groups dropped at higher levels. Then for every level i, except the bottom level, the natural maps: τ_{i} : Rlim(y, a) → J_{i}(y, a) and η_{i} : J_{i}(y, a) → J_{i+1}(y, a), are isomorphisms.

(iv) no QH vertex group that appears in P enMGCore(< y, a >, MGRes_{2}), and not in the core, MGCore(< y, a >, MGRes_{2}), belongs to the bottom two levels of the penetrated core, P enMGCore(< y, a >, MGRes_{2}).

(v) Let {(t_{n}, v_{n}, a)} be a test sequence of the ambient resolution Res(t, v, a). Let m be a level of the ambient resolution Res(t, v, a). The subgroup associated with the m-th level of the resolution P enMGCore(< y, a >, MGRes_{2}) inherits a free decomposition from the abelian decompositions associated with all the levels of the ambient resolution Res(t, v, a) that lie above the m-th level. Let P en_{m} be one of the factors in this free decomposition of the limit group associated with the m-th level of P enMGCore(< y, a > , MGRes_{2}). Then either there exists a (multi-graded) core, MGCore(< y, a >, MGRes_{2}), that is not of maximal complexity, or there exists a firm test sequence of P enMGCore(< y, a >, MGRes_{2}), for which the sequence of specializations of the subgroup P en_{m} corresponding to the given test sequence, converge into a (possibly trivial) action of this subgroup on a real tree Y. The action of the group P en_{n} on the real tree Y is either trivial or it is faithful and geometric, and it is composed from finitely many orbits of discrete and IET components.

Proof: MGCore(< y, a >, MGRes_{2}) is a core resolution, hence, it is a firm geometric subresolution of the multi-graded resolution MGRes_{2}. The resolution, P enMGCore(< y, a >, MGRes_{2}) is a resolution induced by the subgroup associated with the core, MGCore(< y, a >, MGRes_{2}), hence, it is a firm geometric subgroup of the multi-graded resolution MGRes_{2}, and since it is an induced resolution, the abelian decompositions associated with its various levels are inherited from the abelian decompositions associated with the various levels of the multi-graded resolution MGRes_{2}. Therefore, P enMGCore(< y, a >, MGRes_{2}) is a penetrated core resolution, and we get part (i) of the proposition.

Let H be the limit group associated with the formed part of the core MGCore(< y, a >, MGRes_{2}). Each of the abelian decompositions associated with the various levels of P enMGCore(< y, a >, MGRes_{2}), except the bottom two levels, degenerates to a free decomposition H ∗ F^{i+1} ∗ F̂^{i}, where F̂^{i} is the free factor that is dropped in the i-th level of the penetrated core resolution. Since the penetrated core, P enMGCore(< y, a >, MGRes_{2}), is a firm subresolution, for every level i of P enMGCore(< y, a >, MGRes_{2}), except the bottom two levels, the map η_{i} from the limit group associated with the i-th level to the i + 1-th level of P enMGCore(< y, a >, MGRes_{2}), is an isomorphism η_{i} : H ∗ F^{i+1} → H ∗ F^{i+1}, so that for every h ∈ H, η_{i}(h) = h. Hence, for every level i, except the bottom level, the map η_{i} is an isomorphism, which implies part (iii) of the proposition, and, in particular, every QH vertex group in an abelian decomposition associated with one of the levels of P enMGCore(< y, a >, MGRes_{2}), except the two bottom levels, is necessarily of of minimal rank, so we get part (ii) of the proposition.

To prove part (iv), note that if there exists a QH vertex group Q that appears in P enMGCore(< y, a >, MGRes_{2}), and not in the core, MGCore(< y, a > , MGRes_{2}), and Q belongs to one of the bottom two levels of the penetrated core, P enMGCore(< y, a >, MGRes_{2}), then necessarily:

rk(P enMGCore(< y, a >, MGRes_{2}))+factor(P enMGCore(< y, a >, MGRes_{2})) ≤

≤ rk(MGCore(< y, a >, MGRes_{2})) + factor(MGCore(< y, a >, MGRes_{2})) − 1

that clearly implies that the core, MGCore(< y, a >, MGRes_{2}), is not a firm subresolution of the multi-graded resolution MGRes_{2}, a contradiction to our assumptions.

To prove part (v), note that the properties of a test sequence with respect to a given (well-structured) resolution guarantee that the corresponding specializations of a limit group associated with each of the levels of the resolution converge into a faithful action of that limit group on a real tree, and this action is either discrete, or it contains a unique IET component. Let {(t_{n}, v_{n}, a)} be a test sequence of the ambient resolution Res(t, v, a) that satisfies the properties of the construction used in proving proposition 4.10 with respect to the floating QH vertex groups associated with the various levels of the resolution P enMGCore(< y, a >, MGRes_{2}), and let Rlim_{m} be the limit group associated with the m-th level in the ambient resolution Res(t, v, a). Clearly, P en_{m} < Rlim_{m}. Suppose that the sequence of the specializations of the limit group Rlim_{m} corresponding to the test sequence converges to an action of Rlim_{m} on a real tree Y. If the action of Rlim_{m} on the real tree Y is discrete, the corresponding sequence of specializations of its subgroup P en_{m} converge to either trivial or a discrete faithful action on some subtree of Y.

Suppose that the action of Rlim_{m} on the real tree Y contains a unique IET component, and the action of P en_{m} on the real tree Y is non-trivial. Let Θ_{m} be the abelian decomposition associated with Rlim_{m} in the ambient resolution Res(t, v, a), and let T_{m} be the Bass-Serre tree corresponding to the abelian decomposition Θ_{m}. By our assumptions, the abelian decomposition Θ_{m} contains a unique QH vertex group, and (perhaps) several non-QH, non-abelian vertex groups that are all connected to the (unique) QH vertex group. We fix a generating set for the subgroup P en_{m}, P en_{m} =< g_{1}, …, g_{r} >, and look at the finite tree spanned by the points: {t_{0}, g_{1}(t_{0}), …, g_{r}(t_{0})} in the tree T_{m}, where t_{0} ∈ T_{m} is its base point. We denote this finite tree T′_{m}.

Suppose that the finite tree T′_{m} contains a floating QH vertex group with respect to the subgroup P en_{m}, that we denote Q. By lemma 1.5 of [Se4], with the floating QH vertex group Q we can associate a (possibly trivial) free decomposition of the subgroup P en_{m}, P en_{m} = B_{1} ∗ …, B_{k} ∗ F_{s}. Since Q is a floating surface with respect to the subgroup P en_{m}, the construction used to prove proposition 4.10, implies that the action of the subgroup P en_{m} on the real tree Y contains only discrete and IET components if and only if the actions of the various factors B_{1}, …, B_{k} on the real tree Y are either trivial or they contain only discrete and IET components, hence, to complete our argument we may assume that the finite tree T′_{m} contains no floating QH vertex groups with respect to the subgroup P en_{m}.

Suppose that all the QH vertex groups in the finite tree T′_{m} are absorbed QH vertex groups with respect to the subgroup P en_{m}. If all the QH vertex groups in the finite tree T′_{m} are intersected by the subgroup P en_{m} in a subgroup of finite index, the action of the subgroup P en_{m} on the real tree Y is composed from finitely many orbits of IET components. Let Q be an absorbed QH vertex group in the finite tree T′_{m} that does not intersect the subgroup P en_{m} in a subgroup of finite index. We set Ŷ to be a real tree which is the quotient of the real tree Y, obtained by collapsing each of the IET component in Y that are not in the orbit of the IET component associated with Q under the action of P en_{m}. P en_{m} acts on the real tree Ŷ, and by lemma 1.5 of [Se4], P en_{m} inherits an abelian decomposition from its action of the real tree Ŷ that gives rise to a free decomposition: P en_{m} = W_{1} ∗ … ∗ W_{r} ∗ F_{s}. Similarly, we set T̂_{m} to be the discrete tree which is the quotient of the Bass-Serre tree T_{m}, by collapsing all the edges that are not connected to a vertex that is in the orbit of the vertex stabilized by Q under the action of the subgroup P en_{m}. By the standard bass-Serre theory, P en_{m} inherits an abelian decomposition from its action on the discrete tree T′_{m}, which gives rise to a free decomposition: P en_{m} = R_{1} ∗ … ∗ R_{ℓ} ∗ F_{h}. Clearly, s ≥ h, and each of the factors W_{i} can be conjugated into one of the factors R_{j}. If s > h, or s = h and ℓ < r, then a non-trivial subgroup of P en_{m} fixes the vertex stabilized by Q in the Bass-Serre tree T_{m}, which combined with our assumption on Q being an absorbed QH vertex group with respect to the subgroup P en_{m}, implies that we necessarily reduce the rank of the (multi-graded) core if we look at the core associated with the subgroup < P en_{m}, Q >, which implies that there exists a multi-graded core MGCore(< y, a >, MGRes_{2}) that is not of maximal complexity. If s = h and ℓ = r, then necessarily each of the factors W_{i} is conjugate to one of the factors R_{j}, and the action of P en_{m} on the real tree Ŷ is discrete. Hence, in this case the action of P en_{m} on the real tree Y is composed from discrete and IET components if and only if the action of each of the factors W_{1}, …, W_{r} on the real tree Y is either trivial or it is composed from discrete and IET components. Repeating the argument for all the (orbits of) QH vertex groups in the finite tree T′_{m}, we get part (v) of the proposition.

□

Finally, under the assumptions of proposition 4.20, we call the resolution, P enMGCore(< y, a >, MGRes_{2}), the penetrated core resolution of the subgroup < y, a > in the maximal complexity resolution MGRes_{2}.

§5. The Sieve Method (general case)

In the first section of this paper we presented a couple of terminating iterative procedures that together imply that the Boolean algebra of AE sets is invariant under projection in the minimal rank case. Given a set of the form EAE(p), i.e., a set which is a projection of a set that is in the Boolean algebra of AE sets, the first (terminating) iterative procedure described in that section constructs the associated tree of stratified sets, which gives us only finitely many proof systems that are required in order to prove that a specific specialization of the defining parameters is indeed in the elementary set EAE(p) we have started with. The second terminating iterative procedure, constructs a (finite) sequence of bundles of proof statements, that reduces the question of the existence of a possible witness for any given specialization of the defining parameters, to the structure of the bases of these bundles of proof statements. Since by section 3 of [Se3] the base of such a bundle is divided into finitely many stratas and each strata is in the Boolean algebra of AE sets, we were able to finally conclude that the original EAE set is in the Boolean algebra of AE sets.

Our approach towards the analysis of a general EAE set is conceptually similar to the one applied in the minimal rank case, although it is technically much more involved. In the second section of this paper, we used the techniques and the iterative procedure presented in section 4 of [Se4] for validation of a general AE sentence, and presented a (terminating) iterative procedure for the construction of the tree of stratified sets for general EAE predicates, omitting the minimal rank assumption. In the third section we have generalized some of the notions presented in the minimal rank case, and showed that in few cases the procedure used to construct the tree of stratified sets can be slightly modified to give a terminating iterative procedure for the analysis of the bundles of proof statements, which implies that in these cases the EAE sets we started the procedures with are in the Boolean algebra of AE sets. In the fourth section we have presented the (graded, multi-graded) core resolution which is the major additional tool needed in order to further modify the iterative procedure used for the construction of the tree of stratified sets to give a (terminating) iterative procedure for the analysis of the bundles of proof statements for general EAE sets, which finally implies that every EAE set is in the Boolean algebra of AE sets.

Since the iterative procedure we use for the sieve method generalizes the ones used to prove theorem 3.2 in few special cases, we will use the notions and notation used in section 3 of this paper, and continue with the proof of theorem 3.2 for general sets EAE(p). Since the iterative procedure we use is rather involved we present it as we did in presenting the iterative procedure for validation of a sentence in section 4 of [Se4]. We start by presenting the initial step of the procedure, then the second step, the general step, and finally we prove its termination, which allows us to complete the proof of theorem 3.2. Then we apply the procedure we constructed to conclude the proof of theorems 1.4 and 1.3, which finally shows that the Boolean algebra of AE sets is invariant under projection, which is the main goal of this paper.

As in the minimal rank case, and in the few cases presented in section 3, our goal in the analysis of the set T_{1}(p) in the general case, is to find finitely many modular blocks associated with each of the P S limit groups P SHGH, modular blocks in which a generic specialization of the variables w is a witness for the corresponding specialization p_{0} of the defining parameters P =< p >. To achieve this goal, i.e., to find the finitely many modular blocks associated with each of the P S limit groups P SHGH, we present a ”trial and error” procedure, based on the one used to construct the tree of stratified sets (in the general case). The output of the iterative ”trial and error” procedure, i.e., the finitely many modular blocks associated with each P S limit group P SHGH, are later used to derive a predicate in the Boolean algebra of AE predicates that describes the set T_{1}(p).

The initial step of the sieve procedure

Let P =< p > be the group of defining parameters. We start the initial step of the sieve procedure with each of the P S limit groups P SHGH_{i} in parallel, hence, we omit its index. Given a P S limit group P SHGH, we associate its canonical taut graded Makanin-Razborov diagram (with respect to the parameter subgroup P) with it, which contains finitely many graded resolutions which we denote P SHGHRes_{j}, and each graded resolution P SHGHRes_{j} is defined over the rigid or solid limit group P B_{j}(b, p, a). In the sequel, we will treat each strata in the singular locus of the graded resolutions P SHGHRes separately, and do it in parallel.

As we did in the minimal rank case and in the few cases presented in section 3, we start our analysis of the set of valid P S statements by associating with (the completion of) the graded P S resolution P SHGHRes a (canonical) finite collection of Non-Rigid and Non-Solid P S limit groups (definition 1.25), which we call the non-rigid P S limit groups associated with the P S resolution P SHGHRes_{j}, NRgdP S_{1}^{j}, …, NRgdP S_{q}^{j}, and the non-solid P S limit groups associated with P SHGHRes_{j}, NSldP S_{1}^{j}, …, NSldP S_{r}^{j}.

Recall (definition 1.25) that the graded formal closures associated with the collection of non-rigid and non-solid P S limit groups, determine those ”generic” specializations that factor through and are taut with respect to the various P S resolutions P SHGHRes_{j}, but fail to be valid P S statements with respect to the (fixed) proof system because certain specializations that are required to be rigid or strictly solid actually factor through closures that are associated with flexible quotients of the corresponding rigid or solid limit groups. We continue by collecting all the test sequences that factor through the completion of one of the P S resolutions P SHGHRes_{j}, Comp(P SHGHRes_{j}), and for which for at least one of the tuples: (h_{t}^{2}(n), g_{t}^{1}(n), h_{1}(n), w_{n}, p_{n}, a) there exists some specialization g_{m}^{2}(n) so that the (combined) specialization: (g_{t}^{2}(n), h_{t}^{2}(n), g_{t}^{1}(n), h_{1}(n), w_{n}, p_{n}, a) factors through (at least) one of the limit groups W P (HG)^{2}, or there exists a specialization g_{1}(n) so that the (combined) specialization: (g_{1}(n), h_{1}(n), w_{n}, p_{n}, a) factors through (at least) one of the limit groups W P HG, which is not associated with one of the paths associated with our fixed proof system (definition 1.31). The collection of all these (graded) test sequences factor through a (canonical) collection of maximal left P S limit groups: LeftP S_{1}^{j}, …, LeftP S_{m_{j}}^{j}. The analysis of graded formal limit groups presented in section 3 of [Se2] associates (canonically) with each Left PS limit group LeftP S_{i}^{j} a graded formal Makanin-Razborov diagram, and each such graded formal resolution is in fact a one level graded resolution, which is a graded formal closure of the graded resolution P SHGHRes_{j}, GF Cl(P SHGHRes_{j}). Clearly, no specialization that factors through the completion of the resolution P SHGHRes_{j}, Comp(P SHGHRes_{j}), and which is a valid P S statement with respect to our fixed proof system, factors through one of the LeftP S limit groups LeftP S_{1}^{j}, …, LeftP S_{m_{j}}^{j}, and their associated resolutions.

”Generic” specializations that factor through the P S resolutions P SHGHRes can fail to be valid P S statements also if there exist additional rigid or strictly solid specializations of the limit groups P SHG(g_{1}, h_{1}, w, p, a) that are not specified by the given specializations. As in the minimal rank case and the few cases treated in section 3, the ”generic” specializations for which there exists ”surplus” in rigid or strictly solid specializations are collected in extra P S (graded) limit groups and graded resolutions (definition 1.26). We denote the extra P S limit groups associated with the graded P S resolution P SHGHRes_{j}, ExtraP S_{1}^{j}, …, ExtraP S_{ℓ_{j}}^{j}.

The extra P S limit groups and their associated graded formal closures collect all the ”generic” specializations (i.e., all the test sequences) of the P S limit groups P SHGH for which there exist rigid or strictly solid families in addition to those specified by the generic specializations. For a general specialization of the P S limit groups P SHGH, i.e., a specialization which is not necessarily ”generic”, it may as well be that the additional rigid or strictly solid specializations, collected by the extra P S limit groups and their associated graded formal closures, do become flexible or do coincide with the rigid or strictly solid families of the various specializations (g_{t}^{1}, h_{0}^{1}, w_{0}, p_{0}). Let ExtraP SRes be one of the extra P S graded resolutions associated with one of the extra P S graded limit groups ExtraP S_{i}^{j}. We will say that a specialization that factors through and is taut with respect to the extra P S graded resolution ExtraP SRes is collapsed if the variables added for each of the additional rigid or solid specializations (i.e., the ones that were not specified by the proof system) satisfy one of the following:

(1) A specialization of the variables added for one of the additional rigid specializations becomes flexible.

(2) A specialization of the variables added for one of the additional rigid specializations becomes equal to one of rigid specializations specified by the proof system, i.e., with one of the specializations g_{t}^{1} in the specialization: (u, v, (h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}).

(3) A specialization of the variables added for one of the additional strictly solid family of specializations becomes flexible.

(4) A specialization of the variables added for one of the additional strictly solid families of specializations belongs to one of strictly solid families of specializations specified by the proof system, i.e., with a family of one one of the strictly solid family of specializations g_{t}^{1} in the specialization: (u, v, (h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}).

Note that by definition there are only finitely many ways in which a specialization that factors through and is taut with respect to the Extra P S resolution, ExtraP SRes, can become a collapsed specialization. We will call each way a specialization can become collapsed a collapse form.

Defining the finitely many possibilities for collapse forms, we collect all the test sequences of specializations that factor through an extra P S resolution, ExtraP SRes, and are collapsed specializations, in finitely many closures of the resolution P SHGHRes_{j} we have started with, which we call formal collapsed Extra P S (graded) resolutions, precisely as we did in the minimal rank case, and in the cases presented in section 3 (definition 3.6). Collecting all ”collapsed” test sequences in a finite collection of formal collapsed extra P S resolutions, we still need to collect all the collapsed specializations that factor and are taut with respect to a given extra P S resolution ExtraP SRes.

Lemma 5.1. Let ExtraP SRes be one of the extra P S graded resolutions associated with one of the extra P S graded limit groups ExtraP S_{i}^{j}. ExtraP SRes is a graded formal closure of the P S resolution P SHGHRes_{j} we have started the analysis with, so with the resolution ExtraP SRes we can associate its canonical collection of graded auxiliary resolutions (see definition 4.1 in [Se4]). Hence, with the extra P S resolution, ExtraP SRes, we associate one of its associated graded auxiliary resolutions and one of its possible collapse forms. Then:

(i) If the subgroup corresponding to an extra rigid specialization is not elliptic along the entire graded auxiliary resolution, then in any specialization that factors through the extra P S resolution and through its associated graded auxiliary resolution, the corresponding specialization of the subgroup corresponding to that extra rigid specialization is not rigid (i.e., it is flexible).

(ii) If the subgroup corresponding to a non-abelian, non-QH vertex group or to an edge group in the abelian decomposition associated with an extra solid specialization is not elliptic along the entire graded auxiliary resolution, then in any specialization that factors through the extra P S resolution and through its associated graded auxiliary resolution, the corresponding specialization of the subgroup corresponding to that extra solid specialization is not strictly solid.

(iii) Suppose that the subgroup associated with an extra rigid specialization is elliptic along the graded auxiliary resolution. Then for any specialization that factors through our fixed graded auxiliary resolution, and in which the specialization of this extra rigid specialization becomes flexible or coincides with one of the rigid specializations specified by the given collapse form, the corresponding specialization of the graded auxiliary limit group satisfies an additional non-trivial relation.

(iv) Suppose that the subgroup corresponding to all non-abelian, non-QH vertex group in the abelian decomposition associated with an extra solid specialization is elliptic along the entire graded auxiliary resolution. Then it is possible to add some conjugating elements, and elements that correspond to flexible quotients of the associated solid limit groups W P HG, so that for any specialization that factors through our fixed graded auxiliary resolution, and in which the specialization of this extra solid specialization becomes trivial or non strictly solid or belongs to one of the solid families of specializations specified by the given collapse form, in the corresponding specialization of the subgroup generated by the graded auxiliary limit group, the additional conjugating elements, and the elements that correspond to the flexible quotients of the associated solid limit groups W P HG, satisfy relations that correspond to the fixed collapse form.

Proof: If the subgroup corresponding to an extra rigid specialization is not elliptic along the entire graded auxiliary resolution, the graded limit group associated with the auxiliary resolution and the particular extra rigid specialization is necessarily flexible, and we get part (i). By the argument used in proving theorem 1.6 in [Se3], if there exists a non-abelian, non-QH vertex group in the abelian decomposition associated with an extra solid specialization, then the entire collection of specializations that correspond to that extra solid specialization and factor through the graded auxiliary resolution are not strictly solid, which gives part (ii).

Part (iii) follows by definition. If a specialization associated with an extra solid specialization coincides with one of the solid families of specializations that are specified by the proof system, then it is possible to use elements that conjugate the specializations of the non-QH, non-abelian vertex groups in the abelian decomposition of the corresponding solid limit group W P HG associated with the extra solid specialization, into the the specializations of the same vertex groups associated with the corresponding family specified by the proof system. If a specialization associated with an extra solid specialization becomes non-strictly solid, then it is possible to demonstrate that it factors through one of the given closures associated with a flexible quotient of the corresponding solid limit group W P HG, and we get part (iv).

□

We go over all the graded auxiliary resolutions and all the collapse forms associated with the extra P S resolution, ExtraP SRes. Given a graded auxiliary resolution and a collapse form, we add elements that represent the primitive roots of pegs of abelian groups associated with the extra solid resolution, ExtraP SRes, and conjugating elements and the elements corresponding to flexible quotients associated with the given collapse form. We also add new elements that are set to be the primitive roots of pegs of (pegged) abelian groups along the extra P S resolution, ExtraP SRes, for all such pegs that do not belong to a non-QH, nonabelian vertex groups in the terminal (rigid or solid) limit group of the extra P S resolution, ExtraP SRes. The graded abelian decomposition of our given graded auxiliary limit group can get degenerate in finitely many ways (cf. section 11 of [Se1] and section 3 of [Se3]), each possible degeneracy gives rise to a set of equations on the specializations of the graded auxiliary limit group, hence, with each possible degeneracy we associate a (canonical) finite collection of maximal proper quotients of the given graded auxiliary limit group, and treat each of these proper quotients in parallel.

Given the auxiliary limit group or one of its degenerate quotients, and the additional conjugating elements and variables corresponding to flexible quotients according to the given collapse form, and elements that represent primitive roots of pegs of abelian vertex groups, we look at all the specializations of the graded auxiliary limit group (or its degenerate quotient) associated with the given graded auxiliary resolution, and corresponding specializations of primitive roots of the pegs along the extra P S resolution, ExtraP SRes, for which there exist corresponding specializations of the additional elements corresponding to the conjugating elements and flexible quotients, so that the (combined) specializations satisfy the system of equations associated with the given collapse form associated with the extra P S resolution ExtraP SRes. By our standard method presented in section 5 of [Se1], this collection of specializations factor through a canonical (finite) collection of maximal limit groups, which we call collapsed extra P S limit groups, and denote:

CollapseExtraP S_{1}, …, CollpaseExtraP S_{d}.

By construction, if p_{0} ∈ T_{1}(p) then there must exist a valid P S statement of the form: ((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) that factors through one the of the P S resolutions P SHGHRes_{j} constructed with respect to all proof systems of depth 1. By proposition 3.8, the sets T SP S(p) associated with the various P S resolutions P SHGHRes, i.e., the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of the P S resolutions P SHGHRes_{j}, is in the Boolean algebra of AE sets. By theorem 3.7, if there exists a valid P S statement that factors through a P S resolution P SHGHRes_{j}, then either there exists a test sequence of valid P S statements that factor through the P S resolution, or there must exist a specialization that factors through one of the graded auxiliary resolutions associated with one of the P S resolutions P SHGHRes, and one of its associated collapsed extra P S limit groups.

To analyze the remaining set of valid P S statements we construct an iterative procedure that produces a sequence of multi-graded well-separated resolutions, their completions and core resolutions. The iterative procedure is based on the iterative procedure used for the construction of the tree of stratified sets which is based on the iterative procedure for validation of a sentence, presented in section 4 of [Se4]. We start the iterative procedure with each of the P S limit groups P SHGH in parallel, and with each P S limit group we associate its collection of P S resolutions, P SHGHRes_{j}, their associated extra P S resolutions, ExtraP SRes, the associated graded auxiliary resolutions and graded auxiliary limit group, and the associated Collapsed extra P S limit groups CollapseExtraP S. As in the iterative procedure for the construction of the tree of stratified sets and the iterative procedure for validation of a sentence, we divide the construction of the graded developing resolution and the associated anvil constructed in the initial step of our sieve procedure into several cases, depending on the structure of the extra P S resolution we start the initial step with, and the structure of the multi-graded resolutions constructed along the initial step.

(1) Let CollapseExtraP S(t, v, h_{2}, g_{1}, h_{1}, w, p, a) be the collapse extra P S resolution we have started (this branch of the) initial step with. Let Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) be the the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the collapse extra P S limit group CollapseExtraP S(t, v, h_{2}, g_{1}, h_{1}, w, p, a). If Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of the P S limit group P SHGH(h_{2}, g_{1}, h_{1}, w, p, a) we have started this process with, we continue this branch of the iterative procedure, by starting the initial step of the procedure with the limit group Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a).

(2) At this stage we may assume that Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the P S limit group P SHGH(h_{2}, g_{1}, h_{1}, w, p, a). We set the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} to be the non-abelian, non-QH vertex groups in the abelian decomposition associated with the top level of the extra P S resolution ExtraP SRes. Note that the parameter subgroup, AP =< p, a >, is, by definition, a subgroup of Base_{2,1}^{1}. With the collapse extra P S limit group, CollapseExtraP S(t, v, h_{2}, g_{1}, h_{1}, w, p, a), and its subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} we associate the completions of the various resolutions that appear in its taut multi-graded Makanin-Razborov diagram:

MGQRes_{1}(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), …
…, MGQRes_{q}(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a).

We continue with each of the multi-graded resolutions in parallel, hence, we omit the index of the specific resolution we continue with.

Since the extra P S resolution ExtraP SRes, we have started the initial step with, is well-separated, with each QH vertex groups in one of the abelian decompositions associated with ExtraP SRes there is an associated collection of s.c.c. that are mapped to the trivial element in the next level of the extra P S resolution ExtraP SRes. Each QH vertex group in the graded abelian decomposition associated with the top level of the extra P S resolution ExtraP SRes naturally inherits a sequence of abelian decompositions from a multi-graded resolution MGQRes_{j}. If for some such QH vertex group Q, this sequence of multigraded abelian decompositions is not compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the next level of the extra P S resolution ExtraP SRes we omit the multi-graded resolution MGQRes_{j} from the list of completions of resolutions of the collapse extra P S limit group CollapseExtraP S.

By theorem 4.12, the complexity of the multi-graded core of the resolution MGQRes:

MGCore(< h_{2}, g_{1}, h_{1}, g_{1}, w, p, a >, MGQRes)

is bounded by the complexity of the abelian decomposition associated with the top level of the extra P S resolution ExtraP SRes we have started the initial step with, and if the complexities are equal, then the structure of the core:

MGCore(< h_{2}, g_{1}, h_{1}, g_{1}, w, p, a >, MGQRes)

is identical to the structure of the abelian decomposition associated with the top level of the extra P S resolution ExtraP SRes. In this part of the initial step of the procedure we will also assume that the complexity of the core:

MGCore(< h_{2}, g_{1}, h_{1}, g_{1}, w, p, a >, MGQRes)

is strictly smaller than the complexity of the abelian decomposition associated with the top level of the extra P S resolution. The case of equality in these complexities is treated in later parts of the initial step. In parallel with proposition 2.2, the image of the limit group Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the terminal rigid or solid limit group of the multi-graded resolution, MGQRes, is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a).

Proposition 5.2. Let MGQRes(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be one of the multi-graded resolutions constructed above. By construction, the limit group Q^{1}(h_{2}, g_{1}, h_{1}, , w, p, a) is mapped into the limit group associated with each of the levels of the multi-graded resolution MGQRes. Let Q_{term}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) be the image of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the terminal (rigid or solid) limit group of MGQRes. Then the terminal limit group of the multi-graded resolution MGQRes can be replaced by a collection of finitely many terminal multi-graded limit groups in which Q_{term}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a), and a finite collection of terminal rigid or solid graded limit groups which are rigid or solid with respect to the parameter subgroup P =< p >.

Proof: Identical to the proof of proposition 2.2.

□

By proposition 5.2 we can either omit the graded resolution MGQRes from our list of graded resolutions, or we may assume that either the image of the subgroup Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the terminal graded limit group of MGQRes, Q_{term}^{1}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a), or the terminal graded limit group of MGQRes, is rigid or solid with respect to the parameter subgroup P =< p >. To continue our treatment of multi-graded resolutions that are not of maximal complexity we also need the following lemma, that is similar to lemma 2.3.

Lemma 5.3. Let MGQRes(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be one of the multi-graded resolutions in our list that is not of maximal complexity. Let Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a), Q^{1}(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), be the subgroups generated by < h_{2}, g_{1}, h_{1}, w, p, a >, < t, v, h_{2}, g_{1}, h_{1}, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a >, in correspondence, in the collapse P S limit group associated with the multi-graded resolution CollapseExtraP S(t, v, h_{2}, g_{1}, h_{1}, w, p, a). Let

Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a), Q_{2}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a)

be the images in the limit group associated with the second level of MGQRes, GQlim_{2}(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), of the subgroups: Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) and Q^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), in correspondence. Then either Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) or Q_{2}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a).

Proof: The lemma is simply a basic property of a multi-graded resolution.

□

Suppose that the image of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the multi-graded limit group associated with the second level of the multi-graded resolution MGQRes, Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a). In this case we do the following. With the subgroup Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) we associate the graded resolutions that appear in its graded taut Makanin-Razborov diagram with respect to the parameter subgroup P =< p >:

GQRes_{1}(h_{2}, g_{1}, h_{1}, w, p, a), …, GQRes_{t}(h_{2}, g_{1}, h_{1}, w, p, a).

We continue with each of the graded resolutions GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) in parallel. If the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with the resolution GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a), we replace the graded resolution GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) by starting part (2) of the the initial step with the multi-graded resolution obtained from the top level of the multi-graded resolution MGQRes, by replacing its second-level limit group Q_{2}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) with the maximal limit groups obtained from all specializations that factor through both Q_{2}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) and the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with the graded resolution GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a). Hence, for the rest of this part we may assume that the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a).

Let T MGQRes be a one level resolution corresponding to the top level of the multi-graded resolution MGQRes. Let CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) be the graded resolution obtained from the resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the core, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a > , T MGQRes) followed by the graded resolution GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a). If the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the graded resolution CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a), we replace the graded resolution GQRes_{j} by starting the initial step of the procedure with the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the resolution CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a). Hence, we may assume that for the rest of this part the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a).

We now treat each of the graded resolutions GQRes_{j} and their associated resolution CRes_{j} in parallel. Let P B^{2}(b_{2}, p, a) be the terminal rigid or solid limit group of the graded resolution GQRes_{j} (which is also the terminal rigid or solid limit group of its associated resolution CRes_{j}). The graded resolution CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) is composed from the resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the core of the multi-graded resolution T MGQRes followed by the resolution GQRes_{j}. In particular, with every QH vertex group in a graded abelian decomposition associated with one of the levels of the graded resolution CRes_{j}, we can naturally associate either a QH vertex group in an abelian decomposition associated with one of the levels of GQRes_{j}, or a finite index subgroup of a QH vertex group in the abelian decomposition associated with (the core resolution of) the multi-graded resolution T MGQRes. Therefore, with each QH vertex group of CRes_{j} we can naturally associate a modular group, which is the modular group of the QH vertex group in GQRes_{j} or T MGQRes it is associated with. This assignment of modular groups to the QH vertex groups of CRes_{j} naturally extends to (reduced) modular groups associated with each of the levels of the resolution CRes_{j}. With the graded resolution CRes_{j}, equipped with these reduced modular groups, we associate a finite collection of framed resolutions.

Definition 5.4. Let Res(y, p, a) be a well-structured, complete graded resolution with reduced modular groups. By construction, with each QH vertex group Q in one of the levels of Res(y, a), there is an associated QH subgroup Q′, so that Q embeds into Q′ as a subgroup of finite index. Let (y_{0}, p_{0}, a) be a specialization that factors through the graded resolution Res(y, p, a) with its reduced modular groups, and let Q be a QH subgroup in an abelian decomposition associated with one of the levels of Res(y, p, a), and let Q′ be the QH subgroup that contains Q as a subgroup of finite index. The specialization (y_{0}, p_{0}, a) naturally gives rise to a specialization of the QH vertex Q, that may possibly be extended to some subgroup Q̂, Q < Q̂ < Q′.

Let Q_{1}, …, Q_{ℓ} be the collection of QH subgroups associated with the various levels of Res(y, p, a) and let Q′_{1}, …, Q′_{ℓ} be the subgroups containing them as subgroups of finite index. Let Q̂_{1}, …, Q̂_{ℓ} be subgroups for which Q_{i} < Q̂_{i} < Q′_{i}, for every i, 1 ≤ i ≤ ℓ. We say that the collection Q̂_{1}, …, Q̂_{ℓ} is a frame for a specialization (y_{0}, p_{0}, a), if the specialization (y_{0}, p_{0}, a) extends to the subgroups Q̂_{1}, …, Q̂_{ℓ}, but not to any subgroup Q̃_{i} < Q′_{i} that properly contains Q̂_{i}, for any i, 1 ≤ i ≤ ℓ. We denote the extension of the specialization (y_{0}, p_{0}, a) to the subgroups Q̂_{1}, …, Q̂_{ℓ}, (q_{0}, y_{0}, p_{0}, a), and say that it is a framed specialization (with respect to the frame Q̂_{1}, …, Q̂_{ℓ}). We say that a test sequence that factors through Res(y, p, a) is framed with respect to the frame Q̂_{1}, …, Q̂_{ℓ}, if each specialization in the test sequence is framed with respect to the given frame.

Clearly, there are finitely many frames associated with the graded resolution Res(y, p, a). Fixing a frame Q̂_{1}, …, Q̂_{ℓ} associated with the resolution Res(y, p, a), we look at the collection of framed test sequences associated with it. By the techniques presented in [Se2], the entire collection of framed test sequences with respect to the given frame Q̂_{1}, …, Q̂_{ℓ}, factor through a (canonical) finite collection of graded resolutions obtained from closures of the graded resolution Res(y, p, a) by enlarging each of the subgroups Q_{i} to be Q̂_{i}, and for each QH vertex group Q̂_{i} its image in the lower levels of the corresponding closure of the resolution Res(y, p, a). We call the (finite) collection of the graded resolutions obtained from the entire collection of framed test sequences with respect to all possible frames, framed resolutions associated with the resolution Res(y, p, a). With each framed resolution we naturally associate its frame.

With the graded resolution CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) we have associated reduced modular groups. Hence, with CRes_{j} we can associate a (canonical) finite collection of framed resolutions. We treat the framed resolutions associated with CRes_{j} in parallel. Let F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a) be such a framed resolution. The top part of F rmCRes corresponds to the (core of the) multi-graded resolution T MGQRes, and its bottom part corresponds to the various levels in the resolution GQRes_{j}. If the limit group associated with the top level of GQRes_{j} in the framed resolution F rmCRes is a proper quotient of the limit group Q_{2}^{1}, we replace the framed resolution F rmCRes by starting part (2) of the initial step with the graded resolution obtained from T MGQRes by replacing its terminal limit group Q_{2}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) with the finite collection of maximal limit groups obtained from all specializations that factor through both Q_{2}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) and the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with the top level of GQRes_{j} in the framed resolution F rmCRes. Hence, for the continuation we may assume that the limit group associated with the top level of GQRes_{j} in the framed resolution F rmCRes is isomorphic to the limit group Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a).

Repeating what we did with the P S resolution P SHGH_{j} we have started the procedure with, we associate with the framed resolution F rmCRes a (canonical) finite collection of Non-Rigid and Non-Solid P S resolutions, a collection of Left P S resolutions, extra P S resolutions, and formal collapse extra P S resolutions. If every test sequence that factors through an extra P S resolution associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a), factors through a framed resolution with a bigger frame than the one associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a), we exclude this extra P S resolution from the finite collection of P S resolutions associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a). With each of the extra P S resolutions associated with the framed resolution F rmCRes, we associate a (canonical) finite collection of graded auxiliary resolutions, and with each graded auxiliary resolution we associate its terminal graded limit group, which we call graded auxiliary limit group. With each of the extra P S resolutions we associate finitely many collapse forms. Given an extra P S resolution associated with the framed resolution, F rmCRes, one of its associated graded auxiliary resolutions, and one of its associated collapse forms, we construct a (canonical) finite collection of collapse extra P S limit groups, precisely as we associated collapse extra P S limit groups with an extra P S resolution associated with the resolution P SHGHRes_{j} we have started our iterative procedure with.

As in our treatment of the graded P S resolutions P SHGHRes_{j} we have started this procedure with, if a valid P S statement factors through the framed resolution F rmCRes then either there exists a valid P S statement that factors through one of the collapsed extra P S limit groups and their associated graded auxiliary resolutions, or there exists a ”generic” valid P S statement, i.e., a test sequence of valid P S statements that factor through the resolution F rmCRes.

Lemma 5.5. Suppose that there exists a valid P S statement:

((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a)

that factors through the framed resolution F rmCRes. Then one of the following holds:

(1) there exists a test sequence of valid P S statements (with respect to our fixed proof system):

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{ν(ps)}^{2}(n), g_{ν(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

that factor through the resolution CRes_{j} (note that the specialization p_{0} of the defining parameters p are fixed along the entire test sequence).

(2) the valid P S statement factors through one of the graded auxiliary resolutions associated with an extra P S resolution associated with one of the framed resolutions associated with the resolution CRes_{j} and one of its associated collapsed extra P S limit groups.

Proof: Similar to the proof of theorems 3.7 and 1.32.

□

As in our treatment of the P S resolutions P SHGHRes, at this point we analyze the set of specializations of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through the various resolutions CRes_{j}.

Proposition 5.6. Let T SP S_{j}(p) be the set of specializations p_{0} of the defining parameters P =< p >, for which there exists a test sequence of valid P S statements:

(v_{n}, (h_{1}^{2}(n), g_{1}^{1}(n)), …, (h_{ν(ps)}^{2}(n), g_{ν(ps)}^{1}(n)), h_{1}(n), w_{n}, p_{0}, a))

that factor through the completion of the P S resolution CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a), Comp(CRes_{j}). Then T SP S_{j}(p) is in the Boolean algebra of AE sets.

Proof: Identical to the proof of propositions 3.8 and 1.33.

□

By lemma 5.5 and proposition 5.6, given an extra P S resolution associated with the framed resolution F rmCRes, and one of its associated collapse forms, we need to continue only with those valid P S statements that factor though that extra P S resolutions, and through one of the collapse extra P S limit groups associated with it. To collect these valid P S statements, we start with the extra P S resolutions, ExtraP SRes, associated with the framed resolution, F rmCRes, and work with them in parallel. With such an extra P S resolution, ExtraP SRes, we associate its canonical multi-graded taut Makanin-Razborov diagram with respect to the non-QH, non-abelian vertex groups and edge groups in the graded abelian decomposition (with respect to the parameter subgroup P =< p >) associated with its (second level) limit group Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a). We call each of the (finitely many) multi-graded resolutions in this multigraded taut Makanin-Razborov diagram, a graded auxiliary resolution, and its terminal limit group, a graded auxiliary limit group. With each of the extra P S resolutions we associate finitely many collapse forms. Given an extra P S resolution associated with the framed resolution F rmCRes, one of its graded auxiliary resolutions, and one of its associated collapse forms, we construct a (canonical) finite collection of collapse extra P S limit groups, precisely as we associated collapse extra P S limit groups with an extra P S resolution associated with one of the resolutions P SHGHRes_{j} we have started our iterative procedure with. We further associate a (canonical) collection of graded auxiliary resolutions and graded auxiliary limit groups with the resolution MGQRes, with respect to the non-QH, non-abelian vertex groups and edge groups in the graded abelian decomposition associated with its (second level) limit group Q_{2}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a).

We continue with each of the graded auxiliary resolutions and limit groups associated with the extra P S resolutions, and their associated collapse extra P S limit groups in parallel. Given a graded auxiliary limit group associated with an extra P S resolutions, ExtraP SRes, one of its associated collapse extra P S limit groups, CollapseExtraP S, and a graded auxiliary limit group associated with MGQRes, we do the following. We set the developing resolution, Dvlp(MGQRes), to be the extra P S resolution, ExtraP SRes. With this developing resolution we associate a finite collection of anvils. First we associate the group GComb to be the group generated by the collapse extra P S limit group associated with the graded auxiliary resolution that is associated with the extra P S resolution, ExtraP SRes, and the given graded auxiliary limit group associated with MGQRes, by identifying the images of the subgroups associated with the levels that lie above the level associated with Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) (including the level associated with Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a)) in the collapse extra P S limit group associated with the extra P S resolution, ExtraP SRes, with the image of these subgroups in the given graded auxiliary limit group associated with MGQRes. Then we look at the (canonical) finite collection of maximal limit groups that contain the entire collection of valid P S statements, that factor through the amalgamated product GComb. Clearly, each such maximal limit group is a quotient of the group GComb. We set each of these maximal limit groups to be an anvil, Anv(MGQRes), associated with the extra P S resolution, ExtrapSRes, and its associated graded auxiliary resolution AuxGRes, and with the multi-graded resolution MGQRes.

Suppose that Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a). In this case we continue to the next level of the multi-graded quotient resolution MGQRes. Note that by corollary 4.15, since the multi-graded resolution MGQRes is not of maximal complexity, the (multi-graded) core associated with each of its levels is not of maximal complexity as well. If for some level j of the multi-graded resolution MGQRes, the image of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with this level, Q_{j}^{1}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a), then from the highest such level j, we can continue as in case Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a), and associate with the multi-graded resolution MGQRes a finite collection of resolutions composed from the resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the top j levels of the multi-graded resolution MGQRes, followed by each of the various resolutions in the taut graded Makanin-Razborov diagram of Q_{j}^{1}(h_{2}, g_{1}, h_{1}, w, p, a). With each such resolution, we (canonically) associate a finite collection of framed resolutions, and with each framed resolution, a canonical collection of Non-Rigid and Non-Solid P S resolutions, a collection of Left P S resolutions, extra P S resolutions, and formal collapse extra P S resolutions. We set each the extra P S resolutions to be a developing resolution, and with it we associate a (canonical) finite collection of graded auxiliary resolutions, collapse extra P S limit groups, and anvils, precisely as we did in case Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a).

Finally, suppose that for every level j, the image of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the j-th level of the multi-graded resolution MGQRes, Q_{j}^{1}(h_{2}, g_{1}, h_{1}, w, p, a), is isomorphic to Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a). In this case, by lemma 5.2, the terminal limit group of the multi-graded resolution MGQRes, Q_{term}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), is rigid or solid with respect to the parameter subgroup P =< p >. Let P B^{1}(b_{1}, p, a) be the terminal rigid or solid limit group of the graded resolution MGQRes. We set the graded resolution CRes(h_{2}, g_{2}, h_{1}, w, p, a), to be the resolution induced by the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > from the sequence of core resolutions associated with the various levels of the resolution MGQRes. With the graded resolution CRes(h_{2}, g_{1}, h_{1}, w, p, a) we associate a finite (canonical) collection of framed resolutions (see definition 5.4). With each of the framed resolutions associated with CRes(h_{2}, g_{1}, h_{1}, w, p, a) we associate a (canonical) finite collection of Non-Rigid and Non-Solid P S resolutions, a collection of Left P S resolutions, extra P S resolutions, and formal collapse extra P S resolutions. If every test sequence that factors through an extra P S resolution associated with the framed resolution F rmCRes, factors through a framed resolution with a bigger frame than the one associated with the framed resolution F rmCRes, we exclude this extra P S resolution from the finite collection of P S resolutions associated with the framed resolution F rmCRes. With each of the extra P S resolutions associated with the framed resolution F rmCRes, we associate a (canonical) finite collection of graded auxiliary resolutions and collapse extra P S limit groups, precisely as we associated collapse extra P S resolution with an extra P S resolution associated with the resolutions P SHGHRes_{j} we have started our iterative procedure with. We set the developing resolution, Dvlp(MGQRes), to be the extra P S resolution, ExtraP SRes. With each couple of a developing resolution and a collapse extra P S limit group, we associate a finite collection of anvils, precisely as we did in case Q_{2}^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a).

(3) By part (1) we may assume that Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the P S limit group P SHGH we have started the initial step with. Part (2) treat the case in which the multi-graded core of the multi-graded resolution MGQRes is not of maximal possible complexity. Hence, the only case left in presenting the initial step of our procedure is the case of a multi-graded core: MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQRes) of maximal possible complexity. In this case, by theorem 4.12, the maximal complexity graded core has the same structure as the graded abelian decomposition associated with the top level of the extra P S resolution, ExtraP SRes, we have started the initial step with. We treat this case as we treated it in the minimal rank case, and in the procedure for validation of a sentence. In case the structure of the core: MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQRes) and the top level of the extra P S resolution, ExtraP SRes, have the same structure, we use the modular groups associated with the abelian decomposition associated with the top level of ExtraP SRes, that are identical to the modular groups associated with the multi-graded core MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQRes), to map the subgroup Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) into its image in the subgroup associated with the second level of the extra P S resolution, ExtraP SRes. We now set the subgroups Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1} to be the subgroups of the extra P S resolution, ExtraP SRes, corresponding to the non-abelian, non-QH vertex groups in the multi-graded abelian decomposition associated with the second level of the extra P S resolution, ExtraP SRes.

At this point we analyze the image of the subgroup Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the terminal limit group of the multi-graded resolution MGQRes,

Q_{term}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a)

precisely as we analyzed the image of the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > in the collapse P S limit group we have started the initial step with. We start by associating with the terminal limit group Q_{term}^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) its canonical collection of taut multi-graded resolutions with respect to the subgroups: Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1}. Then we analyze each of the obtained multigraded resolutions according to parts (1)-(3). If the core of one of these multi-graded resolutions is of maximal complexity, i.e., if the multi-graded core associated with such a multi-graded resolution has the same structure as the graded abelian decomposition associated with the second level of the extra P S resolution, ExtraP SRes, we have started the initial step with, we continue by analyzing the image of the subgroup Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) in the terminal limit group of that multi-graded resolution. As long as the multi-graded core resolutions associated with the constructed multi-graded resolutions are of maximal possible complexity, i.e., their structure is identical to the structure of the graded abelian decomposition associated with the corresponding level of the extra P S resolution, ExtraP SRes, we have started the initial step with, we continue analyzing multi-graded resolutions associated with the next level of the extra P S resolution, ExtraP SRes.

If there exists a level for which one of the parts (1)-(2) applies, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions, a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(2) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolutions, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that the sequence of multi-graded core resolutions of the multigraded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the graded abelian decomposition associated with the corresponding level of the extra P S resolution, ExtraP SRes, we have started the initial step of the procedure with. We set SCRes(h_{2}, g_{1}, h_{1}, w, p, a) to be the graded resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the multi-graded resolutions constructed along the various levels of the process described above. If every ungraded resolution that factors through the graded resolution SCRes(h_{2}, g_{1}, h_{1}, w, p, a), factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the initial step of the sieve procedure with, we do not continue to the next step of the sieve procedure with the given sequence of multi-graded resolutions we have constructed, and call them a terminal resolution. If there are ungraded resolutions that factor through SCRes, but do not factor through any of the Non-Rigid, Non-Solid nor Left P S resolutions nor the formal collapse extra P S resolutions associated with the extra P S resolutions we have started the initial step of the procedure with, we do the following.

We start by replacing the core resolutions associated with the multigraded resolutions constructed in the various levels, and with the subgroup, < h_{2}, g_{1}, h_{1}, w, p, a >, by the corresponding penetrated core resolutions (see definition 4.19). Note that by construction, the original core resolutions are embedded into the corresponding penetrated core resolutions. We set the graded resolution P enSCRes(r, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated in the various levels with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. From the corresponding collection of the constructed multi-graded resolutions, we naturally obtain a graded resolution, GRes(t, v, h_{2}, g_{1}, h_{1}, w, p, a), which is graded with respect to the defining parameter subgroup P =< p >. By construction, the resolution P enSCRes is embedded into the completion of the graded resolution GRes.

With the graded resolution GRes, we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We further associate with the graded resolution GRes the graded resolution SCRes, which we call first sculpted resolution, and denote both as SCRes_{1}^{1,1}(h_{2}, g_{1}, h_{1}, w, p, a) and SCRes_{1}^{1,2}(h_{2}, g_{1}, h_{1}, w, p, a), and the graded resolution P enSCRes, which we call penetrated sculpted resolution, and denote P enSCRes_{1}^{1,2}(r, h_{2}, g_{1}, h_{1}, w, p, a). Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, and a finite collection of collapse extra P S limit groups, which are set to be the anvils.

II:The second step of the sieve procedure

In the initial step we have constructed finitely many multi-graded resolutions and their associated multi-graded core resolutions, and with each multi-graded resolution we have associated a (canonical) finite collection of Non-rigid, Non-solid, Left, and formal collapse P S resolutions, a finite collection of extra P S resolutions, which are set to be the developing resolutions, and finally a finite collection of collapse P S limit groups and anvils, and possibly, if all the multi-graded core resolutions are of maximal possible complexity (the last case), a sculpted resolution. In this part we present the second step of our iterative sieve procedure. We start the second step of the sieve procedure with the (finite) collection of multi-graded resolutions constructed in the previous step, and their associated developing (and possible sculpted) resolutions, collapse P S limit groups, graded auxiliary resolutions and anvils. With each developing resolution and anvil we further associate the collection of multi-graded resolutions and their multi-graded core resolutions, and the multi-graded limit groups associated with their various levels.

In each step of the iterative sieve procedure we analyze the limit groups associated with the anvils that are associated with developing resolutions constructed in the previous steps of the procedure, where these developing resolutions are not terminal resolutions. Since we treat the anvils in parallel, we present the second step of the sieve procedure with one of the anvils, Anv(MGQRes)(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a). We construct the developing resolutions and anvils associated with the anvil, Anv(MGQRes), iteratively, based on the way we have analyzed quotient resolutions in the general step of the procedure for the construction of the tree of stratified sets. The analysis we carry out in the second step depends on the structure of the multi-graded resolution, MGQRes, its multi-graded core resolution, and its associated developing resolution. As in the initial step of the sieve procedure, our aim is to obtain a strict decrease in either the Zariski closure or the complexity of the core resolution associated with some level of the staggered data structure we construct.

(1) Let Dvlp(MGQRes) be such a developing resolution, and let:

Anv(MGQRes)(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a)

be an anvil associated with it. Let: Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) be the graded limit group generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the anvil Anv(MGQRes). If Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of the P S limit group,

P SHGH(h_{2}, g_{1}, h_{1}, w, p, a)

we have started the initial step with, we continue this branch of the iterative procedure, by starting the initial step of the sieve procedure with the graded limit group Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) instead of the P S limit group P SHGH.

(2) At this stage we may assume that Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to P SHGH(h_{2}, g_{1}, h_{1}, w, p, a). At this part we assume that the core of the multi-graded resolution MGQRes(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQRes), is of maximal possible complexity, i.e., it has the same structure as the graded abelian decomposition associated with the top level of the extra P S resolution, ExtraP SRes, we have started the initial step with. Let:

MGQ^{2}Res_{1}(u, t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), …

…, MGQ^{2}Res_{q}(u, t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

be the completions of the multi-graded resolutions in the taut multi-graded Makanin-Razborov diagram of the anvil, Anv(MGQRes), with respect to the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}. We will treat the multi-graded resolutions MGQ^{2}Res_{j} in parallel, hence, we omit their index.

Since the extra P S resolution ExtraP SRes(v, h_{2}, g_{1}, h_{1}, w, p, a), we have started the initial step with, is well-separated, with each QH vertex groups in one of the abelian decompositions associated with ExtraP SRes there is an associated collection of s.c.c. that are mapped to the trivial element in the next level of the extra P S resolution ExtraP SRes. Each QH vertex group in the graded abelian decomposition associated with the top level of the extra P S resolution ExtraP SRes naturally inherits a sequence of abelian decompositions from a multi-graded resolution MGQ^{2}Res_{j}. If for some such QH vertex group Q, this sequence of multigraded abelian decompositions is not compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the next level of the extra P S resolution ExtraP SRes, we omit the multi-graded resolution MGQ^{2}Res_{j} from the list of completions of resolutions of the anvil, Anv(MGQRes).

By theorem 4.12 the complexity of the core resolution associated with the (second) multi-graded resolution MGQ^{2}Res, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a > , MGQ^{2}Res), is bounded by the complexity of the graded abelian decomposition associated with the top level of the extra P S resolution, ExtraP SRes, we have started the initial step with, and if these complexities are equal, then the structure of the core of MGQ^{2}Res is identical to the structure of the graded abelian decomposition associated with the top level of ExtraP SRes. In this part of the second step of the sieve procedure we will also assume that the complexity of the core of the (second) multi-graded resolution MGQ^{2}Res is strictly smaller than the complexity of the graded abelian decomposition associated with the top level of the extra P S resolution, ExtraP SRes, we have started the initial step with.

In this case, we treat the (second) multi-graded resolution MGQ^{2}Res precisely as we treated the multi-graded resolutions MGQRes in the initial step of sieve procedure, i.e., according to parts (1)-(2) of the initial step.

(3) At this part we may assume that Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the P S limit group P SHGH(h_{2}, g_{1}, h_{1}, w, p, a) we have started with. We also assume that the complexity of the core resolution, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a > , MGQRes), is strictly smaller than the complexity of the graded abelian decomposition associated with the top level of the extra P S resolution, ExtraP SRes, we have started the initial step with. Let Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2} be the the non-abelian, non-QH vertex groups and edge groups in the abelian decomposition associated with the top level of the anvil, Anv(MGQRes), i.e., the non-abelian, non-QH vertex groups and edge groups in the graded abelian JSJ decomposition associated with the collapsed P S limit group we have started the initial step with. Note that the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} are naturally mapped into the subgroups Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}. Let:

MGQ^{2}Res_{1}(u, t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{2}, …, , Base_{2,v_{2}}^{2}, a), …

…, MGQ^{2}Res_{q}(u, t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a)

be the completions of the resolutions in the taut multi-graded Makanin-Razborov diagram of the limit group associated with the anvil, Anv(MGQRes), with respect to the subgroups Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}. Since we treat these multi-graded resolutions in parallel, we will omit their indices in the sequel.

If the graded core associated with the graded abelian decomposition associated with the top level of the multi-graded resolution, MGQ^{2}Res, is trivial, we replace the anvil, Anv(MGQRes), by its proper quotient, which is the limit group associated with the second level of the multi-graded resolution MGQ^{2}Res, and change the multi-graded resolution MGQ^{2}Res by collapsing its top level into the limit group associated with the second level. Repeating this procedure inductively, we may assume that the multi-graded core associated with the top level of MGQ^{2}Res is not trivial.

Let Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) be the subgroup generated by < t, v, h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the (second) multi-graded resolution MGQ^{2}Res. By construction, Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a quotient of the collapse Extra P S limit group, CollapseExtraP S(t, v, h_{2}, g_{1}, h_{1}, w, p, a), we have started the initial step of the sieve procedure with. If Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of CollapseExtraP S, we replace the (second) multigraded resolution MGQ^{2}Res, by starting the initial step of the procedure with the subgroup Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a). By theorem 4.17 only multi-graded resolutions

MGQRes(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

with maximal complexity multi-graded core, i.e., with multi-graded core that have the same structure as the graded abelian decomposition associated with the top level of the extra P S resolution, ExtraP SRes, we have started the initial step with, can be reference resolutions to those multigraded resolutions in the taut multi-graded Makanin-Razborov diagram of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) that have maximal complexity multi-graded core. Hence, all the specializations of the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > that factor through the completion of a multi-graded resolution of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) that have maximal complexity multi-graded core, factor through at least one of the completions of the multi-graded resolutions MGQRes in the taut multi-graded Makanin-Razborov diagram of P SHGH(h_{2}, g_{1}, h_{1}, w, p, a) that have maximal complexity core. Since in this part we have assumed that the multi-graded resolution MGQRes associated with the anvil we have started the second step with, does not have a maximal complexity multigraded core resolution, we may replace the second multi-graded resolution MGQ^{2}Res, by only those multi-graded resolutions in the taut graded Makanin-Razborov diagram of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), for which their associated multi-graded core are not of maximal complexity. We analyze these multi-graded resolutions according to steps (1)-(2) of the initial step of the sieve method.

(4) With the notation of part (3), at this part we assume that the multi-graded resolution MGQRes is not of maximal possible complexity, and that for our given second multi-graded resolution MGQ^{2}Res, the corresponding subgroup Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the collapse P S limit group we have started the initial step with.

Since the multi-graded resolution MGQRes, used in the initial step to construct the anvil, Anv(MGQRes), is well-separated, with each QH vertex groups in one the abelian decompositions associated with MGQRes, there is an associated collection of s.c.c. that are mapped to the trivial element in the next level of the multi-graded resolution MGQRes. Each QH vertex group in the graded abelian decomposition associated with the top level of the multi-graded resolution MGQRes naturally inherits a sequence of abelian decompositions from a (second) multi-graded resolution MGQ^{2}Res. If for some such QH vertex group Q, this sequence of multi-graded abelian decompositions is not compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the next level of the multi-graded resolution MGQRes, we omit the multi-graded resolution MGQ^{2}Res from the list of completions of (multi-graded) resolutions of the anvil, Anv(MGQRes).

By theorem 4.13, the complexity of the (multi-graded) core of the multigraded resolution: MGQ^{2}Res, is bounded by the complexity of the core of the multi-graded resolution MGQRes. In this part, we further assume that:

Cmplx(MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQ^{2}Res)) <
< Cmplx(MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, T MGQRes))

where T MGQRes is the one level resolution corresponding to the top level of the multi-graded resolution MGQRes. The case of maximal complexity core will be treated in the next parts of the second step of the sieve procedure. To treat a (second) multi-graded resolution which is not of maximal possible complexity we need the following two observations, which are similar to lemmas 5.2 and 5.3.

Lemma 5.7. Let MGQ^{2}Res be one of the resolutions in our list of multi-graded resolutions that is not of maximal possible complexity. Let:

MGQ^{2}lim_{term}(u, t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a)

be the terminal rigid or solid limit group of the multi-graded resolution MGQ^{2}Res, and let Q_{term}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) and Q_{term}^{2}(t, v, h_{2}, g_{1}, w, p, a) be the images of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) and Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) in the limit group MGQ^{2}lim_{term} in correspondence. Then the terminal limit group of the multi-graded resolution MGQ^{2}Res can be replaced by a collection of finitely many terminal multi-graded limit groups in which either Q_{term}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), or Q_{term}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), or the terminal limit group MGQ^{2}lim_{term} is rigid or solid with respect to the parameter subgroup P =< p >.

Proof: Identical to the proofs of lemma 2.2 and lemma 4.6 of [Se4].

□

Lemma 5.8. Let MGQ^{2}Res be one of the resolutions in our list of (second) multi-graded resolutions. Let Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a), Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), and Q_{2}^{2}(u, t, v, h_{2}, g_{1}, h_{1}, w, p.a), be the images of the subgroups Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), and Q^{2}(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a) in the limit group:

MGQ^{2}lim_{2}(u, t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a)

associated with the second level of the multi-graded resolution MGQ^{2}Res. Then either Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), or Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), or Q_{2}^{2}(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a).

Proof: The claim is simply one of the basic properties of a multi-graded resolution.

□

Suppose that Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a). In this case we continue as we did in part (2) of the initial step of the sieve procedure. Let T MGQ^{2}Res(u, t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a)) be the resolution that corresponds to the top level of the multi-graded resolution MGQ^{2}Res. By corollary 4.15, since the multi-graded resolution MGQ^{2}Res is not of maximal complexity:

Cmplx(MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, T MGQ^{2}Res)) <
< Cmplx(MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, T MGQRes))

where T MGQRes is the one level resolution that corresponds to the top level of the multi-graded resolution MGQRes.

With Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) we associate the resolutions that appear in its taut Makanin-Razborov diagram. Given a taut graded (Makanin-Razborov) resolution GRes(h_{2}, g_{1}, h_{1}, w, p, a) of Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a), we associate with it to a taut graded resolution CRes(h_{2}, g_{1}, h_{1}, w, p, a), constructed from the multi-graded resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the multi-graded core, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, T MGQ^{2}Res), followed by the graded resolution GRes. As we did in part (2) of the initial step of the sieve procedure, with the graded resolution CRes we associate a (canonical) finite collection of framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a), and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, extra P S resolutions, and with each extra P S resolution, a finite collection of formal collapse P S resolutions. We set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of collapse P S limit groups, graded auxiliary resolutions and anvils, precisely as we did in part

(2) of the initial step of the sieve procedure.

Suppose that Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), and Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a). In this case we do the following. With the subgroup Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) we associate the collection of completions of the multi-graded resolutions that appear in its taut multi-graded Makanin-Razborov diagram with respect to the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}:

MGQRes_{1}(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), …
…, MGQRes_{e}(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a).

We continue with each of the multi-graded resolutions MGQRes_{j} in parallel.

We set the multi-graded resolution

DRes_{j}(t, v, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a)

to be the multi-graded resolution induced by the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a > from (the corresponding core of) the multi-graded resolution T MGQ^{2}Res, followed by the multi-graded resolution MGQRes_{j}. If the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the resolution DRes_{j}, Q_{D}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of the P S limit group P SHGH(h_{2}, g_{1}, h_{1}, w, p, a), we have started with, we replace the multi-graded resolution MGQRes_{j} by starting a new branch of the sieve procedure with Q_{D}(h_{2}, g_{1}, h_{1}, w, p, a) instead of P SHGH(h_{2}, g_{1}, h_{1}, w, p, a). If the subgroup generated by < t, v, h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the resolution DRes_{j}, Q_{D}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of the collapse extra P S limit group CollapseExtraP S(t, v, h_{2}, g_{1}, h_{1}, w, p, a), we have started the initial step with, we replace the multi-graded resolution MGQRes_{j} by starting the initial step of the procedure with the limit group Q_{D}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) instead of the collapse extra P S limit group CollapseExtraP S.

Since the extra P S resolution ExtraP SRes(v, h_{2}, g_{1}, h_{1}, w, p, a), we have started the initial step with, is well-separated, with each QH vertex group in one of the abelian decompositions associated with ExtraP SRes there is an associated collection of s.c.c. that are mapped to the trivial element in the next level of the extra P S resolution ExtraP SRes. Each QH vertex group in the graded abelian decomposition associated with the top level of the extra P S resolution ExtraP SRes naturally inherits a sequence of abelian decompositions from a multi-graded resolution DRes_{j}. If for some such QH vertex group Q, this sequence of multi-graded abelian decompositions is not compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the next level of the extra P S resolution ExtraP SRes, we omit the multi-graded resolution MGQRes_{j} from the list of completions of resolutions of the limit group Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose that the core of a multi-graded resolution MGQRes_{j}, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQRes_{j}), is of maximal possible complexity, i.e., the core has the same structure as the multi-graded abelian decomposition associated with the top level of the extra P S resolution, ExtraP SRes, we have started the initial step of the sieve procedure with. By theorem 4.14, since the core of the multi-graded resolution MGQRes_{j} is assumed to be of maximal complexity, so is the multi-graded core of the constructed multi-graded resolution DRes_{j}, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a > , DRes_{j}. Hence, by theorem 4.17 every specialization of the P S limit group P SHGH(h_{2}, g_{1}, h_{1}, w, p, a) that can be completed to a specializations that factors through the multi-graded resolution DRes_{j}, factors through at least one of the multi-graded resolutions with maximal complexity core MGQRes, of the collapse P S limit group Q^{1}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), we have started the initial step of the sieve procedure with. Since the multi-graded resolution MGQRes associated with the anvil we have started the second step of the sieve procedure with is assumed to have a core which is not of maximal complexity, we may omit those multi-graded resolutions of Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), MGQRes_{j}, that have a maximal complexity core.

Suppose that the multi-graded resolution MGQRes_{j} does not have a maximal complexity core resolution. If the image of the subgroup Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the multi-graded resolution MGQRes_{j} is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), we associate with it a finite collection of developing resolutions, graded auxiliary resolutions and anvils precisely as we did in case the subgroup Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), i.e., as in case (2) of the initial step of the sieve procedure. Since we may assume that the image of the subgroup Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the multi-graded resolution MGQRes_{j} is isomorphic to Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), and the core associated with the multigraded resolution MGQRes_{j} is not of maximal possible complexity, we can apply the construction that appears in part (2) in the initial step of the sieve procedure, and associate with the multi-graded resolution MGQRes_{j} a finite collection of graded resolutions:

GRes_{1}(h_{2}, g_{1}, h_{1}, w, p, a), …, GRes_{r}(h_{2}, g_{1}, h_{1}, w, p, a).

With each of these graded resolutions GRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a), we associate a graded resolution CRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a), constructed from the resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the (corresponding core of the) multi-graded resolution T MGQ^{2}Res (which is the top part of the multi-graded resolution MGQ^{2}Res), followed by the graded resolution GRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a). With each graded resolution CRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a) we can naturally associate a finite collection of framed resolutions as we did in part (2) of the initial step of the sieve procedure. We continue with each of these framed resolutions in parallel, and denote each of them F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a).

If the limit group generated by < h_{2}, g_{1}, h_{1}, w, p, a > associated with a framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), we replace the framed resolution F rmCRes by starting the initial step of the sieve procedure with that limit group (which is a proper quotient of the P S limit group we have started the initial step with). Otherwise, we follow what we did in part (2) of the initial step of the sieve procedure, and associate with the framed resolution F rmCRes a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, extra P S resolutions, and with each extra P S resolution, a finite collection of formal collapse P S resolutions. We set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of collapse P S limit groups, graded auxiliary resolutions and anvils, precisely as we did in part (2) of the initial step of the sieve procedure.

Suppose that Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), and suppose further that Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a). In this case we continue to the next level of the (second) multi-graded quotient resolution MGQ^{2}Res. Note that by corollary 4.15, since the multi-graded resolution MGQ^{2}Res is not of maximal complexity, the (multi-graded) core associated with each of its levels is not of maximal complexity as well. If for some level j of the multigraded resolution, the image of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with this level, Q_{j}^{2}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), or the image of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with this level, Q_{j}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), then from the highest such level j, we can continue as in case Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), or Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), in correspondence, and associate with the (second) multi-graded resolution MGQ^{2}Res a (canonical) finite collection of framed resolutions, and with each framed resolution a canonical finite collection of Non-Rigid and Non-Solid P S resolutions, a collection of Left P S resolutions, extra P S resolutions, and formal collapse extra P S resolutions. With each extra P S resolution we further associate a finite collection of framed resolutions. We set each the framed resolutions to be a developing resolution, and with it we associate graded auxiliary resolutions, collapse P S limit groups and anvils, precisely as we did in case Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), or Q_{2}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a).

Finally, suppose that for every level j, the image of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the j-th level of the multi-graded resolution MGQ^{2}Res, Q_{j}^{2}(h_{2}, g_{1}, h_{1}, w, p, a), is isomorphic to Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a), and for every level j, the image of Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the j-th level of the multi-graded resolution MGQ^{2}Res, Q_{j}^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), is isomorphic to Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a). In this case, by lemma 5.7, the terminal limit group of the multi-graded resolution MGQ^{2}Res, Q_{term}^{2}(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a), is rigid or solid with respect to the parameter subgroup P =< p >.

We continue as in part (2) of the initial step of the sieve procedure. Let P B^{2}(b_{2}, p, a) be the terminal rigid or solid limit group of the multi-graded resolution MGQ^{2}Res. We set the graded resolution CRes(h_{2}, g_{2}, h_{1}, w, p, a), to be the resolution induced by the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > from the sequence of core resolutions associated with the various levels of the multi-graded resolution MGQRes. With the graded resolution CRes we associate a finite (canonical) collection of framed resolutions (see definition 5.4). With each of the framed resolutions associated with CRes we associate a (canonical) finite collection of Non-Rigid and Non-Solid P S resolutions, a collection of Left P S resolutions, extra P S resolutions, and formal collapse extra P S resolutions. If every test sequence that factors through an extra P S resolution associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a), factors through a framed resolution with a bigger frame than the one associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a), we exclude this extra P S resolution from the finite collection of P S resolutions associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a). With each of the extra P S resolutions associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a), we associate a (canonical) finite collection of graded auxiliary resolutions, and with each graded auxiliary resolution we associate a (canonical) finite collection of collapse extra P S limit groups, precisely as we associated collapse extra P S resolution with an extra P S resolution associated with the one of the resolutions P SHGHRes_{j} we have started our iterative procedure with. We set the developing resolution, Dvlp(MGQRes), to be the extra P S resolution, ExtraP SRes. With each couple of a developing resolution and a collapse extra P S limit group, we associate a finite collection of anvils, precisely as we did in case Q_{2}^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a).

(5) By part (1) we may assume that Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the limit group Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) associated with the anvil, Anv(MGQRes). Parts (2)-(4) treat all the cases in which the core associated with the second multi-graded resolution MGQ^{2}Res is not of maximal complexity. In this part we assume that the core associated with the multi-graded resolution MGQRes associated with the anvil, Anv(MGQRes), is not of maximal complexity, and the core associated with the second multi-graded resolution MGQ^{2}Res is of maximal possible complexity. i.e., the core of the multigraded resolution MGQ^{2}Res has the same structure as the core associated with the top level of the multi-graded resolution MGQRes. By part (3), we may assume that Q^{2}(t, v, h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the subgroup Q(t, v, h_{2}, g_{1}, h_{1}, w, p, a) associated with the anvil, Anv(MGQRes).

We treat this case by slightly modifying the way we treated it in the initial step of the sieve procedure. In case the core associated with MGQ^{2}Res and the core associated with the top level of MGQRes have identical structure, we use the modular groups associated with the abelian decomposition associated with the core associated with the second multi-graded resolution MGQ^{2}Res to map this core into its image in the limit group associated with the (appropriate) next level of the multi-graded resolution MGQRes (which is naturally mapped into the anvil, Anv(MGQRes)). At this point we continue analyzing the remaining part of the anvil in the same way we have analyzed the second multi-graded resolution, according to parts (1)-(5) above. As long as the core associated with the constructed multi-graded resolutions are of maximal possible complexity we continue analyzing multi-graded resolutions associated with various parts of the anvil we have started with, until one of the parts (1)-(4) applies to the corresponding multi-graded resolution. If there exists a level for which one of the parts (1)-(4) applies, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions, a graded resolution, composed from the resolutions induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions, and with this graded resolution we associate a (canonical) finite collection of framed resolutions. With each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(4) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that the sequence of multi-graded core resolutions of the multigraded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the core associated with the corresponding level of the developing resolution, in the initial step of the procedure, which means that none of the parts (1)-(4) applies to any of these multi-graded resolutions. In this case we examine the structure of the corresponding developing resolution. The developing resolution is a framed resolution of a resolution built from a sequence of resolutions induced from corresponding core resolutions. Each of the induced resolutions is a resolution induced by the (image of the) subgroup < h_{2}, g_{1}, h_{1}, w, a > from the corresponding core resolution, and with each level of the induced resolution there is an associated (framed) multi-graded abelian decomposition (see section 3 of [Se4] for the construction of the induced resolution, and definition 5.4 for the construction of framed resolutions).

Proposition 5.9. Suppose that all the core resolutions associated with the multigraded resolutions used for the construction of the developing resolution are of maximal possible complexity. Let < q, h_{2}, g_{1}, h_{1}, w, p, a > be the subgroup generated by the closure of the developing resolution in the anvil Anv(MGQRes). From each of the core resolutions associated with the multi-graded resolutions used to construct the developing resolution, there is a resolution induced by the (image of the) subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >.

Then either the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various multi-graded resolutions used to construct the developing resolution, is identical to the structure of the developing resolution we have started the second step with, or there exists some level j, so that the structure of the graded abelian decompositions associated with the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > above level j are identical to the structure of graded abelian decompositions associated with the corresponding levels of the developing resolution, and in level j, either the number of factors in the (graded) free decomposition associated with the graded abelian decomposition associated with the resolution induced by < q, h_{2}, g_{1}, h_{1}, w, p, a > is strictly smaller than the number of factors in the corresponding (graded) free decomposition associated with the corresponding level of the developing resolution, and in case of equality in the number of factors, the complexity of the graded abelian decomposition associated with the resolution induced by < q, h_{2}, g_{1}, h_{1}, w, p, a > is strictly smaller than the complexity of the graded abelian decomposition associated with that level in the developing resolution.

Proof: Identical to the proof of proposition 4.11 of [Se4].

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If the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions used to construct the developing resolution in the second step of the procedure, is not identical to the structure of the developing resolution associated with the anvil Anv(MGQRes), proposition 5.9 implies that there exists some level j for which the structure of the graded abelian decomposition associated with the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > above level j are identical to the structure of the graded abelian decompositions associated with the corresponding levels in the developing resolutions we have started the second step with, and in level j, either the number of factors in the graded free decomposition associated with the graded abelian decomposition associated with the resolution induced by < q, h_{2}, g_{1}, h_{1}, w, p, a > is strictly smaller than the number of factors in the corresponding graded free decomposition associated with the corresponding level in the developing resolution we have started the second step with, and in case of equality in the number of factors, the complexity of the graded abelian decomposition associated with the resolution induced by < q, h_{2}, g_{1}, h_{1}, w, p, a > is strictly smaller than the complexity of the graded abelian decomposition associated with the corresponding level of that developing resolution. In this case we do the following. With the graded resolution constructed from the various resolutions induced by the subgroups < q, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various multi-graded resolutions constructed along the various levels of the second step of the sieve procedure, we associate a (canonical) finite collection of framed resolutions (definition 5.4). With each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions used to construct the developing resolution in the second step of the procedure, has the same structure as that of the developing resolution associated with the anvil Anv(MGQRes). We set SCRes(s, h_{2}, g_{1}, h_{1}, w, p, a) to be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions used to construct the developing resolution in the second step of the procedure, and suppose that the structure of the graded resolution SCRes is identical to the structure of the developing resolution associated with the anvil Anv(MGQRes).

With the graded resolution SCRes we naturally associate a (canonical) collection of framed resolutions (definition 5.4). If every ungraded resolution that factors through the (graded) resolution SCRes, factors through a framed resolution F rmSCRes associated with it, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes, we call the sequence of multigraded resolutions from which SCRes was constructed, a terminal resolution, and do not continue with it to the next step of the sieve procedure. If every ungraded resolution that factors through the graded resolution SCRes, factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the second step of the procedure with, or through a framed resolution, F rmSCRes, associated with SCRes, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes, we call the sequence of multi-graded resolutions from which the graded resolution SCRes was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we do the following.

Let < c, h_{2}, g_{1}, h_{1}, w, p, a > be the collapsed P S limit group associated with the anvil, Anv(MGQRes). With each multi-graded resolution constructed in one of the levels, we associate its core resolution with respect to the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. We replace the core resolutions associated with the multi-graded resolutions constructed in the various levels, and with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >, by the corresponding penetrated core resolutions (see definition 4.19), of the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > with respect to the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. Note that by construction, the original core resolutions are embedded into the corresponding penetrated core resolutions. Then we study the structure of the graded resolutions composed from resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the corresponding penetrated resolutions, in a similar way to proposition 5.9.

Proposition 5.10. Let < q, h_{2}, g_{1}, h_{1}, w, p, a > be the subgroup generated by the developing resolution in the anvil Anv(MGQRes). From each level of the penetrated core resolutions associated with the multi-graded resolutions constructed along the levels along the second step of the procedure, and with the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >, there is a resolution induced by the (image of the) subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >. Let Λ_{1}, …, Λ_{r} be the graded abelian decompositions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the penetrated core resolutions associated with the various levels.

Then for each i, 1 ≤ i ≤ r, either the graded abelian decomposition Λ_{i} is identical to the structure of the corresponding graded abelian decomposition in the developing resolution we have started the second step with, or the number of factors in the (graded) free decomposition associated with Λ_{i} is strictly smaller than the number of factors in the corresponding (graded) free decomposition associated with the corresponding level of the developing resolution, and in case of equality in the number of factors, the complexity of the graded abelian decomposition Λ_{i} is strictly smaller than the complexity of the graded abelian decomposition associated with that level in the developing resolution.

Proof: Identical to the proof of propositions and 5.9 and proposition 4.11 of [Se4].

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Suppose that the structure of at least one of the graded abelian decompositions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, is not identical to the structure of the corresponding abelian decompositions in the developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). Then we do the following.

We set the graded resolution GRes_{1}(r, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated in the various levels with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. With the resolution GRes_{1} we associate a finite collection of framed resolutions, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We further associate with each framed resolution the graded resolution SCRes, which we denote SCRes_{1}^{2,1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) and call the first sculpted resolution (of width 1). We set each of the extra P S resolutions to be both a first sculpted resolution of width 2, which we denote: SCRes_{1}^{2,2}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and a penetrated sculpted resolution, which we denote P enSCRes_{1}^{2,2}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate a finite collection of graded auxiliary resolutions and limit groups, and a finite collection of anvils. We set the first sculpted resolution of width 2 (the extra P S resolution) associated with the anvil, to be the developing resolution. Finally, We associate with the anvil the sequence of core resolutions associated with the multi-graded resolutions constructed in the various levels, and the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >, which we call a Carrier, and denote it Carrier_{2}^{2} (the top index refer to the step number and the bottom index refer to the width, which is the index of the corresponding sequence of core resolutions).

Suppose that the structure of all the graded abelian decompositions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, are identical to the structure of the corresponding abelian decompositions in the developing resolution. Recall that we denote the collapse P S limit group associated with the anvil, Anv(MGQRes), by < c, h_{2}, g_{1}, h_{1}, w, p, a >. We set the resolution GRes_{2}(e, c, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions constructed in the various levels and the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. Note that GRes_{2} is graded with respect to the defining parameter subgroup P =< p >, and the resolution GRes_{1} is embedded into the graded resolution GRes_{2}. With the graded resolution GRes_{2} we associate a (canonical) finite collection of framed resolutions, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We further associate with the graded resolution GRes_{2} the graded resolution SCRes which we call first sculpted resolution, and denote both as SCRes_{1}^{2,1}(s, h_{2}, g_{1}, h_{1}, w, p, a) and SCRes_{1}^{2,2}(s, h_{2}, g_{1}, h_{1}, w, p, a), and the resolution GRes_{1} which we call first penetrated sculpted resolution, and denote P enSCRes_{1}^{2,2}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a). Finally, we set each of the extra P S resolutions to be a developing resolution, which we denote Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate a finite collection of graded auxiliary resolutions and limit groups, and a finite collection of collapse extra P S limit groups, which are set to be the anvils.

Before we continue to part (6) of the second step of the sieve procedure, we present a basic property of framed resolutions of graded resolutions composed from resolutions induced from penetrated core resolutions, which will be used in the sequel.

Proposition 5.11. Let Q be a QH vertex group in one of the abelian decompositions associated with the various levels of the first penetrated sculpted resolution GRes_{1}, and suppose that Q is a subgroup of finite index in a QH subgroup Q′ in an abelian decomposition associated with one of the levels of the associated ambient multi-graded resolution MGQ^{2}Res. Suppose that Q is not a QH vertex groups in the (multi-graded) core resolution associated with MGQ^{2}Res, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQ^{2}Res).

Let F rmGRes be a framed resolution associated with the graded resolution GRes_{1}. Then the QH subgroup containing Q in the framed resolution F rmGRes is Q itself, i.e., Q is not properly contained in a QH vertex group in the framed resolution F rmGRes.

Proof: By theorem 4.20, in a penetrated core resolution, each of the QH vertex groups that do not belong to the formed part of the core resolution associated with the penetrated core, is mapped isomorphically into the limit group associated with the terminal level of the penetrated core. Furthermore, each of the abelian decompositions associated with the various levels of the penetrated core, gives rise to a corresponding abelian decomposition of the limit group associated with its terminal level. Hence, the QH vertex group Q is mapped onto a QH vertex group Q_{t} in some abelian decomposition of the limit group associated with the terminal level of the penetrated core.

Since the limit group associated with the terminal level of the penetrated core is a free product of the formed part of the core resolution with some additional f.g. free group, in any framed resolution of the resolution GRes_{1}, non-trivial elements of the subgroup Q_{t} that do not have non-trivial roots in Q_{t}, do not have non-trivial roots in the subgroup associated with the framed resolution. If, in a framed resolution F rmGRes of GRes_{1}, the QH vertex group Q in GRes_{1} is a finite index subgroup in a QH vertex group Q′ in the framed resolution F rmGRes, and Q′ properly contains Q, then there are non-trivial elements in Q that do not have non-trivial roots in Q but have non-trivial roots in Q′, hence, there are non-trivial elements in Q_{t} that do not have non-trivial roots in Q_{t} but have non-trivial roots in the limit group associated with the framed resolution F rmGRes, a contradiction. Hence, the QH vertex group Q is not properly contained in a QH vertex group Q′ in a framed resolution of the resolution GRes_{1}.

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(6) By part (1) we may assume that Q^{2}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the limit group Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) associated with the anvil, Anv(MGQRes). Parts (2)-(4) treat all the cases in which the core associated with the second multi-graded resolution MGQ^{2}Res is not of maximal complexity. Part (5) treat the case in which the core associated with the second multigraded resolution MGQ^{2}Res is of maximal complexity, but the core associated with the multi-graded resolution MGQRes associated with the anvil, Anv(MGQRes), is not of maximal complexity. In this part we treat the case in which the core associated with the second multi-graded resolution MGQ^{2}Res is of maximal complexity. We further assume that at least one of the core resolutions associated with the multi-graded resolutions associated with the various levels of the anvil, Anv(MGQRes), is not of maximal possible complexity. i.e., we assume that no sculpted resolution is associated with the anvil, Anv(MGQRes).

We treat this case in a similar way to what we did in part (5). In case the core associated with MGQ^{2}Res and the core associated with the top level of MGQRes are identical, we use the modular groups associated with the abelian decomposition associated with the core associated with the second multi-graded resolution MGQ^{2}Res to map this core into its image in the limit group associated with the (appropriate) next level of the multi-graded resolution MGQRes (which is naturally mapped into the anvil, Anv(MGQRes)). At this point we continue analyzing the remaining part of the anvil in the same way we have analyzed the second multi-graded resolution, according to parts (1)-(5) above. As long as the core associated with the constructed multi-graded resolutions are of maximal possible complexity we continue analyzing multi-graded resolutions associated with various parts of the anvil we have started with, until one of the parts (1)-(4) applies to the corresponding multi-graded resolution. If there exists a level for which one of the parts (1)-(4) applies, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions, a graded resolution, composed from the resolutions induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions. With this graded resolution we associate a (canonical) finite collection of framed resolutions. With each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(4) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolutions, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that the sequence multi-graded core resolutions of the multigraded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the core associated with the corresponding level of the developing resolution, in the initial step of the procedure, which means that none of the parts (1)-(4) applies to any of these multi-graded resolutions. In this case we proceed as in part (5) and examine the structure of the corresponding developing resolution. The developing resolution is a framed resolution of a resolution built from a sequence of resolutions induced from corresponding core resolutions. Each of the induced resolutions is a resolution induced by the (image of the) subgroup < h_{2}, g_{1}, h_{1}, w, a > from the corresponding core resolution, and with each level of the induced resolution there is an associated (framed) multi-graded abelian decomposition. If the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions used to construct the developing resolution in the second step of the procedure, is not identical to the structure of the developing resolution associated with the anvil Anv(MGQRes), we follow what we did in part (5) and associate with the graded resolution constructed from the various resolutions induced by the subgroups < q, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various multi-graded resolutions constructed along the various levels of the second step of the sieve procedure, a (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that the sequence of multi-graded core resolutions of the multigraded resolutions constructed by the process described above, are all of maximal complexity. Suppose further that the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions, used to construct the developing resolution in the second step of the procedure, is identical to the structure of the developing resolution associated with the anvil, Anv(MGQRes). In this case, we follow what we did in part (5) and either declare the corresponding sequence of multi-graded resolutions terminal resolutions, or we associate with them penetrated core resolutions with which we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, with which we associate developing and sculpted resolutions, anvils, penetrated sculpted resolutions and (possibly) a carrier, precisely as we did in part (5) of the second step of the sieve procedure.

(7) Parts (2)-(4) treat all the cases in which the core associated with the second multi-graded resolution MGQ^{2}Res is not of maximal complexity. Parts (5) and (6) treat the case in which the core associated with the second multi-graded resolution MGQ^{2}Res is of maximal complexity, but the core associated with at least one of the multi-graded resolutions associated with the anvil, Anv(MGQRes), is not of maximal complexity, i.e., parts (1)-(6) treat all the cases in which no sculpted resolution is associated with the anvil, Anv(MGQRes). In this part we treat the remaining case. We assume that all the core resolutions associated with the multi-graded resolutions associated with the various levels of the anvil, Anv(MGQRes), are of maximal complexity, i.e., we assume that a sculpted resolution is associated with the anvil, Anv(MGQRes). We further assume that the core associated with the second multi-graded resolution MGQ^{2}Res is of maximal complexity, i.e., its core has the same structure as the graded abelian decomposition associated with the top level of sculpted resolution associated with the anvil, Anv(MGQRes), that is identical to the structure of graded abelian decomposition associated with the top level of the the extra P S resolution, ExtraP SRes, we have started the initial step of the sieve procedure with.

Suppose that the structure of the core associated with the second multigraded resolution MGQ^{2}Res, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQ^{2}Res, is identical to the structure of the corresponding core associated with the multi-graded resolution MGQRes. With the anvil, Anv(MGQRes), we have associated a developing resolution, Dvlp(t, v, h_{2}, g_{1}, h_{1}, w, p, a). Parts (1)-(7) studied multi-graded resolutions with respect to the sculpted resolution, and the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. At this point we apply parts (1)-(6) to study the multi-graded resolutions of the anvil, Anv(MGQRes), with respect to the developing resolution, Dvlp(t, v, h_{2}, g_{1}, h_{1}, w, p, a), and the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >. i.e., we study multi-graded resolutions, with respect to the parameter subgroups that are associated with the various levels of the developing resolution Dvlp(t, v, h_{2}, g_{1}, h_{1}, w, p, a), where, in particular, the subgroups Base_{2,1}^{1}, …, Base_{2,v_{2}}^{1}, and the corresponding parts of the formed core resolution, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQ^{2}Res, are assumed to be elliptic throughout the analysis, i.e., they are assumed to be elliptic in all the associated multi-graded resolutions. With each such sequence of multi-graded resolutions we associate a multigraded resolution, which we still denote, MGQ^{2}Res, and since in this part we study multi-graded resolutions with core resolutions of maximal possible complexity, we continue only with those multi-graded resolutions, MGQ^{2}Res, for which the associated core resolution MGCore(< h_{2}, g_{1}, h_{1}, w, p, a > , MGQ^{2}Res, is of maximal possible complexity (cf. theorem 4.18).

At this point we continue analyzing the remaining part of the anvil, level after level, in the same way we have analyzed the second multigraded resolution. i.e., we first check if one of the parts (1)-(4) applies to the multi-graded resolution associated with the corresponding part of the anvil, and if the associated core is of maximal possible complexity, we analyze multi-graded resolutions with respect to the corresponding level of the developing resolution, Dvlp(t, v, h_{2}, g_{1}, h_{1}, w, p, a) and the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >. As long as the core associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > and the constructed multi-graded resolutions are of maximal possible complexity, we continue analyzing multi-graded resolutions associated with various parts of the anvil we have started with, until one of the parts (1)-(4) applies to the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > and the corresponding multi-graded resolution. If there exists a level for which one of the parts (1)-(4) applies, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions, a (canonical) finite collection framed resolutions and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(4) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that none of the parts (1)-(4) applies to any of the multi-graded resolutions associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. In particular, the core resolutions associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the core associated with the corresponding level of the sculpted resolution, hence, the same structure as the graded abelian decomposition associated with the corresponding level of the extra P S resolution, ExtraP SRes, we have started the initial step of the sieve procedure with.

Let SCRes_{1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions used to construct the developing resolution in the second step of the procedure. If every ungraded resolution that factors through the graded resolution SCRes_{1}, factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the second step of the procedure with, we call the sequence of multi-graded resolutions from which the graded resolution SCRes_{1} was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we examine the core resolutions and the multi-graded resolutions associated with the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a > and the developing resolution Dvlp(t, v, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose that there exists a level for which one of the parts (1)-(4) applies to the multi-graded resolutions associated with the developing resolution Dvlp and the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >. With each multi-graded resolution constructed in one of the levels, we associated its core resolution with respect to the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >. We replace the core resolutions associated with the multi-graded resolutions constructed in the various levels, and with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >, by the corresponding penetrated core resolutions (see definition 4.19), of the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > with respect to the core resolutions associated with the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >. Note that by construction, the original core resolutions are embedded into the corresponding penetrated core resolutions. We set the graded resolution GRes_{1}(r, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated in the various levels with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. We associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions (with respect to the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >, a (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(4) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate the resolution SCRes_{1}, that we now denote both as SCRes_{1}^{2,1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) and SCRes_{1}^{2,2}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and call first sculpted resolution, and the resolution GRes_{1}, which we denote P enSCRes_{1}^{2,2}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a) and call penetrated sculpted resolution, and a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that none of the parts (1)-(4) applies to any of the multi-graded resolutions associated with the developing resolution, Dvlp, and the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >. In particular, the core resolutions associated with the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the core associated with the corresponding level of the developing resolution, Dvlp. We set SCRes_{2}(s_{2}, h_{2}, g_{1}, h_{1}, w, p, a) to be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < q, t, v, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions used to construct the developing resolution in the second step of the procedure. If every ungraded resolution that factors through the graded resolution SCRes_{2}, factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the second step of the procedure with, we call the sequence of multi-graded resolutions from which the graded resolution SCRes_{2} was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we do the following. We start by replacing the core resolutions associated with the multi-graded resolutions constructed in the various levels, and with both subgroups, < h_{2}, g_{1}, h_{1}, w, p, a > and < t, v, h_{2}, g_{1}, h_{1}, w, p, a >, by the corresponding penetrated core resolutions (see definition 4.19). Note that by construction, the original core resolutions are embedded into the corresponding penetrated core resolutions. Then we study the structure of the graded resolutions composed from resolutions induced by the subgroup < q, t, v, h_{2}, g_{1}, h_{1}, w, p, a >.

We set the graded resolution GRes_{1}(r, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated in the various levels with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. We set the graded resolution:

GRes_{2}(r, t, v, h_{2}, g_{1}, h_{1}, w, p, a)

to be the resolution composed from the resolutions induced by the subgroup < q, t, v, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated in the various levels with the subgroup < t, v, h_{2}, g_{1}, h_{1}, w, p, a >.

From the corresponding collection of the constructed multi-graded resolutions, we naturally obtain a graded resolution, GRes(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a), which is graded with respect to the defining parameter subgroup P =< p >. By construction, both resolutions GRes_{1} and GRes_{2} are embedded into the completion of the graded resolution GRes(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a). With the graded resolution GRes(u, t, v, h_{2}, g_{1}, h_{1}, w, p, a), we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We further associate with the graded resolution GRes the graded resolutions SCRes_{1}, which we denote as SCRes_{1}^{2,j}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) for j = 1, 2, 3, and call first sculpted resolution, and the graded resolution GRes_{1} which we call first penetrated sculpted resolution, and denote P enSCRes_{1}^{2,3}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a), the graded resolution SCRes_{2}, which we denote SCRes_{2}^{2,j}(s, h_{2}, g_{1}, h_{1}, w, p, a) for j = 2, 3, and call second sculpted resolution, and the graded resolution GRes_{2} which we call second penetrated sculpted resolution, and denote P enSCRes_{2}^{2,3}(r_{2}, h_{2}, g_{1}, h_{1}, w, p, a). Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, and a finite collection of collapse extra P S limit groups, which are set to be the anvils.

III: The general step of the sieve procedure

In the first two steps of the sieve procedure, we have finally obtained finitely many developing resolutions and anvils, with which we have possibly associated a finite collection of sculpted resolutions, penetrated sculpted resolutions and a carrier, and a finite collection of associated Non-rigid and Non-solid and Left P S resolutions, extra P S resolutions, formal collapse P S resolutions, collapsed extra P S limit groups, and graded auxiliary resolutions. After presenting the initial steps, we finally present the general step of the sieve procedure, and then prove it terminates after finitely many steps.

We define the general step of the sieve procedure inductively. For brevity we denote the multi-graded resolutions that were obtained in the previous steps of the sieve procedure, MGQ^{m}Res(t, h_{2}, g_{1}, h_{1}, w, p, a), where m is the index of the step in which they were constructed. With each such multi-graded resolution there is an associated developing resolution, which we denote Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), graded auxiliary resolution, and an anvil, which we denote Anv(MGQ^{m}Res)(t, h_{2}, g_{1}, h_{1}, w, p, a). With the developing resolution and the anvil, we associate a positive integer that denotes the number of sequences of core resolutions associated with the anvil, which we call the width of the anvil, and denote width(m). width(m)=1 if and only if no sculpted resolution is associated with the anvil. In case width(m) > 1, for each width d, 1 ≤ d ≤ width(m), i.e., with the collection of the first d sequences of core resolutions associated with the anvil, there are associated sc(m, d) sculpted resolutions, that we denote, SCRes_{e}^{m,d}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) for e = 1, …, sc(m, d), and corresponding penetrated sculpted resolutions, that we denote P enSCRes_{e}^{m,d}(r_{e}, h_{2}, g_{1}, h_{1}, w, p, a). At each width d, there is possibly an associated carrier, that we denote Carrier_{d}^{m}, that represent the d-th sequence of core resolutions associated with the anvil. We start the general step of the sieve procedure with the (finite) collection of multi-graded resolutions constructed in the previous step, and their associated widths, developing resolutions, graded auxiliary resolutions, sculpted resolutions and their penetrated resolutions and carriers, and anvils. The ultimate goal of the general step of the iterative procedure is to obtain a strict reduction in either the complexity of certain decompositions and resolutions or a strict reduction in the Zariski closures of certain limit groups associated with the anvils constructed in the previous steps of the procedure. The strict reduction in complexity and Zariski closures will finally guarantee the termination of the iterative procedure after finitely many steps.

Since we treat the anvils in parallel, we present the general (n-th) step of the procedure with one of the anvils, Anv(MGQ^{n−1}Res)(t, h_{2}, g_{1}, h_{1}, w, p, a). We construct the developing resolutions, (penetrated) sculpted resolutions and carriers associated with the anvil and the n-th quotient limit group Q^{n}Rlim(w, y, a) iteratively. The analysis we carry out in the general step depends on the structure of the multi-graded resolutions, the developing resolutions, the (penetrated) sculpted resolutions, the carriers, and the graded auxiliary resolutions constructed in the previous steps of the sieve procedure, in a similar way to what we did in the first two steps of the sieve procedure.

(1) Let Q^{n−1}(h_{2}, g_{1}, h_{1}, w, p, a) be the restricted limit group generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the anvil Anv(MGQ^{n−2}Res)(t, h_{2}, g_{1}, h_{1}, w, p, a). Let Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) be the limit group generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the anvil Anv(MGQ^{n−1}Res)(t, h_{2}, g_{1}, h_{1}, w, p, a). If Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of the subgroup Q^{n−1}(h_{2}, g_{1}, h_{1}, w, p, a), we continue this branch of the iterative procedure, by starting the initial step of the procedure with the graded limit group Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) instead of the graded (P S) limit group Q^{n−1}(h_{2}, g_{1}, h_{1}, w, p, a).

(2) At this stage we may assume that Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{n−1}(h_{2}, g_{1}, h_{1}, w, p, a). Along the sieve process used to construct the anvil, Anv(MGQ^{n−1}Res), we enlarge the parameter subgroups each time the complexity of the core associated with the corresponding multi-graded resolution is being reduced. At step m, 1 ≤ m ≤ n − 1, we set the parameter subgroups to be: Base_{2,1}^{s(m)}, …, Base_{2,v_{s(m)}}^{s(m)} and the corresponding multi-graded resolution to be:

MGQ^{m}Res(t_{m}, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s(m)}, …, Base_{2,v_{s(m)}}^{s(m)}, a).

For each index s, 1 ≤ s ≤ s(n − 1), we set f(s) to be the minimal index m, 1 ≤ m ≤ n − 1, for which s = s(m), and ℓ(s) to be the maximal index m for which s = s(m). For each couple of indices m_{1}, m_{2}, 1 ≤ m_{1} ≤ m_{2} ≤ n, let Q^{m_{2}}(t_{m_{1}}, h_{2}, g_{1}, h_{1}, w, p, a) be the subgroup generated by < t_{m_{1}}, h_{2}, g_{1}, h_{1}, w, p, a > in the limit group Q^{m_{2}}(t_{m_{2}}, h_{2}, g_{1}, h_{1}, w, p, a), which is the m_{2} − 1 anvil, Anv(MGQ^{m_{2}−1}Res).

In this part of the general step of the sieve procedure we assume that the core associated with the multi-graded resolution:

MGQ^{n−1}Res(t, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a)

is of maximal possible complexity, i.e., it has the same structure as the core associated with the top level of:

MGQ^{n−2}Res(t, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s(n−2)}, …, Base_{2,v_{s(n−2)}}^{s(n−2)}, a).

Suppose that for some index s, 1 ≤ s ≤ s(n−1)−1, Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{ℓ(s)}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), and let s be the minimal index for which this happens. Then we omit the anvil, Anv(MGQ^{n−1}Res), from the list of anvils we have started the n-th step with, and replace it by going back to the ℓ(s)-th step of the sieve procedure, and start it with the limit group Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) instead of the ℓ(s)-th anvil, Anv(MGQ^{ℓ(s)−1}Res)(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), that was used in the s-th step of the process that lead to the construction of the anvil, Anv(MGQ^{n−1}Res). Since, by definition of the index ℓ(s), the parameter subgroups were enlarged at step ℓ(s) + 1, in analyzing the limit group Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) we need to take into account only those multi-graded resolutions for which their associated core resolutions are not of maximal complexity, i.e., only those multi-graded resolutions for which their core resolution does not have the same structure as the core associated with MGQ^{ℓ(s−1)}Res.

Suppose that for s(n−1)−1 (hence, for every index s, 1 ≤ s ≤ s(n−1)−1), Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Anv(MGQ^{ℓ(s)}Res)(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a). We set s(n) = s(n − 1). Let:

MGQ^{n}Res_{1}(t, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), …

…, MGQ^{n}Res_{q}(t, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

be the completions of the multi-graded resolutions in the multi-graded taut Makanin-Razborov diagram of Anv(MGQ^{n−1}Res) with respect to the parameter subgroups Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}. We will treat the multigraded resolutions MGQ^{n}Res_{j} in parallel, hence, we omit their index.

Since the multi-graded resolution MGQ^{ℓ(s(n)−1)}Res used in step ℓ(s(n)−

1) to construct the anvil, Anv(MGQ^{n−1}Res), is well-separated, with each QH vertex group in one the abelian decompositions associated with MGQ^{ℓ(s(n)−1)}Res there is an associated collection of s.c.c. that are mapped to the trivial element in the next level of the multi-graded resolution MGQ^{ℓ(s(n)−1)}Res. Each QH vertex group in the graded abelian decomposition associated with the top level of the multi-graded resolution MGQ^{ℓ(s(n)−1)}Res naturally inherits a sequence of abelian decompositions from an n-th multi-graded resolution MGQ^{n}Res. If for some such QH vertex group Q, this sequence of multi-graded abelian decompositions is not compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the next level of the multi-graded resolution MGQ^{ℓ(s(n)−1)}Res, we omit the multi-graded resolution MGQ^{n}Res from the list of completions of resolutions of the anvil, Anv(MGQ^{n−1}Res).

By theorem 4.13 the complexity of the core associated with the n-th multi-graded resolution MGQ^{n}Res are bounded by the complexity of the core associated with the multi-graded resolution MGQ^{n−1}Res, and if the complexity of the two core resolutions are equal, then the structure of the core associated with MGQ^{n}Res is identical to the structure of the core associated with MGQ^{n−1}Res. In this part of the n-th step of the sieve procedure we will also assume that the complexity of the core associated with the multi-graded resolution MGQ^{n}Res is strictly smaller than the complexity of the core associated with MGQ^{n−1}Res, hence, we are able to treat the multi-graded resolution MGQ^{n}Res according to part (4) of step n − 1 of the sieve procedure.

(3) At this stage we may assume that Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{n−1}(h_{2}, g_{1}, h_{1}, w, p, a). At this part we assume that the core associated with the multi-graded resolution MGQ^{n−1}Res is not of maximal possible complexity. Suppose that for some index s, 1 ≤ s ≤ s(n−1), Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{ℓ(s)}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), and suppose that s is the minimal index for which this happens. Then we omit the anvil, Anv(MGQ^{n−1}Res), from our list of anvils, and replace it by going back to the ℓ(s)-th step of the sieve procedure, and start it with the limit group Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), which is the subgroup generated by < t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a > in the anvil, Anv(MGQ^{n−1}Res), instead of the ℓ(s)-th limit group Q^{ℓ(s)}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) used in the ℓ(s)-th step of the process that lead to the construction of the anvil, Anv(MGQ^{n−1}Res). Since, by definition of the index ℓ(s), in case ℓ(s) < n − 1 the parameter subgroups were enlarged at step ℓ(s) + 1, and in case ℓ(s) = n − 1 the core associated with the multi-graded resolution MGQ^{n−1}Res is assumed not of maximal possible complexity, in analyzing the limit group Q^{n}Rlim(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) we need to take into account only those multi-graded resolutions for which their associated core are not of maximal complexity, i.e., those multi-graded resolutions for which their associated core resolutions have strictly smaller complexity than the complexity of the core associated with the multi-graded resolution MGQ^{ℓ(s)−1}Res used in the process of the construction of the anvil Anv(MGQ^{n−1}Res).

(4) At this part we assume that the core associated with the multi-graded resolution MGQ^{n−1}Res is not of maximal complexity, and that Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{n−1}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a). We set s(n) = s(n−1)+1, and the parameter subgroups Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)} to be the non-abelian, non-QH vertex groups and edge groups in the multigraded abelian decomposition associated with the top level of the anvil Anv(MGQ^{n−1}Res), i.e., the non-abelian, non-QH vertex groups and edge groups in the multi-graded abelian decomposition associated with Anv(MGQ^{n−2}Res). Let:

MGQ^{n}Res_{1}(t, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), …
…, MGQRes_{q}(w, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

be the completions of the resolutions in the taut multi-graded Makanin-Razborov diagram of Anv(MGQ^{n−1}Res) with respect to the parameter subgroups Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}. We analyze the n-th multi-graded resolutions MGQ^{n}Res_{j} in parallel, hence, we will omit their index.

Since the multi-graded resolution MGQ^{n−1}Res used in step n−1 to construct the anvil, Anv(MGQ^{n−1}Res), is well-separated, with each QH vertex group in one the abelian decompositions associated with MGQ^{n−1}Res there is an associated collection of s.c.c. that are mapped to the trivial element in the next level of the multi-graded resolution MGQ^{n−1}Res. Each QH vertex group in the graded abelian decomposition associated with the top level of the multi-graded resolution MGQ^{n−1}Res naturally inherits a sequence of abelian decompositions from an n-th multi-graded resolution MGQ^{n}Res. If for some such QH vertex group Q, this sequence of multi-graded abelian decompositions is not compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the next level of the multi-graded resolution MGQ^{n−1}Res, we omit the multi-graded resolution MGQ^{n}Res from the list of completions of resolutions of the anvil, Anv(MGQ^{n−1}Res).

At this part we will also assume that the core associated with the n-th multi-graded resolution MGQ^{n}Res is not of maximal possible complexity, i.e., it does not have the same structure as the core associated with the top level of the n − 1 multi-graded resolution: MGQ^{n−1}Res. The case of maximal complexity will be treated in the next parts of the general step. To treat an n-th multi-graded resolution that is not of maximal complexity, we need the following two observations, that are similar to lemmas 5.2 and 5.3.

Lemma 5.12. Let MGQ^{n}Res be an n-th multi-graded resolution that is not of maximal complexity. By construction, the limit group Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) is mapped into the limit group associated with each of the levels of the multigraded resolution MGQ^{n}Res. Let Q_{term}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) be the image of Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) in the terminal (rigid or solid) limit group of MGQ^{n}Res. Then the terminal limit group of the multi-graded resolution MGQ^{n}Res can be replaced by a collection of finitely many terminal multi-graded limit groups in which Q_{term}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), and a finite collection of terminal rigid or solid graded limit groups which are rigid or solid with respect to the parameter subgroup P =< p >.

Proof: Identical to the proof of lemma 5.2.

□

Lemma 5.13. Let MGQ^{n}Res be an n-th multi-graded resolution that is not of maximal complexity. By construction, the limit group Q^{n}(t, h_{2}, g_{1}, h_{1}, w, p, a) is mapped onto the limit group associated with each of the levels of the multi-graded resolution MGQ^{n}Res. Let Q_{2}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), Q_{2}^{n}(t_{n}, h_{2}, g_{1}, h_{1}, w, p, a) be the images of the subgroups generated by Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), Q^{n}(t_{n}, h_{2}, g_{1}, h_{1}, w, p, a) in correspondence, in the limit group associated with the second level of the multigraded resolution MGQ^{n}Res. Then either Q_{2}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of the subgroup Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), or Q_{2}^{n}(t_{n}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of the subgroup Q^{n}(t_{n}, h_{2}, g_{1}, h_{1}, w, p, a).

Proof: The claim of the proposition simply follows from the basic properties of a multi-graded resolution.

□

Suppose first that the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group Q_{2}^{n}(t, h_{2}, g_{1}, h_{1}, w, p, a), denoted Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a). With the subgroup Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a) we associate the completions of the graded resolutions that appear in its graded taut Makanin-Razborov diagram with respect to the subgroup P =< p >: GQRes_{1}(h_{2}, g_{1}, h_{1}, w, p, a), …, GQRes_{c}(h_{2}, g_{1}, h_{1}, w, p, a). We continue with each of the graded resolutions GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) in parallel.

Let T MGQ^{n}Res be the (one level) resolution corresponding to the top level of the multi-graded resolution MGQ^{n}Res. Note that by corollary 4.15, the complexity of the (multi-graded) core of T MGQ^{n}Res, is strictly smaller than the complexity of the core of the multi-graded resolution MGQ^{n−1}Res, i.e., the multi-graded resolution T MGQ^{n}Res is not of maximal complexity. If the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with the resolution GQRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a), we replace the graded resolution GQRes_{j} by starting part (4) of the the n-th step with the multi-graded resolution obtained from T MGQ^{n}Res by replacing its terminal limit group Q_{2}^{n}(t, h_{2}, g_{1}, h_{1}, w, p, a) with the maximal limit groups obtained from all specializations that factor through both Q_{2}^{n}(t, h_{2}, g_{1}, h_{1}, w, p, a) and the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with GQRes_{j}. Hence, for the rest of this part we may assume that the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group associated with GQRes_{j} is isomorphic to Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a).

Let CRes_{j}(h_{2}, g_{1}, h_{1}, w, p, a) be the graded resolution obtained from the resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the (core of the) multi-graded graded resolution T MGQ^{n}Res followed by the graded resolution GQRes_{j}. With the resolution CRes_{j} we associate a finite collection of framed resolutions (see definition 5.4), which we denote F rmCRes, and continue with each of them in parallel. If the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to a framed resolution F rmCRes associated with the graded resolution CRes_{j}, is a proper quotient of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a), we replace the framed resolution F rmCRes by starting the initial step of the procedure with the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the framed resolution F rmCRes. If the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the terminal level of the n-th multi-graded resolution T MGQ^{n}Res in the framed resolution F rmCRes, Q_{F} (h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a), we replace the framed resolution F rmCRes by starting part (4) of the the n-th step with the multigraded resolution obtained from T MGQ^{n}Res by replacing its terminal limit group Q_{2}^{n}(t, h_{2}, g_{1}, h_{1}, w, p, a) with the maximal limit groups obtained from all specializations that factor through both Q_{2}^{n}(t, h_{2}, g_{1}, h_{1}, w, p, a) and Q_{F} (h_{2}, g_{1}, h_{1}, w, p, a). Hence, we may assume that for the rest of this part the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to F rmCRes is isomorphic to Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a), and Q_{F} (h_{2}, g_{1}, w, p, a) is isomorphic to Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a). We now treat each of the framed resolutions F rmCRes in a similar way to our treatment of multi-graded resolutions in part (4) of the second step of the sieve procedure. With the framed resolution F rmCRes we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, extra P S resolutions, and with each extra P S resolution, a finite collection of formal collapse P S resolutions. We set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection graded auxiliary resolutions, collapse P S limit groups and anvils, precisely as we did in part (4) of the second step of the sieve procedure.

Suppose that Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a). In this case we set s, 1 ≤ s ≤ s(n) − 1, to be the minimal index for which: Q_{2}^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) = Q^{ℓ(s)}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a). Let:

MGQRes_{1}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), …
…, MGQRes_{d}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

be the completions of the resolutions in the taut multi-graded Makanin-Razborov diagram of Q_{2}^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) with respect to the parameter subgroups Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}. We continue with each of the multigraded resolutions MGQRes_{j} in parallel.

We set the multi-graded resolution

DRes_{j}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a)

to be the multi-graded resolution induced by the subgroup < t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a > from the n-th multi-graded resolution T MGQ^{n}Res, followed by the multigraded resolution MGQRes_{j}. If the subgroup generated by < h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the resolution DRes_{j}, Q_{D}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of the P S limit group P SHGH(h_{2}, g_{1}, h_{1}, w, p, a), we have started this branch of the sieve procedure with, we replace the multi-graded resolution MGQRes_{j} by starting a new branch of the sieve procedure with Q_{D}(h_{2}, g_{1}, h_{1}, w, p, a) instead of P SHGH(h_{2}, g_{1}, h_{1}, w, p, a). If the subgroup generated by < t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the resolution DRes_{j}, Q_{D}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) = Q^{ℓ(s)}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) we do the following. We set r, 1 ≤ r ≤ s, to be the minimal index for which the subgroup generated by < t_{ℓ(r)}, h_{2}, g_{1}, h_{1}, w, p, a > in the limit group corresponding to the resolution DRes_{j}, Q_{D}(t_{ℓ(r)}, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{n}(t_{ℓ(r)}, h_{2}, g_{1}, h_{1}, w, p, a) = Q^{ℓ(r)}(t_{ℓ(r)}, h_{2}, g_{1}, h_{1}, w, p, a). Then we omit the multi-graded resolution MGQRes_{j} from the list of completions of resolutions of the limit group Q_{term}^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), and replace it by going back to the ℓ(r)-th step of the sieve procedure, and start it with the limit group Q_{D}(t_{ℓ(r)}, h_{2}, g_{1}, h_{1}, w, p, a) instead of the ℓ(r)-th anvil, Anv(MGQ^{ℓ(r)−1}Res), that was used in the ℓ(r)-th step of the process that lead to the construction of the anvil, Anv(MGQ^{n−1}Res). Since, by definition of the index ℓ(r), the parameter subgroups were enlarged at step ℓ(r) + 1, in analyzing the limit group Q_{D}(t_{ℓ(r)}, h_{2}, g_{1}, h_{1}, w, p, a) we need to take into account only those multi-graded resolutions for which their associated core resolutions are not of maximal complexity, i.e., only those multi-graded resolutions for which their core resolution does not have the same structure as the core associated with the top level of the multi-graded resolution MGQ^{ℓ(r−1)}Res.

Suppose that Q_{D}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to:

Q^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a) = Q^{ℓ(s)}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a).

Since the multi-graded resolution MGQ^{ℓ(s−1)}Res used in step ℓ(s − 1) to construct the anvil, Anv(MGQ^{ℓ(s)−1}Res), is well-separated, with each QH vertex group in one the abelian decompositions associated with MGQ^{ℓ(s−1)}Res there is an associated collection of s.c.c. that are mapped to the trivial element in the next level of the multi-graded resolution MGQ^{ℓ(s−1)}Res. Each QH vertex group in the graded abelian decomposition associated with the top level of the multi-graded resolution MGQ^{ℓ(s−1)}Res naturally inherits a sequence of abelian decompositions from the multi-graded resolution DRes_{j}. If for some such QH vertex group Q, this sequence of multi-graded abelian decompositions is not compatible with the collection of s.c.c. on Q that are mapped to the trivial element in the next level of the multi-graded resolution MGQ^{ℓ(s−1)}Res we omit the multi-graded resolution MGQRes_{j} from the list of completions of resolutions of the limit group Q_{term}^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose that the core of a multi-graded resolution MGQRes_{j}, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQRes_{j}), is of maximal possible complexity, i.e., the core has the same structure as the core associated with the top level of the multi-graded resolution MGQ^{ℓ(s−1)}Res, associated with the anvil, Anv(MGQ^{n−1}Res), we have started the n-th step of the sieve procedure with. By theorem 4.14, since the core associated with the multi-graded resolution MGQRes_{j} is of maximal possible complexity, so is the core associated with the multi-graded resolution DRes_{j}. Hence, by theorem 4.17 every specialization of the P S limit group < h_{2}, g_{1}, h_{1}, w, p, a > that can be completed to a specializations that factors through the multi-graded resolution DRes_{j}, factors through at least one of the completions of a multigraded resolution with maximal complexity core, MGQ^{ℓ(s)}Res, associated with the anvil, Anv(MGQ^{ℓ(s)−1}Res). Since the multi-graded resolution MGQ^{ℓ(s)}Res, associated with the anvil, Anv(MGQ^{n−1}Res), is assumed to have a core which is not of maximal complexity, we may omit multigraded resolutions with maximal complexity core from the collection of multi-graded resolutions that appear in the taut Makanin-Razborov diagram of Q_{2}^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose that the multi-graded resolution MGQRes_{j} does not have a maximal complexity core resolution. If the image of the subgroup Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the multi-graded resolution MGQRes_{j} is a proper quotient of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a), we associate with it a finite collection of developing resolutions, graded auxiliary resolutions and anvils precisely as we did in case the subgroup Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a). otherwise, we analyze the multi-graded resolution MGQRes_{j} as we did in step ℓ(s) of our sieve procedure. According to the various parts of step ℓ(s) of the sieve procedure, we associate with the multi-graded resolution MGQRes_{j} a finite collection of graded resolutions:

GRes_{1}(h_{2}, g_{1}, h_{1}, w, p, a), …, GRes_{r}(h_{2}, g_{1}, h_{1}, w, p, a).

With each of these graded resolutions GRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a), we associate a graded resolution CRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a), constructed from the resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the top level of the n-th multi-graded resolution MGQ^{n}Res, T MGQRes, followed by the graded resolution GRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a). With each graded resolution CRes_{i}(h_{2}, g_{1}, h_{1}, w, p, a) we can naturally associate a finite collection of framed resolutions as we did in part (2) of the initial step of the sieve procedure. We continue with each of these framed resolutions in parallel, and denote each of them F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a).

If the limit group generated by < h_{2}, g_{1}, h_{1}, w, p, a > associated with a framed resolution F rmCRes is a proper quotient of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a), we replace the framed resolution F rmCRes by starting the initial step of the sieve procedure with that limit group (which is a proper quotient of the P S limit group we have started the initial step with). Otherwise, we follow what we did in part (2) of the initial step of the sieve procedure, and associate with the framed resolution F rmCRes a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, extra P S resolutions, and with each extra P S resolution, a finite collection of formal collapse P S resolutions. We set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions, collapse P S limit groups and anvils, precisely as we did in part (2) of the initial step of the sieve procedure.

Suppose that Q_{2}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a). In this case we continue to the next level of the n-th multi-graded quotient resolution MGQ^{n}Res. Note that by corollary 4.15, since the multigraded resolution MGQ^{n}Res is not of maximal complexity, the (multigraded) core associated with each of its levels is not of maximal complexity as well. If for some level j of the multi-graded resolution, the image of Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with this level, Q_{j}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), then from the highest such level j, we can continue as in case Q_{2}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), and associate with the n-th multi-graded resolution MGQ^{n}Res a (canonical) finite collection of framed resolutions, and with each framed resolution a canonical finite collection of Non-Rigid and Non-Solid P S resolutions, a collection of Left P S resolutions, extra P S resolutions, and formal collapse extra P S resolutions. With each extra P S resolution we further associate a finite collection of framed resolutions. We set each the framed resolutions to be a developing resolution, and with it we associate graded auxiliary resolutions, collapse P S limit groups and anvils, precisely as we did in case Q_{2}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a).

Finally, suppose that for every level j, the image of Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the j-th level of the multi-graded resolution MGQ^{n}Res, Q_{j}^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), is isomorphic to Q^{n}(t_{n−1}, h_{2}, g_{1}, h_{1}, w, p, a), In this case, by lemma 5.12, the terminal limit group of the multi-graded resolution MGQ^{n}Res, Q_{term}^{n}(t, v, h_{2}, g_{1}, h_{1}, w, p, a), is rigid or solid with respect to the parameter subgroup P =< p >.

We continue as in part (2) of the initial step of the sieve procedure. Let P B^{n}(b_{n}, p, a) be the terminal rigid or solid limit group of the multi-graded resolution MGQ^{n}Res. We set the graded resolution CRes(h_{2}, g_{2}, h_{1}, w, p, a), to be the resolution induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the sequence of core resolutions associated with the various levels of the resolution MGQ^{n}Res. With the graded resolution CRes(h_{2}, g_{1}, h_{1}, w, p, a) we associate a finite (canonical) collection of framed resolutions (see definition 5.4). With each of the framed resolutions associated with CRes(h_{2}, g_{1}, h_{1}, w, p, a) we associate a (canonical) finite collection of Non-Rigid and Non-Solid P S resolutions, a collection of Left P S resolutions, extra P S resolutions, and formal collapse extra P S resolutions. If every test sequence that factors through an extra P S resolution associated with the framed resolution F rmCRes(q, h_{2}, g_{1}, h_{1}, w, p, a), factors through a framed resolution with a bigger frame than the one associated with the framed resolution F rmCRes, we exclude this extra P S resolution from the finite collection of P S resolutions associated with the framed resolution F rmCRes. With each of the extra P S resolutions associated with the framed resolution F rmCRes, we associate a (canonical) finite collection of graded auxiliary resolutions, and with each graded auxiliary resolution we associate a (canonical) finite collection of collapse extra P S limit groups, precisely as we associated collapse extra P S resolution with an extra P S resolution associated with the one of the resolutions P SHGHRes_{j} we have started our iterative procedure with. We set the developing resolution, Dvlp(MGQRes), to be the extra P S resolution, ExtraP SRes. With each couple of a developing resolution and a collapse extra P S limit group, we associate a finite collection of anvils, precisely as we did in case Q_{2}^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is a proper quotient of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a).

(5) By part (1) we may assume that Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is isomorphic to the limit group Q^{n−1}(h_{2}, g_{1}, h_{1}, w, p, a) associated with the anvil, Anv(MGQ^{n−1}Res). Parts (2)-(4) treat all the cases in which the core associated with the multigraded resolution MGQ^{n}Res is not of maximal complexity. In this part we assume that there is no sculpted resolution associated with the anvil, Anv(MGQ^{n−1}Res), i.e., that one of the parts (1)-(4) applied to at least one of the core resolutions associated with one of the multi-graded resolutions constructed along the n − 1 step of the sieve procedure and associated with the anvil Anv(MGQ^{n−2}Res), or that all these core resolutions are of maximal possible complexity, but the developing resolution composed from the various resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from this sequence of core resolutions, is not identical to the corresponding (developing or (penetrated) sculpted) resolution associated with the anvil, Anv(MGQ^{n−2}Res), with which we have started the n − 1 step of the procedure. In this case necessarily, width(n − 1)=1, and no sculpted resolution is associated with the anvil, Anv(MGQ^{n−1}Res).

We treat this case as we treated cases 5-6 in the second step of the sieve procedure. In case the core associated with the multi-graded resolution MGQ^{n}Res and the core associated with the top level of MGQ^{n−1}Res are identical, we use the modular groups associated with the abelian decomposition associated with the core associated with the n-th multi-graded resolution MGQ^{n}Res to map this core into its image in the limit group associated with the next level of the multi-graded resolution MGQ^{n−1}Res (which is naturally mapped into the anvil, Anv(MGQ^{n−1}Res)). At this point we continue analyzing the remaining part of the anvil in the same way we have analyzed the n-th multi-graded resolution, according to parts (1)-(5) above. As long as the core associated with the constructed multi-graded resolutions are of maximal possible complexity we continue analyzing multi-graded resolutions associated with various parts of the anvil we have started with, until one of the parts (1)-(4) applies to the corresponding multi-graded resolution. If there exists a level for which one of the parts (1)-(4) applies, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions, a graded resolution, composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions, and with this graded resolution we associate a (canonical) finite collection of framed resolutions. With each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(4) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that the sequence of multi-graded core resolutions of the multigraded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the core associated with the corresponding level of the developing resolution associated with the anvil, Anv(MGQ^{n−1}Res), which means that none of the parts (1)-(4) applies to any of these multi-graded resolutions. In this case we examine the structure of the corresponding developing resolution according to proposition 5.9. The developing resolution is a framed resolution of a resolution built from a sequence of resolutions induced from corresponding core resolutions. Each of the induced resolutions is a resolution induced by the (image of the) subgroup < q, h_{2}, g_{1}, h_{1}, w, a > from the corresponding core resolution, and with each level of the induced resolution there is an associated (framed) multi-graded abelian decomposition (see section 3 of [Se4] for the construction of the induced resolution, and definition 5.4 for the construction of framed resolutions).

If the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions used to construct the developing resolution in the n-th step of the procedure, is not identical to the structure of the developing resolution associated with the anvil Anv(MGQ^{n−1}Res), proposition 5.9 implies that there exists some level j for which the structure of the graded abelian decomposition associated with the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > above level j are identical to the structure of the graded abelian decompositions associated with the corresponding levels in the developing resolutions we have started the n-th step with, and in level j, either the number of factors in the graded free decomposition associated with the graded abelian decomposition associated with the resolution induced by < q, h_{2}, g_{1}, h_{1}, w, p, a > is strictly smaller than the number of factors in the corresponding graded free decomposition associated with the corresponding level in the developing resolution we have started the n-th step with, and in case of equality in the number of factors, the complexity of the graded abelian decomposition associated with the resolution induced by < q, h_{2}, g_{1}, h_{1}, w, p, a > is strictly smaller than the complexity of the graded abelian decomposition associated with the corresponding level of that developing resolution. In this case we do the following. With the graded resolution constructed from the various resolutions induced by the subgroups < q, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various multi-graded resolutions constructed along the various levels of the n-th step of the sieve procedure, we associate a (canonical) finite collection of framed resolutions (definition 5.4). With each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that the sequence of multi-graded core resolutions of the multigraded resolutions constructed by the process described above, are all of maximal complexity, and the sequence of abelian decompositions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > is identical to the corresponding abelian decompositions associated with the developing resolution, Dvlp, that is associated with the anvil, Anv(MGQ^{n−1}Res). In this case we set SCRes(s, h_{2}, g_{1}, h_{1}, w, p, a) to be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions used to construct the developing resolution in the n-th step of the procedure, and suppose that the structure of the graded resolution SCRes(s, h_{2}, g_{1}, h_{1}, w, p, a) is identical to the structure of the developing resolution associated with the anvil Anv(MGQ^{n−1}Res).

With the graded resolution SCRes we naturally associate a (canonical) collection of framed resolutions (definition 5.4). If every ungraded resolution that factors through the graded resolution SCRes, factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the n-th step of the procedure with, or through a framed resolution F rmSCRes associated with it, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes, we call the sequence of multi-graded resolutions from which SCRes(s, h_{2}, g_{1}, h_{1}, w, p, a) was constructed, a terminal resolution, and do not continue with it to the next step of the sieve procedure. Otherwise, we do the following.

Let < c, h_{2}, g_{1}, h_{1}, w, p, a > be the collapsed P S limit group associated with the anvil, Anv(MGQ^{n−1}Res). With each multi-graded resolution constructed in one of the levels, we associate its core resolution with respect to the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. We replace the core resolutions associated with the multi-graded resolutions constructed in the various levels, and with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >, by the corresponding penetrated core resolutions (see definition 4.20), of the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > with respect to the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. Note that by construction, the original core resolutions are embedded into the corresponding penetrated core resolutions. Then we study the structure of the graded resolutions composed from resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the corresponding penetrated resolutions according to proposition 5.10.

Suppose that the structure of at least one of the graded abelian decompositions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, is not identical to the structure of the corresponding abelian decompositions in the developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). Then we do the following.

We set the graded resolution GRes_{1}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated in the various levels with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. With the resolution GRes_{1} we associate a finite collection of framed resolutions, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We further associate with each framed resolution the graded resolution SCRes, which we denote SCRes_{1}^{n,1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and call the first sculpted resolution of width 1. We set each of the extra P S resolutions to be both a first sculpted resolution of width 2, SCRes_{1}^{n,2}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and a penetrated sculpted resolution, which we denote P enSCRes_{1}^{n,2}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate a finite collection of graded auxiliary resolutions and collapse P S limit groups, and a finite collection of anvils. We set the penetrated resolution associated with the anvil, to be the developing resolution. Finally, we associate with the anvil the sequence of core resolutions associated with the multi-graded resolutions constructed in the various levels, and the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >, which we call a Carrier, and denote it Carrier_{2}^{n}. In this case, width(n) = 2 and sc(n, 1) = sc(n, 2) = 1.

Suppose that the structure of all the graded abelian decompositions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, are identical to the structure of the corresponding abelian decompositions in the developing resolution. Recall that we denote the collapse P S limit group associated with the anvil Anv(MGQ^{n−1}Res) by < c, h_{2}, g_{1}, h_{1}, w, p, a >. We set the resolution GRes_{2}(r_{2}, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions constructed in the various levels and the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. Note that GRes_{2} is graded with respect to the defining parameter subgroup P =< p >, and the resolution GRes_{1} is embedded into the graded resolution GRes_{2}. With the graded resolution GRes_{2}, we associate a (canonical) finite collection of framed resolution, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We further associate with the graded resolution GRes_{2} the graded resolution SCRes which we call first sculpted resolution, and denote SCRes_{1}^{n,j}(s, h_{2}, g_{1}, h_{1}, w, p, a) for j = 1, 2, and the resolution GRes_{1} which we call first penetrated sculpted resolution, and denote P enSCRes_{1}^{n,2}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a). Finally, we set each of the extra P S resolutions to be a developing resolution, which we denote Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and anvils. In this case, width(n) = 2, and sc(n, 1) = sc(n, 2) = 1.

(6) Parts (1)-(5) treat all the cases in which there is no sculpted resolution associated with the anvil, Anv(MGQ^{n−1}Res). In this part we assume that there are (at least one) sculpted resolutions associated with the anvil, Anv(MGQ^{n−1}Res), but no carriers are associated with it, hence, in particular, width(n−1) ≥ 2. We further assume that the core associated with the n-th multi-graded resolution, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQ^{n}Res is of maximal possible complexity.

With the anvil, Anv(MGQ^{n−1}Res), we have associated a finite set of sculpted resolutions and a developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). Parts (1)-(5) studied multi-graded resolutions with respect to the first sculpted resolution, SCRes_{1}^{n−1,1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. At this point we apply parts (1)-(4) to study the multi-graded resolutions of the anvil, Anv(MGQ^{n−1}Res), with respect to the second sculpted resolution, SCRes_{2}^{n−1,2}. i.e., we study multi-graded resolutions, with respect to the parameter subgroups that are associated with the various levels of the second sculpted resolution, SCRes_{1}^{n−1,1}, where, in particular, the subgroups Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, and the corresponding parts of the formed core resolution, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQ^{n}Res, are assumed elliptic throughout the analysis, i.e., they are assumed to be elliptic in all the associated multi-graded resolutions. If none of the parts (1)-(4) applies to the obtained multi-graded resolutions, we study the multi-graded resolutions of the anvil with respect to third sculpted resolution, and so on, until we study the structure of the multi-graded resolutions of the anvil Anv(MGQ^{n−1}Res) with respect to the developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a).

With each such sequence of multi-graded resolutions we associate a multigraded resolution, which we still denote, MGQ^{n}Res. and since in this part we study multi-graded resolutions with core resolutions of maximal possible complexity, we continue only with those multi-graded resolutions, MGQ^{n}Res, for which the associated core resolution MGCore(< h_{2}, g_{1}, h_{1}, w, p, a > , MGQ^{n}Res, is of maximal possible complexity (cf. theorem 4.18).

At this point we continue analyzing the remaining part of the anvil, level after level, in the same way we have analyzed the n-th multi-graded resolution. i.e., we first check if one of the parts (1)-(4) applies to the multigraded resolution associated with the corresponding part of the anvil and the first sculpted resolution, and if the associated core is of maximal possible complexity, we analyze multi-graded resolutions with respect to the second sculpted resolution, third sculpted resolution, and so on, until we analyze the multi-graded resolutions with respect to the corresponding level of the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). As long as the core associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > and the constructed multi-graded resolutions are of maximal possible complexity, we continue analyzing multi-graded resolutions associated with various parts of the anvil we have started with, until one of the parts (1)-(4) applies to the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > and the corresponding multi-graded resolution. If there exists a level for which one of the parts (1)-(4) applies, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions, a (canonical) finite collection of framed resolutions and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(4) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils. In this case the width associated with the anvil, width(n) = 1.

Suppose that none of the parts (1)-(4) applies to any of the multi-graded resolutions associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. In particular, the core resolutions associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the core associated with the corresponding level of the first sculpted resolution, SCRes_{1}^{n−1}. In this case we proceed as in part (5) and examine the structure of the corresponding resolution associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. The first sculpted resolution, SCRes_{1}^{n−1,1} is built from a sequence of resolutions induced by < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions constructed along the various levels. Each of the induced resolutions is a resolution induced by the (image of the) subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, a > from the corresponding core resolution, and with each level of the induced resolution there is an associated (induced) multi-graded abelian decomposition.

We analyze these graded abelian decompositions according to proposition 5.9. If the structure of the resolution composed from the resolutions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various multi-graded resolutions constructed along the various levels in the n-th step of the sieve procedure, is not identical to the structure of the first sculpted resolution associated with the anvil Anv(MGQ^{n−1}Res), SCRes_{1}^{n−1,1}, we follow what we did in part (5) and associate with the graded resolution constructed from the various resolutions induced by the subgroups < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various multi-graded resolutions constructed along the various levels of the n-th step of the sieve procedure, a (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

In this case we set width(n) = 1.

Suppose that the structure of the resolution composed from the resolutions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions constructed in the n-th step of the sieve procedure, is identical to the structure of the first sculpted resolution associated with the anvil Anv(MGQ^{n−1}Res). Let SCRes_{1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded constructed in the n-th step, and suppose that the structure of the graded resolution SCRes_{1} is identical to the structure of the first sculpted resolution SCRes_{1}^{n−1,2} associated with the anvil Anv(MGQ^{n−1}Res).

With the graded resolution SCRes_{1} we naturally associate a (canonical) collection of framed resolutions (definition 5.4). If every ungraded resolution that factors through the graded resolution SCRes(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the n-th step of the procedure with, or through a framed resolution F rmSCRes associated with it, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes, we call the sequence of multigraded resolutions from which the graded resolution SCRes_{1} was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we continue by examining the core resolutions and the multi-graded resolutions associated with the second sculpted resolution, SCRes_{2}^{n−1,2}(s_{2}, h_{2}, g_{1}, h_{1}, w, p, a) associated with the anvil, Anv(MGQ^{n−1}Res).

We start by replacing the core resolutions associated with the multigraded resolutions constructed in the various levels, and with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >, by the corresponding penetrated core resolutions (see definition 4.19), of the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > with respect to the core resolutions associated with the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a >. Note that by construction, the original core resolutions are embedded into the corresponding penetrated core resolutions. Then we study the structure of the graded resolutions composed from resolutions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > according to proposition 5.10.

If the graded abelian decompositions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, is not identical to the structure of the corresponding abelian decompositions in the first sculpted resolution, SCRes_{1}^{n−1,2}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), we do the following. We set the graded resolution GRes_{1}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated in the various levels with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. With the resolution GRes_{1} we associate a finite collection of framed resolutions, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. We further associate with each framed resolution the graded resolution SCRes_{1}, which we denote SCRes_{1}^{n,1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and call the first sculpted resolution of width 1. We set each of the extra P S resolutions to be both a first sculpted resolution of width 2, SCRes_{1}^{n,2}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) and penetrated sculpted resolution, which we denote P enSCRes_{1}^{n,2}(r, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate a finite collection of graded auxiliary resolutions and limit groups, collapse P S limit groups and anvils. We set each of the extra P S resolutions to be the developing resolution. Finally, We associate with the anvil the sequence of core resolutions associated with the multi-graded resolutions constructed in the various levels, and the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a >, which we call a Carrier, and denote it Carrier_{2}^{n}. In this case, width(n) = 2.

Suppose that the graded abelian decompositions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, is identical to the structure of the corresponding abelian decompositions in the first sculpted resolution SCRes_{1}^{n−1,2}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) associated with the anvil, Anv(MGQ^{n−1}Res). In this case we examine the core resolutions and the multi-graded resolutions associated with the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > and the second sculpted resolution, SCRes_{2}^{n−1,2}(s_{2}, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose that there exists a level for which one of the parts (1)-(4) applies to the multi-graded resolutions associated with the second sculpted resolution SCRes_{2}^{n−1,2} and the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a >. With each multi-graded resolution constructed in one of the levels, we associated its core resolution with respect to the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a >. We associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions (with respect to the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a >), a graded resolution GRes(s_{2}, h_{2}, g_{1}, h_{1}, w, p, a) composed from resolutions induced by the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various levels. With the graded resolution GRes(s_{2}, h_{2}, g_{1}, h_{1}, w, p, a) we associate its (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we did in the part (1)-(4) that applies. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate the resolution SCRes_{1}, that we now denote SCRes_{1}^{n,j}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) for j = 1, 2, and call first sculpted resolution, and the resolution GRes_{1}, which we denote P enSCRes_{1}^{n,2}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a) and call penetrated sculpted resolution, and a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils. We set width(n) = 2.

Suppose that none of the parts (1)-(4) applies to any of the multi-graded resolutions associated with the second sculpted resolution, SCRes_{2}^{n−1,2}, and the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a >. In particular, the core resolutions associated with the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed by the process described above, are all of maximal complexity, i.e., each of these multi-graded core resolutions have the same structure as the core associated with the corresponding level of the second sculpted resolution, SCRes_{2}^{n−1,2}. In this case we examine the structure of the corresponding resolution associated with the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > according to proposition 5.9. The second sculpted resolution, SCRes_{2}^{n−1,2}, is a framed resolution of a resolution that is built from a sequence of resolutions induced by < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from corresponding core resolutions. Each of the induced resolutions is a resolution induced by the (image of the) subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, a > from the corresponding core resolution, and with each level of the induced resolution there is an associated (induced) multi-graded abelian decomposition. If the structure of the resolution composed from the resolutions induced by the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions constructed along the various levels, is not identical to the structure of the second sculpted resolution, SCRes_{2}^{n−1,2}, associated with the anvil, Anv(MGQ^{n−1}Res), we repeat what we did in case one of the parts (1)-(4) applies to one of the core resolutions associated with the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed along the process. We associate with the graded resolution constructed from the various resolutions induced by the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions constructed along the various levels of the n-th step of the sieve procedure, a (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. Finally, we set each of the extra P S resolutions to be a developing resolution, and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils. With each anvil, we associate a first sculpted resolution, SCRes_{1}^{n,j}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a) for j = 1, 2, and a penetrated sculpted resolution, P enSCRes_{1}^{n,2}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), precisely as we did in case one of the parts (1)-(4) applies to one of the core resolutions associated with the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed along the various levels. We set width(n) = 2, and sc(n, 1) = sc(n, 2) = 1.

Suppose that the structure of the resolution composed from the resolutions induced by the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions associated with the various levels along the n-th step of the sieve procedure, is identical to the structure of the second sculpted resolution, SCRes_{2}^{n−1,2}, associated with the anvil, Anv(MGQ^{n−1}Res). Let SCRes_{2}(s_{2}, h_{2}, g_{1}, h_{1}, w, p, a) be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions constructed in the n-th step and the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a >. By our assumptions the structure of the graded abelian decompositions associated with the various levels in the graded resolution SCRes_{2} is identical to the structure of the corresponding graded abelian decompositions in the second sculpted resolution SCRes_{2}^{n−1,2} associated with the anvil, Anv(MGQ^{n−1}Res). With the graded resolution SCRes_{2} we naturally associate a (canonical) collection of framed resolutions (definition 5.3). If every ungraded resolution that factors through the graded resolution SCRes_{2}, factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the n-th step of the procedure with, or through a framed resolution F rmSCRes associated with it, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes_{2}, we call the sequence of multi-graded resolutions from which the graded resolution SCRes_{2} was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we continue inductively.

With the anvil, Anv(MGQ^{n−1}Res), we assumed that there are width(n−

1) − 1 associated sculpted resolutions:

SCRes_{1}^{n−1,width(n−1)}, …, SCRes_{width(n−1)−1}^{n−1,width(n−1)}

and no carrier. Suppose that for some index e, 2 < e ≤ width(n−1)−1, the structure of the core resolutions and the abelian decompositions associated with the resolutions associated with the first e − 1 sculpted and penetrated sculpted resolutions and the core resolutions associated with the subgroup < s_{e−1}, h_{2}, g_{1}, h_{1}, w, p, a), are identical to the core resolutions and the abelian decompositions associated with the first e − 1 sculpted resolutions:

SCRes_{1}^{n−1,e−1}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), …, SCRes_{e−1}^{n−1,e−1}(s_{e−1}, h_{2}, g_{1}, h_{1}, w, p, a).

We continue by treating the core resolutions associated with the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >, and the sculpted resolution SCRes_{e}^{n−1,e} in the same way we treated the second sculpted resolution, and the core resolutions associated with it.

We start by replacing the core resolutions associated with the multigraded resolutions constructed in the various levels, and with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >, by the corresponding penetrated core resolutions of the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > with respect to the core resolutions associated with the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >. Then we study the structure of the graded resolutions composed from resolutions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > according to proposition 5.10. If the graded abelian decompositions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, is not identical to the structure of the corresponding abelian decompositions in the first sculpted resolution, SCRes_{1}^{n−1,e}, we associate with the graded resolution induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the various penetrated core resolutions a (canonical) finite collection of framed resolutions, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, and formal collapse P S resolutions. We further associate with each framed resolution the the graded resolutions SCRes_{j}(s_{j}, h_{2}, g_{1}, h_{1}, w, p, a) for 1 ≤ j ≤ e−1, which we denote SCRes_{j}^{n,d}(s_{j}, h_{2}, g_{1}, h_{1}, w, p, a) for 1 ≤ j ≤ d ≤ e − 1 in correspondence, and call sculpted resolutions, and with them we associate the corresponding penetrated sculpted resolutions. We set each of the extra P S resolutions to be both the first sculpted resolution of width e, which we denote SCRes_{1}^{n,e}(s_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and penetrated sculpted resolution, which we denote P enSCRes_{1}^{n,e}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate a finite collection of graded auxiliary resolutions and limit groups, and a finite collection of anvils. We set the extra P S resolution associated with the anvil, to be the developing resolution. Finally, We associate with the anvil the sequence of core resolutions associated with the multi-graded resolutions constructed in the various levels, and the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >, which we call a Carrier, and denote it Carrier_{e}^{n}. In this case we set width(n) = e, sc(n, d) = d for d = 1, …, e − 1, and sc(n, e) = 1. If the structure of all the graded abelian decompositions induced by the subgroup < s_{1}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, are identical to the structure of the corresponding abelian decompositions in the first sculpted resolution of width e, SCRes_{1}^{n−1,e}, we associate penetrated core resolutions with the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > and the second sculpted resolution of width e, SCRes_{2}^{n−1,e}, and examine the resolution composed from the resolutions induced by the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated with the various levels according to proposition 5.10. If the abelian decompositions induced by the subgroup < s_{2}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions are not identical to the abelian decompositions associated with SCRes_{2}^{n−1,e}, we continue as in our treatment of SCRes_{1}^{n−1,e}. Otherwise, in case all the induced abelian decompositions are identical to those associated with SCRes_{2}^{n−1,e}, we associate penetrated core resolutions with the subgroup < s_{3}, h_{2}, g_{1}, h_{1}, w, p, a > and the third sculpted resolution of width e, SCRes_{3}^{n−1,e}, and examine the resolution composed from the resolutions induced by the subgroup < s_{3}, h_{2}, g_{1}, h_{1}, w, p, a > from the penetrated core resolutions associated with the various levels according to proposition 5.10. We continue with this process as long as the structures of the abelian decompositions induced by the relevant subgroups are identical to the structures of the graded abelian decompositions of the corresponding sculpted resolution.

Suppose that for every e′, 1 ≤ e′ ≤ e − 1, the structures of all the graded abelian decompositions induced by the subgroup < s_{e′}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels, are identical to the structures of the corresponding abelian decompositions in the e′ sculpted resolution of width e, SCRes_{e′}^{n−1,e′}. Suppose that there exists a level for which one of the parts (1)-(4) applies to the multi-graded resolutions associated with the e-th sculpted resolution SCRes_{e}^{n−1,e}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) and the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >. With each multi-graded resolution constructed in one of the levels, we associated its core resolution with respect to the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >. In this case, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions (with respect to the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >), a graded resolution composed from resolutions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various levels. With the graded resolution we associate its (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, and formal collapse P S resolutions, precisely as we did in the part (1)-(4) that applies. Finally, we set each of the extra P S resolutions to be a developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate the resolutions SCRes_{j}(s_{j}, h_{2}, g_{1}, h_{1}, w, p, a) for 1 ≤ j ≤ e − 1, which we denote SCRes_{j}^{n,d}(s_{j}, h_{2}, g_{1}, h_{1}, w, p, a) for j ≤ d ≤ e, and call sculpted resolutions, together with their corresponding penetrated sculpted resolutions: P enSCRes_{j}^{n,d}(r_{j}, h_{2}, g_{1}, h_{1}, w, p, a) for 1 ≤ j ≤ e − 1, and j ≤ d ≤ e, and a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils. We set width(n) = e, sc(n, e) = e − 1 and sc(n, d) = d for d = 1, …, e − 1.

Suppose that none of the parts (1)-(4) applies to any of the multi-graded resolutions associated with the e-th sculpted resolution, SCRes_{e}^{n−1,e}, and the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >. In particular, the core resolutions associated with the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed by the process described above, are all of maximal complexity. In this case we examine the structure of the corresponding resolution associated with the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > according to proposition 5.9. If the structure of the resolution composed from the resolutions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions constructed along the various levels, is not identical to the structure of the e-th sculpted resolution, SCRes_{e}^{n−1,e}, associated with the anvil, Anv(MGQ^{n−1}Res), we repeat what we did in case one of the parts (1)-(4) applies to one of the core resolutions associated with the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed along the process.

Suppose that the structure of the resolution composed from the resolutions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions associated with the various levels along the n-th step of the sieve procedure, is identical to the structure of the e-th sculpted resolution of width e, SCRes_{e}^{n−1,e}, associated with the anvil, Anv(MGQ^{n−1}Res).

Let SCRes_{e}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions constructed in the n-th step and the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a >. By our assumptions the structure of the graded abelian decompositions associated with the various levels in the graded resolution SCRes_{e}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) is identical to the structure of the corresponding graded abelian decompositions in the e-th sculpted resolution SCRes_{e}^{n−1,e} associated with the anvil Anv(MGQ^{n−1}Res). If every ungraded resolution that factors through the graded resolution SCRes_{e}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a), factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the n-th step of the procedure with, or through a framed resolution F rmSCRes associated with it, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes_{e}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a), we call the sequence of multi-graded resolutions from which the graded resolution SCRes_{e}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we continue inductively until we go over the core resolutions associated with the entire collection of sculpted resolutions.

After studying the penetrated sculpted resolutions associated with the sculpted resolution:

SCRes_{1}^{n−1,width(n−1)}, …, SCRes_{width(n−1)−1}^{n−1,width(n−1)}

associated with the anvil, Anv(MGQ^{n−1}Res), we treat in a similar way the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and the core resolutions associated with it along the various levels. We start by studying the penetrated sculpted resolution associated with the first sculpted resolution, SCRes_{1}^{n−1,width(n−1)}, and the core resolutions associated with the developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a) and the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >, in the same way we studied the penetrated sculpted resolution associated with the first sculpted resolution and the core resolutions associated with the next sculpted resolutions. We continue by studying the penetrated sculpted resolutions associated with the second sculpted resolution, SCRes_{2}^{n−1}, and so on until we get to study the penetrated sculpted resolution associated with the last sculpted resolution, SCRes_{width(n−1)−1}^{n−1,width(n−1)} and the core resolutions associated with the developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a).

Suppose that for every e, 1 ≤ e ≤ width(n−1)−1, the structures of all the graded abelian decompositions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the various levels and the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), are identical to the structure of the corresponding abelian decompositions in the e-th sculpted resolution SCRes_{e}^{n−1,width(n−1)}. Suppose that there exists a level for which one of the parts (1)-(4) applies to the multi-graded resolutions associated with the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). With each multi-graded resolution constructed in one of the levels, we associated its core resolution with respect to the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >. In this case, we associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions (with respect to the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >), a graded resolution composed from resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various levels. With the graded resolution we associate its (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, and formal collapse P S resolutions, precisely as we did in the part (1)-(4) that applies. Finally, we set each of the extra P S resolutions to be a developing resolution, (still denoted) Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), set width(n) = width(n − 1), and sc(n, d) = sc(n − 1, d) for d = 1, …, width(n), and with it we associate the resolutions: SCRes_{e} for 1 ≤ e ≤ width(n − 1) − 1, which we denote SCRes_{e}^{n,d}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) for e ≤ d ≤ width(n − 1), and call sculpted resolutions, together with their corresponding penetrated sculpted resolutions: P enSCRes_{e}^{n,d}(r_{e}, h_{2}, g_{1}, h_{1}, w, p, a), and a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that none of the parts (1)-(4) applies to any of the multi-graded resolutions associated with the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >. In particular, the core resolutions associated with the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed by the process described above, are all of maximal complexity. In this case we examine the structure of the corresponding resolution associated with the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > according to proposition 5.9. If the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions constructed along the various levels, is not identical to the structure of the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), associated with the anvil, Anv(MGQ^{n−1}Res), we repeat what we did in case one of the parts (1)-(4) applies to one of the core resolutions associated with the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed along the process.

Suppose that the structure of the resolution composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions associated with the various levels along the n-th step of the sieve procedure, is identical to the structure of the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), associated with the anvil, Anv(MGQ^{n−1}Res). In this case we set SCRes_{width(n)}(q, h_{2}, g_{1}, h_{1}, w, p, a) to be the graded resolution (graded with respect to the parameter subgroup P =< p >), composed from the resolutions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the various core resolutions associated with the multi-graded resolutions constructed in the n-th step and the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a >.

By our assumptions the structure of the graded abelian decompositions associated with the various levels in the graded resolution SCRes_{width(n)} is identical to the structure of the corresponding graded abelian decompositions in the developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a) associated with the anvil Anv(MGQ^{n−1}Res). With the graded resolution SCRes_{width(n)} we naturally associate a (canonical) collection of framed resolutions (definition 5.4). If every ungraded resolution that factors through the graded resolution SCRes_{width(n)}, factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the n-th step of the procedure with, or through a framed resolution F rmSCRes associated with it, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes_{width(n)}, we call the sequence of multigraded resolutions from which the graded resolution SCRes_{width(n)} was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we continue by studying the core resolutions associated with the collapse P S limit group associated with the anvil, Anv(MGQ^{n−1}Res).

Recall that we denote the collapse P S limit group associated with the anvil Anv(MGQ^{n−1}Res) by < c, h_{2}, g_{1}, h_{1}, w, p, a >. At this point we study the penetrated resolutions associated with core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed along the various steps, in a similar way to our study of the core resolutions associated with the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). We start by studying sequentially the penetrated sculpted resolutions associated with the sculpted resolutions:

SCRes_{1}^{n−1,width(n−1)}, …, SCRes_{width(n−1)−1}^{n−1,width(n−1)}

and the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >, precisely in the same way we studied sequentially the penetrated sculpted resolutions associated with the core resolutions associated with the developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). If for every e, 1 ≤ e ≤ width(n − 1) − 1, the structures of all the graded abelian decompositions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the sculpted resolution, SCRes_{e}^{n−1,width(n−1)}, and the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >, are identical to the structures of the corresponding abelian decompositions in the e-th sculpted resolution SCRes_{e}^{n−1,width(n−1)}, we study the penetrated sculpted resolution associated with the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >, precisely in the same way we studied the penetrated core resolutions associated with the sculpted resolutions SCRes_{e}^{n−1,width(n−1)}. If the structure of all the graded abelian decompositions induced by the subgroup < q, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >, are identical to the structure of the corresponding abelian decompositions in developing resolution Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), we continue as follows.

We set the resolution DRes(e, c, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions constructed in the various levels and the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. Note that DRes(e, c, h_{2}, g_{1}, h_{1}, w, p, a) is graded with respect to the defining parameter subgroup P =< p >. Note that all the penetrated resolutions are embedded into the graded resolution DRes(q, t, v, h_{2}, g_{1}, h_{1}, w, p, a). With the graded resolution DRes(e, c, h_{2}, g_{1}, h_{1}, w, p, a), we associate a (canonical) finite collection of framed resolution, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We set width(n) = width(n − 1) + 1, sc(n, width(n)) = width(n) − 1, and sc(n, d) = d for d = 1, …, width(n) − 1. We further associate with the graded resolution DRes the graded resolutions: SCRes_{e} for 1 ≤ e ≤ sc(n, d), where 1 ≤ d ≤ width(n), which we denote SCRes_{e}^{n,d}, and call sculpted resolutions, together with their corresponding penetrated sculpted resolutions: P enSCRes_{e}^{n,d}. Finally, we set each of the extra P S resolutions to be a developing resolution, (still denoted) Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), and with it we associate a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and anvils.

(7) Parts (1)-(5) treat all the cases in which there is no sculpted resolution associated with the anvil, Anv(MGQ^{n−1}Res), and part (6) treat the case in which there are sculpted resolutions associated with the anvil, Anv(MGQ^{n−1}Res), but there is no carrier associated with it. In this part we treat the general case. We assume that the anvil Anv(MGQ^{n−1}Res) has width, width(n −

1) ≥ 2, and with it there are associated a developing resolution, Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), sculpted resolutions, which we denote SCRes_{e}^{n−1,d}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a), and corresponding penetrated sculpted resolutions, P enSCRes_{e}^{n−1,d}(r_{e}, h_{2}, g_{1}, h_{1}, w, p, a), where 1 ≤ d ≤ width(n − 1), and for each d, 1 ≤ e ≤ sc(n − 1, d), where sc(n − 1, d) is the number of sculpted resolutions in width d and step n − 1. With the anvil, Anv(MGQ^{n−1}Res), and a subset of the width indices d, 1 ≤ d ≤ width(n − 1), there are also associated carriers, Carrier_{d}^{n−1}. We further assume that the core associated with the n-th multi-graded resolution, MGCore(< h_{2}, g_{1}, h_{1}, w, p, a >, MGQ^{n}Res) is of maximal possible complexity.

Let d_{0}, 2 ≤ d_{0} ≤ width(n − 1) be the maximal width for which for every width d, 1 ≤ d ≤ d_{0} − 1, there is no carrier associated with the core resolutions at width d, i.e., for every d ≤ d_{0} − 1, there does not exist a carrier, Carrier_{d}^{n−1}. Since d_{0} is the maximal such index, and we assume that there exists at least one carrier associated with the anvil, at width d_{0} there must be a carrier associated with the anvil, Anv(MGQ^{n−1}Res).

Hence, at width d_{0}, there are sc(n − 1, d_{0}) associated sculpted resolutions, SCRes_{e}^{n−1,d_{0}} for e = 1, …, sc(n−1, d_{0}), corresponding penetrated sculpted resolutions, P enSCRes_{e}^{n−1,d_{0}}, and a carrier, Carrier_{d_{0}}^{n−1}. We start our analysis of the the anvil, Anv(MGQ^{n−1}Res), and its associated core resolutions, sculpted and penetrated sculpted resolutions and carriers, by analyzing the sculpted resolutions associated with the core resolutions of width at most d_{0} − 1, according to the finite process presenting in part (6) of the general step of the sieve procedure, i.e., as in the case in which there are no carriers associated with the anvil.

Suppose that for every width d, 1 ≤ d ≤ d_{0} − 1, and every index e, 1 ≤ e ≤ d, the structures of all the graded abelian decompositions induced by the various subgroups < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > associated with the sculpted and penetrated sculpted resolutions SCRes_{e}^{n−1,d} and P enSCRes_{e}^{n−1,d} from the terminal levels of the various penetrated core resolutions constructed along the analysis of the anvil, Anv(MGQ^{n−1}Res), are identical to the structure of the corresponding abelian decompositions in the sculpted and penetrated sculpted resolutions SCRes_{e}^{n−1,d} and P enSCRes_{e}^{n−1,d}. Suppose further that the collection of sequences of core resolutions of width at most d_{0} − 1, is not a terminal resolution.

Suppose that there exists a level for which one of the parts (1)-(4) applies to the multi-graded resolutions associated with the carrier, Carrier_{d_{0}}^{n−1}. In this case we follow the analysis presented in part (6), with respect to the obtained width-d_{0} sequence of core resolutions (i.e., the sequence of core resolutions associated with the carrier Carrier_{d_{0}}^{n−1}), and the sculpted resolutions associated with width d_{0} −1, SCRes_{e}^{n−1,d_{0}−1} for e = 1, …, d_{0} − 1. Since we have analyzed the sculpted resolutions of width at most d_{0} − 1 according to part (6) of the general step of the sieve procedure, with each sculpted resolution SCRes_{e}^{n−1,d_{0}−1} we associated a graded resolution SCRes_{e} that is composed from resolutions induced by the subgroups < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various levels.

We start by studying the penetrated sculpted resolution associated with the first sculpted resolution, SCRes_{1}^{n−1,d_{0}−1} and the core resolutions associated with the carrier, Carrier_{d_{0}}^{n−1} in the same way we studied the penetrated sculpted resolution associated with the first sculpted resolution and the core resolutions associated with the next sculpted resolutions in part (6) of the general step. We continue by studying the penetrated sculpted resolutions associated with the second sculpted resolution, SCRes_{2}^{n−1,d_{0}−1}, and so on until we get to study the penetrated sculpted resolution associated with the last sculpted resolution of width d_{0} − 1, SCRes_{d_{0}−1}^{n−1,d_{0}−1} and the core resolutions associated with the carrier, Carrier_{d_{0}}^{n−1}.

Suppose that for every e, 1 ≤ e ≤ d_{0} − 1, the structures of all the graded abelian decompositions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions are identical to the structures of the corresponding abelian decompositions in the e sculpted resolution SCRes_{e}^{n−1,d_{0}−1}. Let < c_{d_{0}}, h_{2}, g_{1}, h_{1}, w, p, a > be the collapsed P S limit group associated with the sequence of core resolutions of width d_{0}. We associate with the multi-graded resolutions constructed along the various levels, and their corresponding multi-graded core resolutions of width d_{0} (with respect to the subgroup < c_{d_{0}}, h_{2}, g_{1}, h_{1}, w, p, a >), a graded resolution composed from resolutions induced by the subgroup < c_{d_{0}}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various levels. With the graded resolution we associate its (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, and formal collapse P S resolutions, precisely as we did in the part (1)-(4) that applies. Finally, we set each of the extra P S resolutions to be a developing resolution, (still denoted) Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), set width(n) = d_{0} and sc(n, d_{0}) = d_{0} − 1, and with the developing resolution we associate the resolutions: SCRes_{e} for 1 ≤ e ≤ d_{0} − 1, which we denote SCRes_{e}^{n,d}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) for e ≤ d ≤ d_{0}, and call sculpted resolutions, together with their corresponding penetrated sculpted resolutions: P enSCRes_{e}^{n,d}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a) for 1 ≤ e ≤ d_{0} − 1, and e ≤ d ≤ d_{0}, and a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that none of the parts (1)-(4) applies to the multi-graded resolutions associated with the carrier, Carrier_{d_{0}}^{n−1}, and sc(n − 1, d_{0}) > 1. In this case we start by following the analysis presented in part (6), with respect to the obtained width-d_{0} sequence of core resolutions (i.e., the sequence of core resolutions associated with the carrier Carrier_{d_{0}}^{n−1}), and the sculpted resolutions associated with width d_{0} except for the last sculpted resolution associated with this width: SCRes_{e}^{n−1,d_{0}} for e = 1, …, sc(n − 1, d_{0}) − 1. Since we have analyzed the sculpted resolutions of width at most d_{0} − 1 according to part (6) of the general step of the sieve procedure, with each sculpted resolution SCRes_{e}^{n−1,d_{0}} we associated a graded resolution SCRes_{e}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) that is composed from resolutions induced by the subgroups < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the various levels.

We study the penetrated sculpted resolution associated with the first sculpted resolution, SCRes_{1}^{n−1,d_{0}} and the core resolutions associated with the carrier, Carrier_{d_{0}}^{n−1} in the same way we studied the penetrated sculpted resolution associated with the first sculpted resolution and the core resolutions associated with the next sculpted resolutions in part (6) of the general step. We continue by studying the penetrated sculpted resolutions associated with the second sculpted resolution, SCRes_{2}^{n−1,d_{0}}, and so on until we study the penetrated sculpted resolution associated with the sculpted resolution SCRes_{sc(n−1,d_{0})−1}^{n−1,d_{0}} and the core resolutions associated with the carrier, Carrier_{d_{0}}^{n−1}.

Suppose that sc(n−1, d_{0}) = 1 or that for every e, 1 ≤ e ≤ sc(n−1, d_{0})−1, the structures of all the graded abelian decompositions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions are identical to the structures of the corresponding abelian decompositions in the width-d_{0} e-th sculpted resolution SCRes_{e}^{n−1,d_{0}}. In this case we examine the structure of the (penetrated) resolution corresponding to the sequence of core resolutions associated with the carrier, Carrier_{d_{0}}^{n−1}, and the last sculpted resolution of width d_{0} associated with the anvil Anv(MGQ^{n−1}Res), SCRes_{sc(n−1,d_{0}}^{n−1,d_{0}}), according to proposition 5.10. Let SCRes_{sc(n−1,d_{0})}(s_{sc(n−1,d_{0})}, h_{2}, g_{1}, h_{1}, w, p, a) be the resolution composed from the resolutions induced by the subgroup < s_{sc(n−1,d_{0})}, h_{2}, g_{1}, h_{1}, w, p, a > from the various levels of the penetrated core resolutions associated with the width d_{0} core resolutions, and the subgroup < s_{sc(n−1,d_{0})}, h_{2}, g_{1}, h_{1}, w, p, a >. If the structures of the graded abelian decompositions associated with the various levels of the resolution SCRes_{sc(n−1,d_{0})}(s_{sc(n−1,d_{0})}, h_{2}, g_{1}, h_{1}, w, p, a) are not identical to the structures of the graded abelian decompositions associated with the last width-d_{0} sculpted resolution SCRes_{sc(n−1,d_{0})}^{n−1,d_{0}} associated with the anvil, Anv(MGQ^{n−1}Res), we do the following. With the graded resolution SCRes_{sc(n−1,d_{0})}(s_{sc(n−1,d_{0})}, h_{2}, g_{1}, h_{1}, w, p, a) we associate its (canonical) finite collection of framed resolutions, and with each framed resolution we associate a finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, and formal collapse P S resolutions. We set each of the extra P S resolutions to be a developing resolution, (still denoted) Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a), width(n) = d_{0} and sc(n, d_{0}) = sc(n − 1, d_{0}). We set each of the n − 1 sculpted resolutions SCRes_{e}^{n−1,d} for 1 ≤ e ≤ d ≤ d_{0} − 1 to be the corresponding n sculpted resolution SCRes_{e}^{n,d}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) , and similarly for the corresponding penetrated sculpted resolutions. We further set the resolutions: SCRes_{e} for 1 ≤ e ≤ sc(n − 1, d_{0}) to be the width-d_{0} n sculpted resolutions SCRes_{e}^{n,d}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a), and their corresponding penetrated sculpted resolutions: P enSCRes_{e}^{n,d}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a) for 1 ≤ e ≤ sc(n − 1, d_{0}), and a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils. With the anvil we associate a carrier, Carrier_{d_{0}}^{n}, which is set to be the width d_{0} sequence of core resolutions.

Suppose that the structures of the graded abelian decompositions associated with the various levels of the resolution SCRes_{sc(n−1,d_{0})} are identical to the structures of the graded abelian decompositions associated with the last width-d_{0} sculpted resolution SCRes_{sc(n−1,d_{0})}^{n−1,d_{0}}, associated with the anvil, Anv(MGQ^{n−1}Res). With the graded resolution SCRes_{sc(n−1,d_{0})} we naturally associate a (canonical) collection of framed resolutions (definition 5.4). If every ungraded resolution that factors through the graded resolution SCRes_{sc(n−1,d_{0})}, factors through either one of the Non-Rigid, Non-Solid, Left P S or formal collapse extra P S resolutions associated with the extra P S resolutions we have started the n-th step of the procedure with, or through a framed resolution F rmSCRes associated with it, where the frame associated with the framed resolution F rmSCRes strictly contains the frame associated with the graded resolution SCRes_{sc(n−1,d_{0})}, we call the sequence of multigraded resolutions from which the graded resolution SCRes_{sc(n−1,d_{0})} was constructed, a terminal resolution, and do not continue with it to the next step of the procedure. Otherwise, we continue by either studying the core resolutions associated with the collapse P S limit group associated with the anvil, Anv(MGQ^{n−1}Res), in case width(n − 1) = d_{0}, or by studying the core resolutions of higher width and their associated sculpted resolutions and carriers in case width(n − 1) > d_{0}.

Suppose that width(n − 1) = d_{0}, and recall that we denote the collapse P S limit group associated with the anvil Anv(MGQ^{n−1}Res) by < c, h_{2}, g_{1}, h_{1}, w, p, a >. At this point we study the penetrated resolutions associated with core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a > and the multi-graded resolutions constructed along the various steps, precisely as we did in in part (6) of the general step. We start by studying sequentially the penetrated sculpted resolutions associated with the sculpted resolutions:

SCRes_{1}^{n−1,d_{0}}, …, SCRes_{sc(n−1,d_{0})}^{n−1,d_{0}}

and the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. If for every e, 1 ≤ e ≤ sc(n − 1, d_{0}), the structure of all the graded abelian decompositions induced by the subgroup < s_{e}, h_{2}, g_{1}, h_{1}, w, p, a > from the terminal levels of the various penetrated core resolutions associated with the sculpted resolution, SCRes_{e}^{n−1,d_{0}}, and the core resolutions associated with the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >, is identical to the structure of the corresponding abelian decompositions in the e-th sculpted resolution SCRes_{e}^{n−1,d_{0}}, we continue as follows.

We set the resolution DRes(e, c, h_{2}, g_{1}, h_{1}, w, p, a) to be the resolution composed from the resolutions induced by the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a > from the core resolutions associated with the multi-graded resolutions constructed in the various levels and the subgroup < c, h_{2}, g_{1}, h_{1}, w, p, a >. With the graded resolution DRes(e, c, h_{2}, g_{1}, h_{1}, w, p, a), we associate a (canonical) finite collection of framed resolution, and with each framed resolution we associate a (canonical) finite collection of Non-Rigid, Non-Solid and Left P S resolutions, and a finite collection of extra P S resolutions, precisely as we associated those with the P S resolution P SHGHRes we have started the initial step of the procedure with. With each extra P S resolution we associate a finite collection of formal collapse P S resolutions. We set width(n) = d_{0} + 1, sc(n, d_{0} + 1) = sc(n − 1, d_{0}), and sc(n, d) = sc(n − 1, d) for d = 1, …, d_{0}, and each of the extra P S resolutions to be a developing resolution, (still denoted) Dvlp(q, h_{2}, g_{1}, h_{1}, w, p, a). We set each of the n − 1 sculpted resolutions SCRes_{e}^{n−1,d} for 1 ≤ e ≤ d ≤ d_{0} to be the corresponding n sculpted resolution, SCRes_{e}^{n,d}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a) , and similarly for the corresponding penetrated sculpted resolutions. We further set the n − 1 carrier Carrier_{d_{0}}^{n−1} to be the n carrier Carrier_{d_{0}}^{n}, and the graded resolutions: SCRes_{e} for 1 ≤ e ≤ sc(n−1, d_{0}) to be the width-d_{0} +1, n sculpted resolutions SCRes_{e}^{n,d_{0}+1}(s_{e}, h_{2}, g_{1}, h_{1}, w, p, a), and their corresponding penetrated sculpted resolutions: P enSCRes_{e}^{n,d_{0}+1}(r_{e}, h_{2}, g_{1}, h_{1}, w, p, a) for 1 ≤ e ≤ sc(n − 1, d_{0}), and a finite collection of graded auxiliary resolutions and limit groups, a finite collection of collapse extra P S limit groups, and a finite collection of anvils.

Suppose that width(n − 1) > d_{0}. Then we continue by analyzing the sequence of core resolutions of width d_{0} + 1. If there is no carrier associated with the core resolutions of width d_{0}+1, i.e., there is no carrier Carrier_{d_{0}+1}^{n−1} associated with the anvil, Anv(MGQ^{n−1}Res), we analyze the sequence of core resolutions of width d_{0} + 1 according to part (6) of the general step of the sieve procedure. If there a carrier Carrier_{d_{0}+1}^{n−1} associated with the anvil, Anv(MGQ^{n−1}Res), we analyze the core resolutions of width d_{0} + 1 according to the finite process presented above (part (7) of the general step). By continuing inductively, we analyze the collection of core resolutions of all widths, associated with the various multi-graded resolutions constructed at the n-th step of the sieve procedure.

IV: Termination of the sieve procedure

Defining the initial, second and general steps of the sieve procedure, we are still required to prove its termination. To prove termination of our iterative procedure in the minimal rank case (section 1), we used the strict decrease in the complexity of the resolutions associated with successive steps of the procedure, a strict decrease that forces termination. In the procedures used to construct the tree of stratified sets and to validate a general AE sentence, we proved termination by combining the decrease in either the Zariski closures or the complexities of the resolutions and decompositions associated with successive steps of these procedures. In presenting the general step of the sieve procedure, we have considered the possibility that both the Zariski closures and the complexities of the various core resolutions and developing resolutions associated with an anvil do not decrease. In this case, if the constructed developing resolution is not a terminal resolution, we have associated additional sculpted resolutions, penetrated sculpted resolutions and carriers with the corresponding anvil. As we will see in the sequel, in addition to the arguments used to prove the termination of the procedure for the construction of the tree of stratified sets, in order to prove termination of the sieve procedure, we will also need to obtain a global bound on the number of sculpted resolutions (of given width) associated with an anvil.

Theorem 5.14. The sieve procedure terminates after finitely many steps.

Proof: Suppose that the sieve procedure does not terminate after finitely many steps, hence, the procedure must contain an infinite path. At step n along the infinite path of the sieve procedure, there are associated anvil and developing resolution, a width, width(n), width(n) collections of core resolutions, and possibly sculpted resolutions, penetrated sculpted resolutions, and carriers. To get a contradiction to the existence of an infinite path, we start by applying the arguments used to prove the termination of the iterative procedure for validation of a sentence ([Se4], 4.12), to show that the sequence of core resolutions associated with a given subgroup along an infinite path, can be changed only finitely many times, i.e., parts (1)-(4) of the general step of the sieve procedure, can be applied to a multi-graded resolution and its associated core resolution with respect to the given subgroup, only finitely many times. We continue, by deducing from this ”stabilization” of the sequence of core resolutions associated with a given subgroup, that the number of sculpted resolutions of the same width along the infinite path, i.e., the set of non-negative integers {sc(n, d) | 1 ≤ d ≤ width(n)}_{n=1}^{∞}, is not bounded. To obtain a contradiction, we apply the approach used in [Se3] to obtain a global bound on the number of rigid and strictly solid families of solutions, to get a bound on the number of sculpted resolutions of the same width along the infinite path, that clearly contradicts the unboundedness of the set: {sc(n, d) | 1 ≤ d ≤ width(n)}_{n=1}^{∞}.

Proposition 5.15. Let M be either the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > or a collapse P S limit group < c, h_{2}, g_{1}, h_{1}, w, p, a > associated with an anvil Anv(MGQ^{n_{0}}Res) constructed at step n_{0} along our given infinite path of the sieve procedure. Suppose that for some (fixed) width d_{0}, the width d_{0} core resolutions associated with the anvils Anv(MGQ^{n}Res) for all n > n_{0}, are the core resolutions associated with the (fixed) subgroup M. Then the structure of the core resolutions associated with M (the width d_{0} sequence of core resolutions), can be changed in only finitely many steps along the given infinite path of the sieve procedure, i.e., parts (1)-(4) can be applied only in finitely many steps along the infinite path to the core resolutions associated with the subgroup M.

Proof: Since for all n > n_{0}, the width-d_{0} sequence of core resolutions, is the sequence of core resolutions associated with the subgroup M, either d_{0} = 1, in which case M =< h_{2}, g_{1}, h_{1}, w, p, a >, or the sequences of depth d core resolutions are unchanged for all n > n_{0} and d < d_{0}. Hence, the claim of the proposition for d_{0} > 1 is similar to the case d_{0} = 1. So, to prove the proposition it is enough to consider the case d_{0} = 1, in which case M =< h_{2}, g_{1}, h_{1}, w, p, a >. The argument we apply in this case, is a modification of the argument used to prove the termination of the iterative procedure for validation of a sentence ([Se4],4.12).

Each time part (1) of the general step of the procedure is applied to the subgroup M along the given infinite path of the sieve procedure, the limit group Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) is replaced by its proper quotient. Hence, by the ascending chain condition for limit groups, for the rest of the argument we may assume that part (1) of the general step is not applied to the core resolution associated with the subgroup M along our given path of the procedure. If part (3) is applied to the core resolution associated with the subgroup M along the infinite path, then the limit group Q^{ℓ(s)}(t_{ℓ(s)}, h_{1}, w, p, a), is replaced by its proper quotient, Q^{n}(t_{ℓ(s)}, h_{1}, w, p, a) for some index s, 1 ≤ s ≤ s(n − 1). Hence, by the ascending chain condition for limit groups, for any fixed s, part (3) can be applied to the limit group Q^{n}(t_{ℓ(s)}, h_{1}, w, p, a) (associated with the subgroup M) only finitely many times along the infinite path. Each time part (4) of the general step of the sieve procedure is applied to a multigraded resolution associated with the subgroup M along the infinite path, the index of the based subgroups, s(n), is increased by 1, and the complexity of the associated (multi-graded) core resolution strictly decreases. Since a strictly decreasing sequence of complexities of (multi-graded) core resolutions terminates after finitely many steps, parts (1), (3) and (4), can be applied to the multi-graded resolution associated with the top level of the developing resolution (or anvil) associated with the subgroup M along our given infinite path of the sieve procedure only finitely many times.

Hence, there exists some step n_{0} along the given path of the sieve procedure, so that for every step n > n_{0} and for the (top level) core resolution associated with the subgroup M:

(i) s(n) = s(n_{0}).

(ii) for every 1 ≤ s ≤ s(n_{0})−1, Q^{ℓ(s)}(t_{ℓ(s)}, h_{1}, w, p, a) is isomorphic to Q^{n}(t_{ℓ(s)}, h_{1}, w, p, a). (iii) the multi-graded core resolution associated with the top level of the devel-

oping (or sculpted) resolution and the subgroup M constructed in step n of

the sieve procedure, MGCore(M, MGQ^{n}Res), is of maximal complexity. Since for every n > n_{0}, the multi-graded core MGCore(M, MGQ^{n}Res) asso-

ciated with the top level of the developing resolution and the subgroup M is of maximal complexity, if we restrict our attention to the core resolutions associated with the subgroup M, then for every n > n_{0}, the procedure is effectively applied to the limit group associated with the terminal level of the maximal complexity multigraded resolutions, MGQ^{n}Res. By construction, in the limit group associated with the terminal level of the maximal complexity resolutions MGQ^{n}Res, the image of the limit group Q^{n}(t_{ℓ(s(n_{0})−1)}, h_{1}, w, p, a), Q_{term}^{n}(t_{ℓ(s(n_{0})−1)}, h_{1}, w, p, a), is a proper quotient of Q^{ℓ(s)}(t_{ℓ(s(n_{0})−1)}, h_{1}, w, p, a). If at some step n > n_{0}, the image of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) in the terminal level of MGQ^{n}Res, Q_{term}^{n}(h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a), then for some n_{1}, for n > n_{1}, the infinite path of the iterative procedure is effectively applied to a proper quotient of the limit group Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) we have started with. Otherwise, let s_{1} ≤ s(n_{0})−1 be the minimal index s, for which Q_{term}^{n}(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), is a proper quotient of Q^{ℓ(s)}(t_{ℓ(s)}, y, a) for some step n > n_{0}. Repeating the arguments used for analyzing the multi-graded resolutions associated with the top level of the developing resolution along our infinite path, part (1), (3) and (4) can be applied only finitely many times to the multi-graded resolutions associated with the second level of the developing resolution (anvil) associated with the subgroup M, along our given infinite path of the sieve procedure, hence, there must exists some index n_{1}, so that for every n > n_{1} the multi-graded resolutions associated with the top two levels of the developing resolution (anvil) associated with the subgroup M along the given infinite path of the sieve procedure are of maximal complexity.

Suppose that the core resolutions associated with the subgroup M are being changed at infinitely many steps along our given infinite path of the sieve procedure. By the ascending chain condition for limit groups ([Se1],5.1), if we repeat this argument inductively, we obtain either:

(i) a level ℓ, and an index n_{ℓ}, for which the multi-graded resolutions associated with the top ℓ levels of the associated developing resolutions associated with the subgroup M are of maximal complexity at all steps n > n_{ℓ}, and the image of Q^{n}(h_{2}, g_{1}, h_{1}, w, p, a) in the limit group associated with the ℓ + 1 level of the developing resolution associated with M, is a proper quotient of Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) for all n > n_{ℓ}.

(ii) an infinite sequence of limit groups Q(t_{ℓ(s)}, h_{2}, g_{1}, h_{1}, w, p, a), a corresponding sequence of (multi-graded) core resolutions with respect to the subgroup M, so that the sequence of complexities of these core resolutions is strictly decreasing.

Since a sequence of strictly decreasing complexities of core resolutions terminates after finitely many steps, part (ii) can not exist. Hence, part (i) holds, so there exists some index n_{q} for which for every step n > n_{q} along the infinite path of the sieve procedure, the procedure is effectively applied to a proper quotient of the limit group Q^{1}(h_{2}, g_{1}, h_{1}, w, p, a) we have started the procedure with. Therefore, the ascending chain condition for limit groups, contradicts the existence of an infinite path, for which the core resolutions associated with the subgroup M are being changed at infinitely many steps, and the proposition follows.

□

Proposition 5.15 implies that for each width d, the sequences of core resolutions of width d associated with the various anvils along our given infinite path of the sieve procedure, stabilizes after finitely many steps. The structure of the general step of the sieve procedure guarantees that in such case there is no global bound on the number of sculpted resolutions of the same width associated with the anvils constructed along an infinite path of the sieve procedure.

Proposition 5.16. For each positive integer m, there exists a step n_{m} and width d_{m}, so that the number of n_{m} sculpted resolutions of width d_{m}, sc(n_{m}, d_{m}), satisfies: sc(n_{m}, d_{m}) = m.

Proof: The depth 1 sequence of core resolutions associated with the anvils constructed along the sieve procedure, are, by construction, the sequence of core resolutions associated with the subgroup < h_{2}, g_{1}, h_{1}, w, p, a >. Hence, by proposition 5.15, the structure of the width 1 core resolutions associated with the anvils along our given infinite path of the sieve procedure, stabilizes, i.e., remains unchanged, after finitely many steps. The developing or sculpted resolution associated with the width 1 sequence of core resolutions, is the framed resolution of a resolution that is built from resolutions induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the sequence of width 1 core resolutions. By the construction of the general step of the sieve procedure, after the sequence of width 1 core resolutions stabilizes, the sculpted or developing resolution associated with the width 1 sequence of core resolutions can be modified only according to proposition 5.9, i.e., only if the complexity of at least one of the graded abelian decompositions, induced by the subgroup < h_{2}, g_{1}, h_{1}, w, p, a > from the sequence of width 1 core resolutions, strictly decreases. Since such a strict decrease of complexity can occur only finitely many times, there exists some positive integer ℓ_{1}, for which the sculpted resolution of width 1, SCRes_{1}^{n,1}(h_{2}, g_{1}, h_{1}, w, p, a) remains unchanged at every step n > ℓ_{1} along our given infinite path of the sieve procedure.

Since the width 1 sculpted resolution remains unchanged, the limit group associated with the width 2 sequence of core resolutions, which is the collapse P S limit group associated with the width 1 sculpted resolution, remains unchanged at every step n > ℓ_{1} along the infinite path of the sieve procedure. Hence, proposition 5.15 implies that there exists some positive integer t_{2} > ℓ_{1}, so that the width 2 sequence of core resolutions associated with the anvil remains unchanged at every step n > t_{2} along the infinite path of the sieve procedure. By the structure of the general step of the sieve procedure, for every n > t_{2}, with the width 2 sequence of core resolutions there is an associated carrier, Carrier_{2}^{n}, or two width 2 sculpted resolutions, or a width 2 sculpted resolution a developing resolution.

Suppose that there exists a step n > t_{2} for which there is a carrier, Carrier_{2}^{n}, associated with the anvil, Anv(MGQ^{n}Res). Then for every step n′ > n, there is a carrier, Carrier_{2}^{n′}, associated with the anvil, Anv(MGQ^{n′}Res). Since there is a carrier, Carrier_{2}^{n′}, associated with the anvil at each step n′ > n, there is only one sculpted resolution of width 2 associated with the anvil for every step n′ > n. Since the structure of the sculpted resolution of width 2 can be modified only according to proposition 5.10, i.e., only if the complexity of the graded abelian decompositions associated with at least one of the penetrated core resolutions of width 2 strictly decreases, and such strict decrease can occur only finitely many times, there exists a positive integer ℓ_{2}, so that the structure of the width 2 sculpted resolution remains unchanged for every n > ℓ_{2}.

Suppose that for every n > t_{2}, there is no carrier, Carrier_{2}^{n}, associated with the anvil constructed at step n of our given infinite path of the sieve procedure. Then the structure of the developing or the second width 2 sculpted resolution can be modified only according to proposition 5.9, and since a strict decrease in the complexity of at least one of the graded abelian decompositions induced from the sequence of width 2 core resolutions can occur only finitely many times, there exists a positive integer ℓ_{2}, so that the width 2 sculpted resolutions remains unchanged at every step n > ℓ_{2} along the infinite path of the sieve procedure.

Since both in case there is a carrier, Carrier_{2}^{n}, associated with the anvils, and in case there are 2 width 2 sculpted resolutions (and no carrier), the structure of both the sequence of width 2 core resolutions and the width 2 sculpted resolutions remains unchanged for every step n > ℓ_{2}, the limit group associated with the width 3 sequence of core resolutions, which is the collapse P S limit group associated with the appropriate (first or second, depending on the existence of a carrier) width 2 sculpted resolution, remains unchanged for every n > ℓ_{2} along our given infinite path of the sieve procedure. Hence, by proposition 5.15, there exists an integer, t_{3} > ℓ_{3}, so that at every step n > t_{3} the structure of the sequence of width 3 core resolutions remains unchanged. Since at every step n > t_{3}, the width 3 sculpted resolutions can be modified only according to propositions 5.9 and 5.10, and these propositions can be applied only finitely many times, there exists an integer ℓ_{3} > t_{3}, so that for every step n > ℓ_{3}, the width 3 sculpted resolutions remain unchanged along the given infinite path. Repeating this argument inductively, we get an increasing sequence of integers {ℓ_{d}}_{d=1}^{∞}, so that for every positive integer d, the sequence of width d core resolutions, the sequence of width d sculpted resolutions, and the width d carrier (if it exists), remain unchanged for every step n > ℓ_{d}.

So far we proved the stabilization of the width d sculpted resolutions along the infinite path of the sieve procedure, for every fixed d. Hence, the structure of the ”stabilized” first sculpted resolution of width d can be modified only according to proposition 5.10, i.e., only if there is a ”stabilized” carrier of width d, and the complexity of the graded abelian decompositions associated with at least one of the penetrated core resolutions associated with the ”stabilized” first sculpted resolution of width d strictly decreases. Since such strict decrease can occur only finitely many times, there exists a width d_{1}, so that the ”stabilized” first sculpted resolution of width d remains unchanged for every width d > d_{1}. Hence, for every width d > d_{1}, there are at least 2 ”stabilized” sculpted resolutions of width d. Since for every d > d_{1}, the structure of the ”stabilized” second sculpted resolution of width d can be modified only according to proposition 5.11, there exists a width d_{2} > d_{1}, so that the first two ”stabilized” sculpted resolutions of width d remain unchanged for every width d > d_{2}. Continuing inductively, for every positive integer m, there exists a width d_{m}, so that the first m ”stabilized” sculpted resolutions of width d remain unchanged for every width d > d_{m}. Setting n_{m} = ℓ_{d_{m}}, we obtain a sequence of step numbers and widths, {(n_{m}, d_{m})}_{m=1}^{∞}, for which the corresponding numbers of sculpted resolutions of depth d_{m} satisfy: sc(n_{m}, d_{m}) = m.

□

Proposition 5.16 shows that if the sieve procedure does not terminate after finitely many steps, there is no bound on the number of sculpted resolutions of the same width associated with the anvils along an infinite path of the sieve procedure. Therefore, to conclude the proof of theorem 5.14, i.e., to prove the termination of the sieve procedure after finitely many steps, we need to show the existence of a global bound on the number of sculpted resolutions of the same width associated with the anvils along an infinite path of the procedure.

Theorem 5.17. There exists a global bound on the number of sculpted resolutions of the same width, i.e., on the numbers sc(n, d), associated with an anvil along a given infinite path of the sieve procedure.

Proof: Our approach towards obtaining a bound on the number of sculpted resolutions with the same width along an infinite path of the sieve procedure is based on the argument used to obtain a bound on the number of rigid and strictly solid families of solutions (with respect to a given covering closure) of rigid and solid limit groups, presented in the first two sections of [Se3] (theorems 2.3,2.7 and 2.11 in [Se3]).

Suppose that there is no bound on the number of sculpted resolutions of the same width associated with an anvil along the given infinite path of the sieve procedure. Then for each positive integer m, there exists some index n_{m} and width d_{m}, so that the m-th sculpted resolution of width d_{m} constructed in the n_{m} step along the given path, SCRes_{m}^{n_{m},d_{m}}(s_{m}, g_{1}, h_{1}, w, p, a), contains the m−1 penetrated sculpted resolutions:

P enSCRes_{1}^{n_{m},d_{m}}(r_{1}, h_{2}, g_{1}, h_{1}, w, p, a), …, P enSCRes_{m−1}^{n_{m},d_{m}}(r_{m−1}, h_{2}, g_{1}, h_{1}, w, p, a)

that are naturally embedded in the m-th sculpted resolution SCRes_{m}^{n_{m},d_{m}}(s_{m}, h_{2}, g_{1}, h_{1}, w, p, a). Each sculpted resolution added along the sieve procedure is an extra P S resolution. Hence, in constructing the i-th sculpted resolution, SCRes_{i}^{n_{m},d_{m}}(s_{i}, h_{2}, g_{1}, h_{1}, w, p, a), several (boundedly many) extra rigid and almost shortest strictly solid solutions are being added, while the extra rigid solutions associated with the previously constructed sculpted resolution become flexible, and extra strictly solid solutions associated with previously constructed sculpted resolutions become non-strictly solid. Note that by construction, the variables that represent an extra rigid or solid solution that is being added to the i-th sculpted resolution of width d_{m}, SCRes_{i}^{n_{m},d_{m}}, (x_{i}, h_{1}, w, p, a), differ from the variables that represent any other extra rigid or solid solution that is being added to any of the previous sculpted resolutions of the same width that are embedded into it.

Suppose that for some index i < m, to the i-th sculpted resolution of width d_{m}, SCRes_{i}^{n_{m},d_{m}} constructed along the infinite path, an additional extra rigid solution is being added. Since the sculpted resolution SCRes_{i}^{n_{m},d_{m}} is embedded in the next sculpted resolution of width d_{m}, SCRes_{i+1}^{n_{m},d_{m}}, the extra rigid solution that was added to the i-th sculpted resolution, SCRes_{i}^{n_{m},d_{m}}, does not become flexible when embedded into the i + 1 sculpted resolution, SCRes_{i+1}^{n_{m},d_{m}}, which clearly contradicts the properties of the i + 1 sculpted resolution. Hence, the only extra solutions that are being added to the various sculpted resolutions that are embedded in SCRes_{m}^{n_{m},d_{m}}, are almost shortest strictly solid solutions (with respect to the given covering closures associated with the solid limit groups W P HG, see definition 2.11 in [Se3]).

Since our given path of the sieve procedure is infinite, none of the resolutions constructed in the various steps along the infinite path are terminal resolutions. Hence, at each step of the infinite path, there exists a test sequence that factors through the developing resolution so that the corresponding test sequences that factor through each of the sculpted resolutions associated with the given developing resolution, for which the specialization of at least one of the variables associated with an extra solid solution associated with that sculpted resolution is indeed strictly solid. The extra strictly solid solutions that are being added to the various sculpted resolutions along the infinite path of the sieve procedure are strictly solid solutions of a finite collection of solid limit groups W P HG(g_{1}, h_{1}, w, p, a) associated with the tree of stratified sets. Hence, for each positive integer m, there exists an index n_{m}, width d_{m}, and a sequence of integers:

1 ≤ i_{1} < … < i_{m}

and an associated sequence of (variables corresponding to) extra solid specializations (x_{i_{1}}, h_{1}, w, p, a), …, (x_{i_{m}}, h_{1}, w, p, a) that are being added to the sculpted resolutions of width d_{m}, SCRes_{i_{1}}^{n_{m},d_{m}}, …, SCRes_{i_{m}}^{n_{m},d_{m}} in correspondence, so that these extra solid specializations are extra solid specializations of the same solid limit group W P HG, and there exists a test sequence that factors through the n_{m}-th sculpted resolution of width d_{m}, SCRes_{i_{m}}^{n_{m},d_{m}}, and for each j, 1 ≤ j ≤ m, the specializations of the variables (x_{i_{j}}, h_{1}, w, p, a) along the corresponding test sequence of the sculpted resolution SCRes_{i_{j}}^{n_{m},d_{m}}, are all strictly solid specializations of the (same) solid limit group W P HG.

Given such a test sequence that factors through the sculpted resolution of width d_{m}, SCRes_{i_{m}}^{n_{m},d_{m}}, we naturally associate with it test sequences of the penetrated sculpted resolutions P enSCRes_{i_{1}}^{n_{m},d_{m}}, …, P enSCRes_{i_{m}}^{n_{m},d_{m}}. Given the corresponding test sequence of the penetrated sculpted resolution P enSCRes_{i_{1}}^{n_{m},d_{m}}, we further associate with the penetrated sculpted resolution, P enSCRes_{i_{1}}^{n_{m},d_{m}}, and the corresponding extra solid solution, (x_{i_{1}}, h_{1}, w, p, a), a sequence of strictly solid solutions {(q_{i_{1}}, h_{1}, w, p, a)_{ℓ}} which are almost shortest in the strictly solid families of the solutions {(x_{i_{1}}, h_{1}, w, p, a)_{ℓ}} (where the metric according to which we measure the length of a specialization, is the metric induced by the penetrated sculpted resolution from its embedding into the limit group associated with the sculpted resolution SCRes_{i_{m}}^{n_{m},d_{m}}, and the action of the limit group associated with this sculpted resolution on a real tree obtained from the test sequence of specializations). Since penetrated sculpted resolutions are well-separated, by passing to a subsequence of the given test sequence, we may assume that the sequence {(q_{i_{1}}, h_{1}, w, p, a)_{ℓ}} converge into a subgroup (q_{i_{1}}, h_{1}, w, p, a) of a closure of the penetrated sculpted resolution, P enSCRes_{i_{1}}^{n_{m},d_{m}}, and so are the variables that conjugate the subgroups corresponding to non-QH vertex groups in the extra solid solutions (x_{i_{1}}, h_{1}, w, p, a) and (q_{i_{1}}, h_{1}, w, p, a), i.e., the variables that demonstrate that they belong to the same strictly solid family. Given that subsequence, we associate a similar almost shortest solution, (q_{i_{2}}, h_{1}, w, p, a) in a closure of the penetrated sculpted resolution SCRes_{i_{2}}^{n_{m},d_{m}}, and so on, until we associate an almost shortest solution, (q_{i_{m}}, h_{1}, w, p, a) with the (penetrated) sculpted resolution P enSCRes_{i_{m}}^{n_{m},d_{m}}.

Definition 5.18. Given our test sequences of specializations, we say that a sequence of integers {m_{u}}_{u=1}^{∞}, together with tuples of integers 1 ≤ i_{j_{1}} < i_{j_{2}} < … < i_{j_{u}} ≤ i_{m_{u}}, and test sequences that factor through the various penetrated sculpted resolutions P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}} are a fenced subsequence, if there exists a (fixed) proper quotient W P HG′(g_{1}, h_{1}, w, p, a) of the solid limit group W P HG, so that for every subgroup < q_{i_{j}}, h_{1}, w, p, a > of a penetrated sculpted resolution P enSCRes_{i_{j}}^{n_{mu} ,d_{mu}} that is associated with an extra solid specialization, there exists a subgroup < q̂_{i_{j}}, h_{1}, w, p, a > of a closure of the same penetrated sculpted resolution P enSCRes_{i_{j}}^{n_{mu} ,d_{mu}}, so that the subgroups < q_{i_{j}}, h_{1}, w, p, a > and < q̂_{i_{j}}, h_{1}, w, p, a > represent elements that belong to the same strictly solid class of the solid limit group W P HG, i.e., there exist elements in the corresponding closure of P enSCRes_{i_{j}}^{n_{mu} ,d_{mu}} that conjugate non-QH vertex groups in < q_{i_{j}}, h_{1}, w, p, a > to the corresponding vertex groups in < q̂_{i_{j}}, h_{1}, w, p, a >, and all the subgroups < q̂_{i_{j}}, h_{1}, w, p, a > factor through the (fixed) proper quotient W P HG′ of W P HG.

If the test sequences of specializations we have started with contains a fenced subsequence, we continue with the fenced subsequence, replace the solid limit group W P HG by the proper quotient corresponding to the subsequence which we denote W P HG′, and for each couple of indices (m, i_{j}), we replace the subgroup < q̂_{i_{j}}, h_{1}, w, p, a >, by a subgroup < q̃_{i_{j}}, h_{1}, w, p, a >, obtained as a limit from a test sequence of specializations {(q̃_{i_{j}}, h_{1}, w, p, a)}_{ℓ=1}^{∞} that are the almost shortest specializations in the solid family of the corresponding sequence of specializations {(q̂_{i_{j}}, h_{1}, w, p, a)}_{ℓ=1}^{∞} with respect to the graded abelian decomposition and graded modular groups of the proper quotient W P HG′ of W P HG (where the parameter subgroup is < h_{1}, w, p >). To the subgroups < q_{i_{j}}, h_{1}, w, p, a >, < q̂_{i_{j}}, h_{1}, w, p, a > and < q̃_{i_{j}}, h_{1}, w, p, a >, of (the corresponding closure of) P enSCRes_{i_{j}}^{n_{m},d_{m}} we add variables that conjugate the subgroups corresponding to non-QH, non-abelian vertex groups in the extra solid solutions < q_{i_{j}}, h_{1}, w, p, a >, < q̂_{i_{j}}, h_{1}, w, p, a > and < q̃_{i_{j}}, h_{1}, w, p, a >, i.e., the variables that demonstrate that they belong to the same strictly solid family of the original solid limit group W P HG.

If the obtained sequence contains a further fenced subsequence, we continue with it and replace the obtained limit group W P HG′ by its corresponding proper quotient. Since limit groups satisfy the ascending chain condition, after finitely many replacements, we may assume that our sequence of test sequences contain no fenced subsequence. We call the obtained collection of test sequences with no fence subsequences, and its corresponding limit group (still denoted W P HG), a stable collection and a stable limit group in correspondence. We continue denoting the subgroups corresponding to extra strictly solid solutions in the various penetrated sculpted resolutions P enSCRes_{i_{j}}^{n_{m},d_{m}} that factor through a stable limit group, < q_{i_{j}}, h_{1}, w, p, a >.

Let η be the natural quotient map from the original solid limit group onto the stable limit group W P HG. Let Θ_{0} be the graded abelian decomposition of the original solid limit group with respect to the parameter subgroup AP W H_{1} =< h_{1}, w, p, a >, and let Θ be the graded abelian decomposition of the stable limit group W P HG. Suppose that Θ contains an abelian vertex group V, that is not the distinguished vertex group (i.e., V does not contain the subgroup AP W H_{1}), and the subgroup generated by the edge groups connected to the vertex stabilized by V in Θ_{0} is not of finite index in V. Since the stable limit group W P HG is obtained from a sequence of strictly solid specializations of the original solid limit group, every non-QH, non-abelian vertex group and every edge group in Θ_{0} must be mapped by the natural quotient map η, into either a vertex group or into an edge group in the graph of groups Θ. Furthermore, if a non-QH, non-abelian vertex group or an edge group in Θ_{0} is mapped into (a conjugate of) the abelian vertex group V, then the image of that vertex group or edge group in V intersects the subgroup generated by the edge groups connected to V in a subgroup of finite index.

Since the sequence of test sequences contains no fenced subsequence, we can further pass to a subsequence of test sequences, for which there is no subsequence of integers {m_{u}}_{u=1}^{∞}, together with a sequence of integers, {j_{u}}_{u=1}^{∞}, 1 ≤ j_{u} ≤ m_{u}, and test sequences that factor through the various penetrated sculpted resolutions P enSCRes_{i_{ju}}^{n_{mu} ,d_{mu}}, for which there exists a (fixed) proper quotient W P HG′ of the stable limit group W P HG, so that for every subgroup < q_{i_{ju}} , h_{1}, w, p, a > of a penetrated sculpted resolution P enSCRes_{i_{ju}}^{n_{mu} ,d_{mu}} that is associated with an extra solid specialization, there exists a subgroup < q̂_{i_{ju}} , h_{1}, w, p, a > of a closure of the same penetrated sculpted resolution P enSCRes_{i_{ju}}^{n_{mu} ,d_{mu}}, so that the subgroups < q_{i_{ju}} , h_{1}, w, p, a > and < q̂_{i_{ju}} , h_{1}, w, p, a > represent elements that belong to the same strictly solid class of the solid limit group W P HG, i.e., there exist elements in the corresponding closure of P enSCRes_{i_{ju}}^{n_{mu} ,d_{mu}} that conjugate non-QH vertex groups in < q_{i_{ju}} , h_{1}, w, p, a > to the corresponding vertex groups in < q̂_{i_{ju}} , h_{1}, w, p, a >, and all the subgroups < q̂_{i_{ju}} , h_{1}, w, p, a > factor through the (fixed) proper quotient W P HG′ of the stable limit group W P HG. We continue with this sequence of subsequences, and the corresponding collection of penetrated sculpted resolutions, which we still denote, P enSCRes_{i_{j}}^{n_{m},d_{m}}, for 1 ≤ j ≤ m and 1 ≤ i_{1} < … < i_{m}.

Our approach towards obtaining a global bound on the number of sculpted resolutions of the same width (theorem 5.17), is based on the arguments used to obtain a global bound on the number of rigid and solid solutions presented in the first two sections of [Se3]. In order to modify the arguments presented in [Se3] to prove theorem 5.17, we first need to identify divisible and non-divisible (cyclic) edge groups. Let Θ be the graded abelian decomposition of the stable limit group W P HG with respect to the parameter subgroup AP W H_{1} =< h_{1}, w, p, a >, and let E_{1}, …, E_{d} be the edge groups in Θ. The stable limit group W P HG is mapped into each of the penetrated sculpted resolutions P enSCRes_{i_{j}}^{n_{m},d_{m}}, and we denoted its image < q_{i_{j}}, h_{1}, w, p, a >. In order to be able to modify the arguments presented in [Se3] to prove theorem 5.17, we need to obtain global bounds on the order of roots of images of the maximal abelian subgroups containing the edge groups E_{1}, …, E_{d} in the penetrated sculpted resolutions P enSCRes_{i_{j}}^{n_{m},d_{m}}, when viewed as subgroups of the penetrated sculpted resolution P enSCRes_{i_{m}}^{n_{m},d_{m}}. We start by demonstrating our approach for obtaining global bounds on the order of roots, in case the graded abelian decomposition Θ of the stable limit group W P HG is an amalgamated product W P HG = D ∗_{C} E, where C is cyclic, E is non-abelian, and the parameter subgroup AP W H_{1} is a subgroup of D.

Proposition 5.19. Suppose that the graded abelian JSJ decomposition of the stable limit group W P HG with respect to the parameter subgroup AP W H_{1} =< h_{1}, w, p, a > is W P HG = D ∗_{C} E, where C is cyclic and E is non-abelian. Let C =< c >, and for each couple (m, j), 1 ≤ j ≤ m, let c_{j}^{m} be the image of the element c in the penetrated sculpted resolution SCRes_{i_{j}}^{n_{m},d_{m}}. Let {c_{j}^{m}(ℓ)}_{ℓ=1}^{∞}, be the restriction of our given test sequence of specializations that factors through the penetrated sculpted resolution P enSCRes_{i_{m}}^{n_{m},d_{m}}. Then there exists some global bound b_{0}, and an index m_{0}, so that for every m > m_{0}, and every j, 1 ≤ j ≤ m, there exists some index ℓ(m, j) so that for every ℓ > ℓ(m, j), the order of a root of c_{j}^{m}(ℓ) is bounded by b_{0}.

Proof: Identical to the proof of lemma 4.2 of [Se3]. In case there is no such bound b_{0}, the vertex group E admits a non-trivial abelian decomposition in which the edge group C is elliptic. Such an abelian decomposition allows one to further refine the amalgamated decomposition W P HG = D ∗_{C} E, a contradiction to the canonical properties of the JSJ decomposition.

□

Proposition 5.19 gives a bound on the order of roots of cyclic edge groups in the graded abelian decomposition Θ of the stable limit group W P HG, in case Θ corresponds to a simple amalgamated product. For general graph of groups Θ, we naturally define the notion of a divisible and a non-divisible edge group.

Definition 5.20. Let Θ be the graded abelian JSJ decomposition of the stable limit group W P HG with respect to the parameter subgroup AP W H_{1}. Let E be either an edge group or an abelian vertex group in Θ, let A be the maximal abelian subgroup containing E in the stable limit group W P HG, and for each couple (m, j), 1 ≤ j ≤ m, let A_{j}^{m} be the image of the subgroup A in the penetrated sculpted resolution P enSCRes_{i_{j}}^{n_{m},d_{m}}. Let {a_{j}^{m}(ℓ)}_{ℓ=1}^{∞}, be the generators of the cyclic images of A_{j}^{m} that correspond to the given test sequence of specializations that factors through the penetrated sculpted resolution P enSCRes_{i_{m}}^{n_{m},d_{m}}. We say that the edge group (abelian vertex group) E in the graph of groups Θ is non-divisible, if there exists some global bound b_{E}, and an index m_{E}, so that for every m > m_{E}, and every j, 1 ≤ j ≤ m, there exists some index ℓ(m, j) so that for every ℓ > ℓ(m, j), the order of a root of a_{j}^{m}(ℓ) is bounded by b_{E}. Note that since the limit group W P HG is assumed to be stable, every abelian vertex group V in the graph of groups Θ that is not the distinguished vertex group (i.e., it does not contain the subgroup AP W H_{1}), and for which the subgroup generated by the edge groups connected to V is not of finite index in V, is necessarily a divisible abelian vertex group.

Definition 5.21. Let Θ be the graded abelian decomposition of a stable limit group W P HG, with respect to the parameter subgroup AP W H_{1}. We call the subgraph that contains the distinguished vertex in Θ, together with all the non-divisible abelian vertex groups and all the non-divisible edge groups in Θ, the primary non-divisible subgraph of Θ, and denote it by Λ.

The following theorem generalizes theorem 5.19 to a general stable limit group.

Theorem 5.22. Let Θ be the graded abelian decomposition of a stable limit group W P HG, with respect to the parameter subgroup AP W H_{1}. Then Λ, its primary non-divisible subgraph, is connected and contains all the non-abelian vertex groups in Θ.

Proof: To construct the non-divisible subgraph Λ, we start with the distinguished vertex V_{D} in Θ, i.e., the vertex stabilized by the subgroup AP W H_{1} in the graph of groups Θ. Since W P HG is a limit group, every non-cyclic abelian vertex group in Θ is adjacent to only non-abelian vertex groups in Θ. Hence, if Θ contains no nonabelian vertex groups apart from the distinguished one, Λ is necessarily connected and contains the distinguished vertex V_{D}.

Suppose that there are at least two non-abelian vertex groups in Θ. We start with the distinguished vertex V_{D}. Suppose that all the edge groups that are connected to V_{D} in Θ are divisible. In this case we argue that our sequence of test sequences contains a fenced subsequence (definition 5.18), hence, contradicts the assumption that the limit group W P HG is stable. Let Γ be the subgraph of the graph of groups Θ, obtained from Θ by taking out from it the vertex V_{D}, all the abelian vertex groups A_{j} adjacent to V_{D} in Θ, and all the edges that are connected to the vertices V_{D} and A_{j} in Θ. Note that Γ is not empty, since we assumed that Θ contains at least two non-abelian vertex groups. Let Γ_{1} be a connected component of Γ, and let U_{1} be its fundamental group. By construction, every vertex group in Γ_{1} that is adjacent to V_{D} or one of the vertices A_{j} in Θ, is non-abelian, and an edge group attached to an edge that connects a vertex in Γ_{1} to V_{D} or one of the vertices A_{j} in Θ is divisible.

With the (abelian) graph of groups Γ_{1}, we naturally associate a modular group of automorphisms of its fundamental group U_{1} (that is a subgroup of the modular group associated with the ambient graph of groups Θ), that we denote Mod(U_{1}). Note that the automorphisms in Mod(U_{1}) preserve, in particular, the conjugacy classes of the (divisible) edge groups that connect vertices in Γ_{1} to the vertex V_{D} or one of the (divisible) abelian groups A_{j} connected to it. Given each of the penetrated sculpted resolutions, P enSCRes_{i_{j}}^{n_{m},d_{m}}, a corresponding test sequence of specializations that factor through it, and the associated sequence of specializations: {(q_{i_{j}}, h_{1}, w, p, a)_{ℓ}}, we replace each of the specializations of the limit group U_{1} associated with the specialization: (q_{i_{j}}, h_{1}, w, p, a)_{ℓ}, by a specialization (q̂_{i_{j}}, h_{1}, w, p, a)_{ℓ}, that is almost shortest with respect to the action of the modular group Mod(U_{1}). For each tuple (j, m), 1 ≤ j ≤ m, we further pick a convergent subsequence (still denoted), (q̂_{i_{j}}, h_{1}, w, p, a)_{ℓ}, and we denote the associated limit group < q̂_{i_{j}}, h_{1}, w, p, a >.

If the sequence of subgroups < q̂_{i_{j}}, h_{1}, w, p, a > contains a fenced subsequence (see definition 5.18), we get a contradiction to our assumption that the limit group W P HG, associated with the sequences of specializations {(q_{i_{j}}, h_{1}, w, p, a)_{ℓ}}, is stable. Hence, we may assume that the sequence < q̂_{i_{j}}, h_{1}, w, p, a > contains no fenced subsequence. Since the sequence of limit groups < q̂_{i_{j}}, h_{1}, w, p, a > contains no fenced subsequence, there exists a sequence of indices {m_{u}}_{u=1}^{∞}, and for each integer u an index j(u), where 1 ≤ j(u) ≤ u, for which the ball of radius u in the Cayley graph of the limit group < q̂_{i_{j(u)}}, h_{1}, w, p, a >, associated with the penetrated sculpted resolution, P enSCRes_{i_{j(u)}}^{n_{mu} ,d_{mu}}, is identical to the ball of radius u in the Cayley graph of the stable limit group W P HG.

Each of the limit groups < q_{i_{j}}, h_{1}, w, p, a > is obtained from a convergent sequence of specializations {(q_{i_{j}}, h_{1}, w, p, a)}_{ℓ}, hence, there is an associated (faithful) action of it on some real tree. Hence, by possibly further passing to a subsequence, we may assume that the sequence of actions of the limit groups < q_{i_{j(u)}}, h_{1}, w, p, a > on their associated real trees, converges into a faithful action of the stable limit group W P HG on a real tree. In particular, the sequence of actions images of the subgroup U_{1} in < q_{i_{j(u)}}, h_{1}, w, p, a > converges into a faithful action of U_{1} on the limit tree.

Let E_{1}, …, E_{s} be the edges in the graph of groups Θ, that connect vertices in Γ_{1} to either the vertex V_{D} or one of the (divisible) abelian vertex groups A_{j} connected to it. We set the group RU_{1} to be: RU_{1} = U_{1}∗ < r_{1} > ∗ … ∗ < r_{s} >. For each index u, the penetrated sculpted resolution P enSCRes_{i_{j(u)}}^{n_{mu} ,d_{mu}} is naturally

embedded into the penetrated sculpted resolution P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}}. Hence, for each index u, the specializations of the limit group U_{1} associated with the sequence: {(q_{i_{j(u)}}, h_{1}, w, p, a)_{ℓ}}, can be extended to homomorphisms from the limit group RU_{1}, by sending each of the generators r_{t} of RU_{1} into a primitive root of the specialization of its associated (divisible) edge group. By passing to corresponding subsequences, we may assume that for each index u, the corresponding sequence of homomorphisms of RU_{1} converges into an action of RU_{1} on some limit tree, and by further passing to a subsequence of the indices u, we may assume that this sequence of actions converges into an action of RU_{1} on some real tree Y. We denote the limit group corresponding to this limit action RootU_{1}. Clearly, RootU_{1} is a (proper) quotient of RU_{1}, and the generators r_{t} commute with their associated divisible edge groups in RootU_{1}. Also, by construction, the limit group U_{1} is embedded in RootU_{1}.

The limit group RootU_{1} inherits a (non-trivial) abelian decomposition Δ from its action on the real tree Y. Since the generators r_{t} commute with the divisible edge groups associated with them, and since the elements r_{t} are mapped to primitive roots of the specializations of the divisible edge groups, the generators r_{t} and the divisible edge groups associated with them, are elliptic in Δ. Since U_{1} is embedded in RootU_{1}, and the divisible edge groups that connect U_{1} to the vertex groups V_{D} and A_{j} in Θ are elliptic in Θ, if the graph of groups Δ is not compatible with the graph of groups Γ_{1} of U_{1}, the graded abelian JSJ decomposition Θ of the stable limit group W P HG can be further refined, a contradiction to its canonical properties. Hence, Δ is compatible with the graph of groups Γ_{1} of U_{1}. But this implies that for large enough u, and for large enough ℓ, the specializations (q_{i_{j(u)}}, h_{1}, w, p, a)_{ℓ} are not almost shortest with respect to the action of the modular group Mod(U_{1}), a contradiction. Hence, at least one of the edge groups connected to V_{D} is non-divisible.

Let Λ_{1} be the subgraph of Θ that contains the distinguished vertex V_{D}, the non-divisible edges and abelian vertex groups connected to V_{D}, and the vertices adjacent to those non-divisible edges and abelian vertex groups. Clearly, Λ_{1} ⊂ Λ and Λ_{1} is connected. If there is no non-abelian vertex group in the graph of groups Θ that is not in Λ_{1}, the theorem follows. Suppose there is a non-abelian vertex group in Θ that is not contained in Λ_{1}. Then the same argument used for the edge groups connected to the distinguished vertex group V_{D}, implies that there must exist a non-divisible edge group E in Θ, so that E is adjacent to Λ_{1} and E is not in Λ_{1}. We set Λ_{2} to be the union of Λ_{1} with all the non-divisible edge groups and abelian vertex groups that are adjacent to Λ_{1}. By construction, Λ_{2} is connected and contained in Λ. Repeating the construction iteratively, we can still enlarge the obtained subgraph Λ_{j} as long as there exists a non-abelian vertex group in Θ that is not contained in Λ_{j}. Hence, for some k, Λ = Λ_{k}, Λ is connected, and contains all the non-abelian vertex groups in Θ.

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Given a primary non-divisible subgraph we still need to analyze the edge groups in Θ that are not contained in it. Note that every abelian vertex group V in Θ for which the subgroup generated by the edge groups connected to V is not of finite index in V, is necessarily divisible, hence, it is not in Λ.

Definition 5.23. Let Θ be the graded abelian decomposition of a stable limit group W P HG, and let Λ be its non-divisible primary subgraph. Let E_{1}, …, E_{d} be the edge groups or abelian vertex groups in Θ that are not in Λ. We go over the the edge groups (abelian vertex groups) sequentially, according to their prescribed order. Let A_{1} be the maximal abelian subgroup containing E_{1} in the stable limit group W P HG, and for each couple (m, j), 1 ≤ j ≤ m, let (A_{1})_{j}^{m} be the image of A_{1} in the penetrated sculpted resolution P enSCRes_{i_{j}}^{n_{m},d_{m}}.

Suppose that there exists a constant b_{1} and a subsequence of indices {m_{u}}_{u=1}^{∞}, with corresponding indices, 1 ≤ j_{1} < … < j_{u} ≤ m_{u}, so that for each u there exists a subsequence of the test sequence associated with the penetrated sculpted resolution P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}}, for which for every f, 1 ≤ f ≤ u, all the generators of the (cyclic) images of the maximal abelian subgroups (A_{1})_{i_{j_{f}}}^{m_{u}} in F_{k}, which we denote {a_{i_{j_{f}}}^{m_{u}}(ℓ)}_{ℓ=1}^{∞}, the order of a root of a_{i_{j_{f}}}^{m_{u}}(ℓ) is bounded by b_{1}. In this case we call the edge group (abelian vertex group) E_{1}, non-divisible. We further restrict our analysis to the sequence of penetrated sculpted resolutions, P enSCRes_{i_{j_{f}}}^{n_{m},d_{m}}, and their associated test sequences, and continue with these subsequences to analyze the next edge group (abelian vertex group) in our given list.

Suppose that such a constant b_{1} does not exist. Then there exists a subsequence of indices {m_{u}}_{u=1}^{∞}, with corresponding indices, 1 ≤ j_{1} < … < j_{u} ≤ m_{u}, so that for each u there exists a subsequence of the test sequence associated with the penetrated sculpted resolution P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}}, for which for every f, 1 ≤ f ≤ u, and every generator of the image of the maximal abelian subgroup (A_{1}) containing E_{1}, which we denote {a_{i_{j_{f}}}^{m_{u}}(ℓ)}_{ℓ=1}^{∞}, the element a_{i_{j_{f}}}^{m_{u}}(ℓ) has a root of order at least u. In this case we call the edge group (abelian vertex group) E_{1}, divisible. We further restrict our analysis to the sequence of penetrated sculpted resolutions, P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}}, and their associated test sequences, and continue with these subsequences to analyze the next edge groups (abelian vertex groups) in our list.

Continuing iteratively with the various edge groups (abelian vertex groups), we finally set each such edge group (abelian vertex group) to be either divisible or nondivisible. We call the subgraph of groups obtained from Θ by eliminating the divisible edges and vertices corresponding to divisible edge groups and divisible abelian vertex groups, the non-divisible subgraph of Θ, and denote it Θ_{ND}. We denote its fundamental group, W P HG_{ND}, and call the non-divisible subgroup of the stable limit group W P HG.

The subgroups < q_{i_{j_{f}}} , h_{1}, w, p, a > were chosen as limits of almost shortest sequences of specializations. In defining almost shortest specializations ([Se3], definition 2.6), we have fixed an order on the edge groups and the QH vertex groups of the graded abelian decomposition Θ of the stable limit group W P HG. Fixing a non-divisible subgraph of the graph of groups Θ of the stable limit group W P HG, for each index u and each f, 1 ≤ f ≤ u, we modify the subgroup < q_{i_{j_{f}}} , h_{1}, w, p, a > by replacing it with a corresponding subgroup that is a limit of a sequence of almost shortest specializations of the penetrated sculpted resolution P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}}, where the order on the edges and QH vertex groups of Θ starts with the edges and QH vertex groups in the non-divisible subgraph Θ_{ND}, and then the divisible edge groups. We still denote the obtained subgroups of the corresponding sculpted resolutions, < q_{i_{j_{f}}} , h_{1}, w, p, a >.

Definition 5.24. Let E_{1}, …, E_{q} be the divisible edge groups in the non-divisible graph of groups Θ_{ND}. We set G^{ND} = W P HG_{ND}∗ < s_{1} > ∗ … ∗ < s_{q} >. With each couple of indices (u, f), 1 ≤ f ≤ u, and each specialization of the penetrated sculpted resolution P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}}, we associate a natural specialization of the group G^{ND}, by mapping its subgroup W P HG_{ND} to the specialization it inherits from the corresponding specialization of the subgroup < q_{i_{j_{f}}} , h_{1}, w, p, a >, and mapping the elements s_{1}, …, s_{q} to the generators in the free group F_{k} of the images of the divisible edge groups E_{1}, …, E_{q} in correspondence.

By our standard arguments presented in section 5 of [Se1], the entire set of specializations of the group G^{ND}, correspond to finitely many (maximal) limit groups G_{1}^{ND}, …, G_{t}^{ND}. Hence, for some limit group G_{j}^{ND} there is a subsequence of couples (still denoted) (u, f), 1 ≤ f ≤ u, and test sequences that factor through the corresponding penetrated sculpted resolutions, (still denoted) P enSCRes_{i_{j_{f}}}^{n_{mu} .d_{mu}},

so that the corresponding specializations of the subgroup G^{ND} all factor through the limit group G_{j}^{ND}. If there exists a subsequence of the couples (u, f), still denoted (u, f), 1 ≤ f ≤ u, and associated test sequences of specializations that factor through the penetrated sculpted sculpted resolutions P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}}, so that the corresponding special-

izations of the subgroup G_{j}^{ND} do all factor through a proper quotient of G_{j}^{ND}, we pass to this subsequence (a fenced subsequence), and replace the limit group G_{j}^{ND} by its corresponding proper quotient. We iteratively replace the obtained subsequence by fenced subsequence, as long as a fenced subsequence exists. By the ascending chain condition for limit groups, such an iterative process terminates after finitely many steps. We call the obtained quotient of the limit group G_{j}^{ND}, a stable rooted limit group, and denote it RootW P HG_{ND}. Since the group W P HG is stable, the non-divisible subgroup W P HG_{ND} that is naturally mapped into RootW P HG_{ND} is actually embedded into it.

Since the sequence of test sequences contain no fenced subsequence, we can further pass to a subsequence of test sequences, for which there is no subsequence of integers u, with a sequence of integers, {f_{u}}_{u=1}^{∞}, 1 ≤ f_{u} ≤ u, and subsequences of the test sequences that factor through the various penetrated sculpted resolutions P enSCRes_{i_{j_{fu}}}^{n_{mu} ,d_{mu}}, for which there exists a (fixed) proper quotient of the stable rooted limit group, RootW P HG_{ND}, so that the all the specializations of RootW P HG_{ND} that correspond to the given test sequences factor through that fixed proper quotient of RootW P HG_{ND}.

In the formulation of the following theorem, recall (definition 1.1 in [Se3]) that if G(x, p, a) is a graded limit group with respect to the parameter subgroup P =< p >, then a specialization (x_{0}, p_{0}, a) of the subgroup G(x, p, a) is said to be R-AP - covered if the specialization of the (fixed) set of generators of G(x, p, a) is covered by the union of translates of the fixed set of generators of the subgroup AP =< p, a > by elements in the R-ball of the graded limit group G(x, p, a).

Theorem 5.25. There exists a (global) constant R, for which given any positive integer u, and a test sequence that factors through the penetrated sculpted resolution P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}} (r_{i_{mu}} , g_{1}, h_{1}, w, p, a) and associated with it, there exists an integer ℓ(u), so that the sets of specializations of the stable rooted group RootW P HG_{ND} corresponding to the specializations:

(q_{i_{j_{1}}} , h_{1}, w, p, a)_{ℓ}, …, (q_{i_{ju}} , h_{1}, w, p, a)_{ℓ}

are R-AP W H_{1}-covered for all ℓ > ℓ(u).

Proof: Suppose that there is no such global constant R. Then for every positive integer t, there exists an index u(t), an index f(t), 1 ≤ f(t) ≤ u(t), and a test sequence that factors through the penetrated sculpted resolution

P enSCRes_{i_{j_{f(t)}}}^{n_{m_{u(t)}} ,d_{m_{u(t)}}} (r_{i_{j_{f(t)}}} , h_{1}, w, p, a), for which the sequence of specializations of the stable rooted group RootedW P HG_{ND} corresponding to the sequence of specializations: {(q_{i_{j_{f(t)}}} , h_{1}, w, p, a)_{ℓ}}_{ℓ=1}^{∞} are not t-AP W H_{1}-covered. For every t, we can pass to a subsequence that converges into an action of the limit group RootW P HG_{ND} on a limit tree Y_{t}, and since each specialization in this sequence is not t-AP W H_{1}-covered, the action of RootW P HG_{ND} on the limit tree Y_{t} is not t-AP W H_{1}-covered.

Since for each index t the action of RootW P HG_{ND} on the tree Y_{t} is not t-AP W H_{1} covered, from the sequence of actions of the limit group RootW P HG_{ND} on the trees Y_{t}, it is possible to extract a subsequence (still denoted Y_{t}), that converges into an action of the stable rooted limit group RootW P HG_{ND} on a limit tree Y. Since the limit group RootW P HG_{ND} is stable, the action of RootW P HG_{ND} on the real tree Y is faithful. By construction, the action of the limit group RootW P HG_{ND} on the limit tree Y satisfies the properties of the action obtained in proving theorem 1.7 of [Se3], i.e., the orbit of the subtree Y_{AP W H_{1}} in the limit tree Y misses either a non-degenerate segment or a germ of a branching point in Y.

The continuation of our argument is a modification of the argument used to prove theorem 1.7 of [Se3]. By the proof of theorem 1.2 of [Se3], from the action of the stable rooted limit group RootW P HG_{ND} on the real tree Y, RootW P HG_{ND} inherits an abelian decomposition Γ, a decomposition in which the image of the subgroup AP W H_{1} =< h_{1}, w, p, a >, is contained in a proper subgraph of groups Γ′. Furthermore, since we have added elements that represent (primitive) roots of each of the divisible subgroups in the graph of groups Θ_{ND} of the non-divisible limit group W P HG_{ND} to the stable rooted limit group RootW P HG_{ND}, each divisible subgroup of W P HG_{ND} is elliptic in the graph of groups Γ as well as the subgroups corresponding to their primitive roots. Hence, in the graph of groups inherited by the non-divisible subgroup W P HG_{ND} from the graph of groups Γ, every divisible subgroup is elliptic as well. Let Δ be the graph of groups obtained from Γ by collapsing Γ′ to a vertex. In Δ the (parameter) subgroup AP W H_{1} is contained in a vertex stabilizer, each divisible subgroup is elliptic, each edge group is a non-trivial abelian subgroup of RootW P HG_{ND} and each vertex group that does not contain a conjugate of AP W H_{1} corresponds to either a point stabilizer, an IET component, or an axial component in Y. Hence, every non-QH vertex group and every edge group in the non-divisible decomposition Θ_{ND} of the non-divisible subgroup W P GH_{ND} of RootW P HG_{ND}, is contained in a non-QH vertex group or an edge group of Δ, and if it contained in an abelian vertex group, then it must intersect the subgroup generated by the edge groups connected to this vertex group in a subgroup of finite index. Furthermore, since the graph of groups Δ is non-trivial, and since RootW P HG_{ND} is obtained from W P HG_{ND} by adding roots, the abelian decomposition inherited by W P HG from Δ is non-trivial as well. Hence, the abelian decomposition inherited by W P HG_{ND} is non-trivial and compatible with Θ_{ND}, so for large enough t, the specializations {(q_{i_{j_{f(t)}}} , h_{1}, w, p, a)_{ℓ}}_{ℓ=1}^{∞} are not almost shortest in their strictly solid families for large enough ℓ, which clearly contradicts our assumptions.

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To prove theorem 5.25 we modified the argument used to prove theorem 1.7 in [Se3]. Given theorem 5.25, to obtain a bound on the number of sculpted resolutions, i.e., to complete the proof of theorem 5.17, we modify the argument used to prove theorems 2.7 and 2.11 of [Se3]. Let B be a ball in the Cayley graph of the rooted stable limit group RootW P HG_{ND} that contains all the defining relations of RootW P HG_{ND}. By theorem 5.25 there exists some constant R′, so that for any positive integer u, and the associated test sequence that factors through the penetrated sculpted resolution P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}}, there exists an index ℓ(u), so that for every ℓ > ℓ(u), the images of the ball B of RootW P HG_{ND} corresponding to the tuple of specializations:

(q_{i_{j_{1}}} , h_{1}, w, p, a)_{ℓ}, …, (q_{i_{ju}} , h_{1}, w, p, a)_{ℓ}

is R′-AP W H_{1}-covered. Let R = 2 · R′. Clearly, there exist only finitely many combinatorial types of R-states, hence, by possibly changing the indexing, we may further assume that for every index u, and every f, 1 ≤ f ≤ u, and every index ℓ, the combinatorial types of the R-states associated with the specializations of RootW P HG_{ND} associated with the specializations (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} are identical. For each index u, and each f, 1 ≤ f ≤ u, the R-state of the rooted stable limit group RootW P HG_{ND} corresponding to the sequences of specializations {(q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ}} converge, after rescaling the metric on them by dilatation constants that depend only on the specializations of the defining parameters (h_{1}, w, p)_{ℓ}, into an R-state, which we denote Rstate_{∞}^{u}(q_{i_{j_{f}}} , h_{1}, w, p, a). By changing the indexing, we may further assume that any sequence of R-states {Rstate_{∞}^{u}(q_{i_{j_{f}}} , h_{1}, w, p, a)}_{u=1}^{∞}, where 1 ≤ f ≤ u, converges in the Gromov-Hausdorff topology on metric spaces, and the limit (finite) tree Y is independent of the particular choice of indices.

Given a specialization of RootW P HG_{ND} corresponding to a specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ}, for some f, 1 ≤ f ≤ u, we set p_{r}(ℓ, f, u) to be a segment in the R-state Rstate(q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} that is mapped to the edge e_{r} in the finite tree Y (p_{r}(ℓ, f, u) is a subword of a specialization of one of the defining parameters). Recall (definition 2.4 of [Se3]) that we say that p_{r}(ℓ, f, u) is pseudo-periodic if:

(i) p_{r}(ℓ, f, u) = v_{1}α^{s}v_{2} and the equality is graphical (i.e., with no cancellations).

(ii) s > 10 and the length of α^{s} is at least three quarters of the length of

p_{r}(ℓ, f, u). If p_{r}(ℓ, f, u) is pseudo-periodic, we set period(p_{r}(ℓ, f, u)) to be the minimal length of an element α ∈ F_{k} that satisfies condition (i) above. We call period(p_{r}(ℓ, f, u)) the pseudo-period of the pseudo-periodic segment p_{r}(ℓ, f, u). We call the maximal possible s that appears in condition (i) above, the pseudo-periodicity of p_{r}(ℓ, f, u).

We continue by going over the segments in the tree Y. If for such segment e_{r}, there exists a subsequence of indices {(f, u)} for which there exist test sequences for which {p_{r}(ℓ, f, u)}_{u,ℓ=1}^{∞} is either not pseudo-periodic, or with bounded pseudoperiodicity, or if for each ℓ, u and f, p_{r}(ℓ, f, u) = v_{1}(ℓ, f, u)α(ℓ, f, u)^{s(ℓ,f,u)}v_{2}(ℓ, f, u), where the equation is graphical, α(ℓ, f, u) is the pseudo-period of p_{r}(ℓ, f, u) and s(ℓ, f, u) is the pseudo-periodicity of p_{r}(ℓ, f, u), and there exists some ϵ > 0 for which: \frac{|v_{1}(ℓ,f,u)|+|v_{2}(ℓ,f,u)|}{|p_{r}(ℓ,f,u)|} > ϵ, we restrict our attention to this subsequence and declare the particular segment in the finite tree Y not truly periodic. After possibly restricting our attention to some subsequence, we may assume that the sequence of specializations corresponding to each segment is either not truly periodic, or it is pseudo-periodic with no bounded subsequence of pseudo-periods. We call these last segments truly periodic.

We have chosen the test sequences of specializations to have the same combinatorial type of R-states, {Rstate_{∞}^{u}(q_{i_{j_{f}}} , h_{1}, w, p, a)}. Given indices u and f and an in-

dex ℓ in a test sequence that factors through and associated with P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}} and converges into Rstate_{∞}^{u}(q_{i_{j_{f}}} , h_{1}, w, p, a)}_{u=1}^{∞} for 1 ≤ f ≤ u, we denote by Mper^{u,ℓ} the maximum among all pseudo-periods of truly periodic segments in the R-state corresponding to the specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} after rescaling the metric on the corresponding R-state (note that this maximum is independent of the index f, 1 ≤ f ≤ u). We denote by Mp_{f,u}^{ℓ} the maximum between Mper^{u,ℓ} and the maximum length (after rescaling the metric) of a segment between branching points in the R-state corresponding to the specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} which is mapped to a point in the limit tree Y. For each two specializations (q_{i_{j_{f_{1}}}} , h_{1}, w, p, a)_{ℓ}, (q_{i_{j_{f_{2}}}} , h_{1}, w, p, a)_{ℓ} we have associated R-states RState(ℓ, f_{1}, u) and RState(ℓ, f_{2}, u) in correspondence. Given a branching point in these states and a copy of one of the defining parameters that passes through this branching point, we can measure the difference in the placement of the branching point on the copy of the defining parameter in the two different R-states RState(ℓ, i_{j_{f_{1}}} , m) and RState(ℓ, i_{j_{f_{2}}} , m). We denote by Mfluct(ℓ, f_{1}, f_{2}, u) the maximum among all these (finite set of) differences. Note that by construction both Mp_{f,u}^{ℓ} and Mfluct(ℓ, f_{1}, f_{2}, u) are asymptotically o(1).

Following definition 2.2 of [Se3], we say that a sequence of indices {u_{t}}_{t=1}^{∞}, together with tuples of integers 1 ≤ f_{1} < f_{2} < … < f_{t} ≤ u_{t}, and test sequences that factor through the various penetrated sculpted resolution P enSCRes_{i_{mu_{t}}}^{n_{mu_{t}} ,d_{mu_{t}}} are tame with respect to segments in the R-state that correspond to edges in the real tree Y, if there exists a constant c for which:

c · Mp_{f_{b_{1}} ,u_{t}}^{ℓ} > Mfluct(ℓ, f_{b_{2}}, f_{b_{3}}, u_{t})

for every possible indices t, ℓ, b_{1}, b_{2}, b_{3}, 1 ≤ b_{1}, b_{2}, b_{3} ≤ u_{t}.

If the test sequences of specializations we have started with contain a tame subsequence, we continue with the tame subsequence of specializations. By possibly passing to a further subsequence, we may assume that the maximum Mp_{f,u}^{ℓ} is always obtained for the same segment in the corresponding R-state or for a pseudo-period of the same truly periodic segment, that the maximal fluctuation Mfluct(ℓ, f_{1}, f_{1}, u) is asymptotically o(Mp_{f,u}^{ℓ}), and that the ratio between the length of segments that are between branching points and pseudo-periods of truly periodic segments in the R-state that are asymptotically Ω(Mp_{f,u}^{ℓ}) converge , where the limit is taken first on the corresponding test sequences (the index ℓ) and then on the indices u of the sculpted resolutions. If the original sequence of specializations is tame with respect to the segments in the R-state corresponding to edges in the tree Y, we declare each of the original segments in Y that are truly periodic, and for which their pseudo-period is Ω(Mp_{f,u}^{ℓ}) as not truly periodic for the continuation. We also check which of the ”newly uncovered” segments are truly periodic, and pass to an appropriate subsequence. We check if the obtained subsequence is tame with respect to segments in the R-state corresponding to both edges in Y and segments and pseudo-periods of truly periodic segments for which their length grow as Ω(Mp_{f,u}^{ℓ}). Since there are only finitely many segments in the R-state, and since there are u distinct specializations for each step u in our sequence, after finitely many ”uncoverings” we obtain a sequence which is not tame with respect to the previously ”uncovered” segments, and pseudo-periods of truly periodic segments in the R-state.

Once we obtain a sequence of specializations with no tame subsequences with respect to the previously ”uncovered” segments and pseudo-periods of truly periodic segments in the corresponding R-states, we can choose a subsequence of indices (u, f), 1 ≤ f ≤ u, together with test sequences that factor through the various penetrated sculpted resolutions P enSCResn_{m_{u}}, d_{m_{u}}_{i_{mu}}, so that for every u and every index ℓ of the subsequence:

u · Mp_{f′,u}^{ℓ} < Mfluct(ℓ, f_{2}, f_{3}, u)

for every possible indices f_{2}, f_{3}, and an index f′ for which the minimal Mp_{f′,u}^{ℓ} is obtained. We call such a sequence of specializations a perturbed sequence.

Given a perturbed sequence, for each index u we construct the R-state corresponding to the specialization (q_{i_{j_{f′}}} , h_{1}, w, p, a)_{ℓ}, for which Mp_{f′,u}^{ℓ} is minimal among all possible indices f, 1 ≤ f ≤ u. Clearly, given such an R-state we can present each of the specializations of the variables q_{i_{j_{f′}}} in terms of fractions of specializations of few of the defining parameters {h_{1}, w, p, a}. Hence, for every integer u we can represent the specialization (q_{i_{j_{f′}}} , h_{1}, w, p, a)_{ℓ} in terms of fractions of specializations of the parameters {h_{1}, w, p, a} that appear in the R-state corresponding to this specialization, and for any other possible index f, we can present the specialization (q_{i_{j_{f}}} h_{1}, w, p, a)_{ℓ} in terms of the same fractions of the specializations of the parameters that appear in the R-state corresponding to the specialization (q_{i_{j_{f′}}} , h_{1}, w, p, a)_{ℓ}, together with elements u_{u,ℓ}(f) that encode the fluctuations in the corresponding branching points along the specializations of the variables q_{i_{j_{f}}} between the R-state associated with the specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} and the ”reference” R-state associated with the specialization (q_{i_{j_{f′}}} , h_{1}, w, p, a).

At this point we continue as we did in the proofs of theorems 2.3, 2.7 and 2.11 of [Se3]. Each relation from the defining relations of the limit group RootW P HG_{ND} corresponds to a close loop on the R-state, and since the combinatorial type of the R-states associated with all specializations from the perturbed sequence are assumed to be identical, the ”combinatorial type” of the loop corresponding to a given relation is identical for all the specializations from the perturbed sequence. Furthermore, by our ”uncovering” procedure, every part of a loop that starts at some point on any of the ”uncovered” segments of the defining parameters, which are not truly periodic, and get back to the same point, represent the identity element in all R-states corresponding to specializations from our perturbed sequence. Also, every part of a loop that starts at some point on any of the ”uncovered” segments of the defining parameters, which are truly periodic, and get back to the same point, represent an element that commutes with a pseudo-period of that truly periodic segment in all R-states corresponding to the specializations from our perturbed sequence. Since for every index u such part of a loop of a defining relation represent either the identity element or an element that commutes with a pseudo-period of a truly periodic ”uncovered” segment in the R-state associated with the (distinguished) specialization (q_{i_{j_{f′}}} , h_{1}, w, p, a)_{ℓ}, and the same holds for every other R-state associated with a specialization (q_{i_{j_{f}}} h_{1}, w, p, a), every such part of a loop of one of the defining relations corresponds to a (parametric) equation involving (possibly) coefficients from the free group F_{k} =< a >, fractions of parameters corresponding to segments in the R-state associated with the (distinguished) specialization (q_{i_{j_{f′}}} , h_{1}, w, p, a)_{ℓ} that have not been uncovered and pseudo-periods of truly periodic ”uncovered” segments, that we denote p̂, and variables corresponding to the ”fluctuations” u_{u,ℓ}(f). Since the combinatorics of the R-states associated with all the specializations from our perturbed sequence is identical, the set of parametric equations corresponding to different parts of (closed) partial loops corresponding to the defining relations is identical for all specializations from the perturbed sequence. We denote this system of equations by Σ_{1}(u, p̂, a) = 1.

To the system of equations Σ_{1}(u, p̂, a) that involves the loops around each of the branching points, we add equations that guarantee partial equivariance. i.e., the additional equations guarantee that the specializations of each of the variables q_{i_{j_{f}}} in all places it appears in the image of the ball of radius R′ = \frac{R}{2}, B_{R′}, are identical. These equations can also be written in terms of the same fractions of the defining parameters p̂ and the variables corresponding to the various fluctuations u that were used in formulating the equations that involve the loops around the branching points. We denote the system of equations that combine the equations corresponding to the loops around branching points and the equations coming from the partial equivariance condition by Σ(u, p̂, a) = 1.

By our standard arguments presented in section 5 of [Se1], the specializations of the fluctuations and the ”uncovered” fractions of parameters {u, p̂, a} corresponding to the test sequences of specializations (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} factor through a (canonical) finite collection of maximal limit groups {MLim_{t}(u, p̂, a)}. Hence, by passing to a further subsequence we may assume that they factor through a unique maximal limit group MLim(u, p̂, a). By theorem 2.3 of [Se3] we may assume that this maximal limit group is not rigid with respect to the parameter subgroup < p̂ >. With the limit group MLim(u, p̂, a) we associate a limit group MP Lim(u, p̂, p′, a) generated by the limit group MLim(u, p̂, a) and the fractions p′ that were ”uncovered” through the process. By construction, there exists a natural map η : RootW P HG_{ND} → MP Lim(u, p̂, p′, a). The limit group MLim(u, p̂, a) was obtained as a limit group from a sequence of specializations of the penetrated sculpted resolutions P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}}, for which the

corresponding specializations of the penetrated sculpted resolutions P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}} were assumed to be almost shortest strictly solid specializations (with respect to the given covering closure) that factor through the stable limit group W P HG. If the limit group MLim(u, p̂, a) admits a non-trivial graded free decomposition, then its associated limit group MP Lim(u, p̂, p′, a) admits a non-trivial graded free decomposition (with respect to the parameter subgroup < p̂, p′, a >), which implies that the rooted stable limit group RootW P HG_{ND} admits a non-trivial graded free decomposition in which all the divisible subgroups can be conjugated into the various factors, which finally implies that the non-divisible subgroup W P HG_{ND} admits a non-trivial graded free decomposition in which all the divisible subgroups can be conjugated into the various factors. Hence, the stable limit group W P HG admits a graded free decomposition, which implies that all the specializations of the penetrated sculpted resolutions P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}} split through epimorphisms: λ : W P HG(g_{1}, h_{1}, w, p, a) → F_{k} ∗ F_{t}, where F_{t} is a non-trivial free group, and so that the epimorphism λ maps the subgroup < h_{1}, w, p, a > onto the free group F_{k}. This clearly contradicts the assumption that the specializations are strictly solid. Hence, we may assume that the limit group MLim(u, p̂, a) admits no non-trivial graded free decomposition. The rooted stable limit group RootW P HG_{ND} is mapped into the limit group MP Lim(u, p̂, p′, a) by the homomorphism η : RootW P HG_{ND} → MP Lim(u, p̂, p′, a). Recall that Θ_{ND} is the (graded) non-divisible abelian decomposition associated with the non-divisible subgroup W P HG_{ND}, and let Γ be the graded abelian JSJ decomposition associated with MP Lim(u, p̂, p′, a). By the construction of the limit group MP Lim(u, p̂, p′, a), the image of the stable rooted limit group RootW P HG_{ND}, η(RootW P HG_{ND}), can not be conjugated into the fundamental group of a proper subgraph of the graph of groups Γ, and every divisible subgroup of W P HG_{ND} (which is naturally a subgroup of RootW P HG_{ND}) is elliptic in Γ (simply since every non-cyclic abelian subgroup is elliptic). Since Θ_{ND} is the graded non-divisible abelian decomposition of the non-divisible subgroup W P HG_{ND}, and every divisible subgroup of W P HG_{ND} is elliptic in Γ, every non-QH vertex group and every edge group in Θ_{ND} are mapped by η into either a non-abelian, non-QH vertex group or into an edge group in Γ, or it is mapped into an abelian vertex group and intersects the subgroup generated by the edge groups connected to it in a subgroup of finite index.

By our assumptions, the specializations of the stable rooted limit group RootW P HG_{ND} corresponding to the specializations {(q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ}} converge into an action of the stable rooted limit group RootW P HG_{ND} on a real tree T. Along our ”uncovering” procedure, we have uncovered fractions p′ of the defining parameters < h_{1}, w, p >, until we reached a stage in which the size of the fluctuations are much bigger than the size of the fractions p̂ of the defining parameters that are not yet uncovered. From the action of the stable rooted limit group RootW P HG_{ND} on the real tree T, it inherits an abelian decomposition Δ_{1}. Each non-QH, non-abelian vertex group V in Δ_{1} acts on real tree T_{V} obtained from the sequences of specializations {(q_{i_{j}}, h_{1}, w, p, a)_{ℓ}}. The vertex group V inherits an abelian decomposition Δ_{2} from this action. By iteratively continuing with non-abelian, non-QH vertex groups in the newly obtained abelian decompositions, we finally reach a level n for which the fluctuations are elliptic in the abelian decompositions Δ_{n} constructed at that level, and they are not all elliptic in the various abelian decompositions Δ_{n+1} constructed at the n + 1 level. Since there are only finitely many orbits of vertex stabilizers in each of the abelian decompositions Δ_{i}, and the abelian decompositions for conjugate vertices are identical, with each level i, we associate only finitely many abelian decompositions Δ_{i}.

Definition 5.26. Let V be a non-QH, non-abelian vertex group in one of the abelian decompositions Δ_{n}. We say that an abelian decomposition Γ_{V} of the vertex group V is a multi-graded compatible abelian decomposition of V, if the following conditions hold:

(i) every edge group connected to V in Δ_{n} is elliptic in Γ_{V}.

(ii) by part (i), we can use the abelian decomposition Γ_{V} to refine the decomposition Δ_{n}, and obtain an abelian decomposition Δ′_{n}. Let Γ_{n} be the abelian decomposition obtained from Δ′_{n} by collapsing all the edge groups in Δ′_{n} that correspond to edges in Δ_{n}, and let V_{n} be the fundamental group of Γ_{n} (which is the fundamental group of Δ_{n}). Note that the structure of Γ_{n} is similar to the structure of Γ_{V}, i.e., there is a one to one correspondence between the edges and the non-QH, non-abelian vertex groups. Then every edge group connected to V_{n} in Δ_{n−1} is elliptic in Γ_{n}.

(iii) We continue by induction. Given the abelian decomposition Γ_{i} of a vertex group V_{i} in the abelian decomposition Δ_{i−1}, we refine the abelian decomposition Δ_{i−1}, to obtain an abelian decomposition Δ′_{i−1}. We further collapse all the edges in Δ′_{i−1} that correspond to edges in Δ_{i}, to obtain an abelian decomposition Γ_{i−1} of the fundamental group of Δ_{i−1}, which we denote V_{i−1}. Then every edge group connected to V_{i−1} in Δ_{i−2} is elliptic in Γ_{i−2}.

(iv) By part (iii) we finally obtain an abelian decomposition Γ_{1} of RootW P HG_{ND}. We further assume that the parameter subgroup < h_{1}, w, p, a > is elliptic in the abelian decomposition Γ_{1}, that every non-cyclic abelian subgroup of RootW P HG_{ND} is elliptic in the abelian decomposition Γ_{1}, and that every divisible subgroup of W P HG_{ND}, which is a subgroup of RootW P HG_{ND}, is elliptic in Γ_{1}.

Given a sequence of decompositions of the rooted stable limit group RootW P HG_{ND}, the collection of multi-graded decompositions of RootW P HG_{ND} that are compatible with the given sequence of abelian decompositions is encoded in a multigraded compatible abelian decomposition of the (graded) stable rooted limit group RootW P HG_{ND}.

Theorem 5.27. Let RootW P HG_{ND} be a multi-graded limit group, where the parameter subgroups is taken to be < h_{1}, w, p > and E_{1}, …, E_{q}, where E_{1}, …, E_{q} are its divisible subgroups. Suppose that RootW P HG_{ND} admits no multi-graded free decompositions, i.e., RootW P HG_{ND} can not be decomposed into a free product in which the subgroups < h_{1}, w, p, a >, E_{1}, …, E_{q} can be conjugated into the factors.

Let Δ_{1} be an abelian decomposition of the limit group RootW P HG_{ND}, viewed as an ungraded limit group. With each non-QH, non-abelian vertex group V in Δ_{1} we further associate an abelian decomposition Δ_{2}, with each non-QH, nonabelian vertex group in one of the abelian decompositions Δ_{2} we associate an abelian decomposition Δ_{3} and so on, until we associate an abelian decomposition Δ_{n} with each non-QH, non-abelian vertex group in each of the abelian decompositions Δ_{n−1}.

Then there exists a reduced (perhaps trivial) splitting of RootW P HG_{ND} with abelian edge groups, which we call a multi-graded compatible abelian JSJ decomposition of RootW P HG_{ND} (with respect to the above sequence of hierarchical decompositions) with the following properties.

(i) The multi-graded compatible abelian JSJ decomposition is compatible with the given sequence of hierarchical decompositions of RootW P HG_{ND}.

(ii) Every (multi-graded compatible) canonical maximal QH subgroup (CMQ) of RootW P HG_{ND} is conjugate to a vertex group in the multi-graded compatible JSJ decomposition. Every (multi-graded compatible) QH subgroup of W P HG_{ND} can be conjugated into one of the (multi-graded compatible) CMQ subgroups of RootW P HG_{ND}. Every vertex group in the multi-graded compatible JSJ decomposition which is not a CMQ subgroup of RootW P HG_{ND} is elliptic in any multi-graded compatible abelian splitting of RootW P HG_{ND}.

(iii) A one edge (multi-graded compatible) abelian splitting RootW P HG_{ND} = D ∗_{A} E or RootW P HG_{ND} = D∗_{A} which is hyperbolic in another such elementary multi-graded compatible abelian splitting is obtained from the multi-graded compatible abelian JSJ decomposition of RootW P HG_{ND} by cutting a 2-orbifold corresponding to a (multi-graded compatible) CMQ subgroup of RootW P HG_{ND} along a weakly essential s.c.c.

(iv) Let Δ be a one edge (multi-graded compatible) splitting along an abelian subgroup RootW P HG_{ND} = D ∗_{A} E or RootW P HG_{ND} = D∗_{A}, which is elliptic with respect to any other one edge (multi-graded compatible) abelian splitting of RootW P HG_{ND}. Then Δ is obtained from the multi-graded compatible JSJ decomposition of RootW P HG_{ND} by a sequence of collapsings, foldings, and conjugations.

(v) If JSJ_{1} is another multi-graded compatible abelian JSJ decomposition of RootW P HG_{ND}, then JSJ_{1} is obtained from the multi-graded compatible abelian JSJ decomposition by a sequence of slidings, conjugations and modifying boundary monomorphisms by conjugations (see section 1 of [Ri-Se2] for these notions)

Proof: Given two multi-graded abelian decompositions Λ_{1} and Λ_{2} of the multigraded limit group RootW P HG_{ND}, the ”machine” for the construction of a JSJ decomposition, presented in [Ri-Se2], constructs a multi-graded abelian decompositions, Λ_{C}, which is a ”common refinement” of Λ_{1} and Λ_{2}. If Λ_{1} and Λ_{2} are compatible with the given sequence of hierarchical decompositions of RootW P HG_{ND}, so is the common refinement produced by the ”JSJ machine”. Therefore, the (canonical) multi-graded abelian decomposition produced by the ”JSJ machine” from the entire collection of multi-graded compatible abelian decompositions of RootW P HG_{ND}, is a multi-graded compatible abelian decomposition of RootW P HG_{ND}. As in the construction of the (multi-graded) abelian JSJ decomposition of a limit group, the (canonical) multi-graded abelian decomposition produced from the entire collection of graded compatible abelian decompositions of RootW P HG_{ND}, satisfies properties (i)-(v) listed in the theorem.

□

As we have already observed, from the action of the stable rooted limit group RootW P HG_{ND} on the real tree T, RootW P HG_{ND} inherits an abelian decomposition Δ_{1}. Each non-QH, non-abelian vertex group V in Δ_{1} acts on a real tree T_{V} obtained from the sequences of specializations {(q_{i_{j}}, h_{1}, w, p, a)_{ℓ}}. The vertex group V inherits an abelian decomposition Δ_{2} from this action. By iteratively continuing with non-abelian, non-QH vertex groups in the newly obtained abelian decompositions, we finally reach a level n for which the fluctuations are contained in non-QH, non-abelian vertex groups in the abelian decompositions Δ_{n} constructed at that level, and they are not all elliptic in the various abelian decompositions Δ_{n+1} constructed at the n + 1 level.

The stable rooted limit group RootW P HG_{ND} is embedded by η into the limit group MP Lim(u, p̂, p′, a). By construction, the graded abelian JSJ decomposition of the limit group MP Lim(u, p̂, p′, a) is non-trivial, hence, the abelian decomposition inherited by the stable rooted limit group RootW P HG_{ND} from its embedding into MP Lim(u, p̂, p′, a), is non-trivial, every divisible subgroup of the non-divisible subgroup W P HG_{ND} (that is embedded in RootW P HG_{ND}) is elliptic in it, and by construction this abelian decomposition of RootW P HG_{ND} is multi-graded (with respect to the parameter subgroup < h_{1}, w, p > and the divisible subgroups) and compatible with the hierarchical sequence of abelian decompositions the Rooted stable limit group W P HG_{ND} inherits from its action on the real tree T. Hence, the multi-graded compatible JSJ decompositions of the rooted stable limit group RootW P HG_{ND} with respect to the sequence of abelian decompositions it inherits from its action on the real tree T is non-trivial, which implies that the multigraded compatible JSJ decomposition of the non-divisible subgroup W P HG_{ND} is non-trivial. Let Γ_{W P HG_{ND}} be the multi-graded abelian decomposition inherited by W P HG_{ND} from its action on the real tree T.

As we did in the proof of theorems 2.7 and 2.11 of [Se3], the properties of the restriction of the embedding η, from the non-divisible limit group W P HG_{ND} we have started with, into the graded limit group MP Lim(u, p̂, p′, a), allow us to bound the reduced complexity of the multi-graded compatible abelian JSJ decomposition Γ_{W P HG_{ND}} in terms of the reduced complexity of the non-divisible graded abelian decomposition Θ_{ND} of W P HG_{ND}. We start by recalling the definition of the reduced complexity of a graded limit group (definition 2.8 of [Se3]).

Let Glim(y, p, a) be a (multi) graded limit group, with respect to the subgroup P =< p >, and suppose Glim(y, p, a) admits no non-trivial graded free decomposition. Let Γ be a graded abelian decomposition of Glim(y, p, a). Let Q_{1}, …, Q_{n} be the QH vertex groups in the graded abelian decomposition Γ, and let S_{1}, …, S_{n} be their corresponding surfaces. We set the complexity of a QH vertex group Q_{j}, to be the ordered couple (|χ(S_{j})|, genus(S_{j})). We set the reduced complexity of the graded abelian decomposition Γ, denoted Rcomp(Γ), to be the tuple:

((χ(S_{1})|, genus(S_{1})), …, (|χ(S_{n})|, genus(S_{n})), d)

where the sequence of couples (χ(S_{1})|, genus(S_{1})), …, (|χ(S_{n})|, genus(S_{n})) is ordered in a decreasing lexicographical order, and the integer d is set to be d = e − a, where e is the number of edges between non-QH vertex groups in Γ, and a is the number of abelian vertex groups adjacent to the collection of these edges.

Let Γ_{1} and Γ_{2} be two graded abelian decompositions of the graded limit groups Glim_{1}(y, p, a) and Glim_{2}(y, p, a) in correspondence, and Glim_{1}(y, p, a), Glim_{2}(y, p, a) admit no non-trivial graded free decompositions. We say that Rcomp(Γ_{1}) ≤ Rcomp(Γ_{2}) if the sequence associated with Γ_{1} is not greater than the sequence associated with Γ_{2} in the natural lexicographical order on the above sequences of (non-negative) integers. Clearly, the reduced complexity of any two such graded decompositions can be compared.

Proposition 5.28. Let Γ_{W P HG_{ND}} be the multi-graded compatible abelian JSJ decompositions of the non-divisible limit group W P HG_{ND} with respect to the sequence of abelian decompositions it inherits from its action on the real tree T, let Θ be the graded abelian decomposition of the stable limit group W P HG, and let Θ_{ND} be the (original) non-divisible graded abelian decomposition of W P HG_{ND}. Then:

Rcomp(Γ_{W P HG_{ND}}) < Rcomp(Θ_{ND}) ≤ Rcomp(Θ).

Proof: The second inequality is immediate from the construction of the nondivisible subgroup. Since Θ_{ND} is the multi-graded abelian JSJ decomposition of W P HG_{ND}, where all the divisible subgroups are supposed to be elliptic, and Γ_{W P HG_{ND}} is such a multi-graded abelian decomposition of W P HG_{ND}, Γ_{W P HG_{ND}} has to be compatible with Θ_{ND}, hence, Rcomp(Γ_{W P HG_{ND}}) ≤ Rcomp(Θ_{ND}). If Rcomp(Γ_{W P HG_{ND}}) = Rcomp(Θ_{ND}) then the graph of groups Γ_{W P HG_{ND}} is identical to Θ_{ND}, which implies that all the QH vertex groups in the graded abelian JSJ decomposition Θ_{ND}, are contained in non-QH, non-abelian vertex groups in the abelian decompositions Δ_{n}, inherited by (subgroups of) W P HG_{ND} from the n-th level of its action on the real tree T. Since Θ_{ND} is the (multi-) graded abelian JSJ decomposition of W P HG_{ND}, all the boundary components of QH vertex groups in Θ_{ND} are mapped by η into non-QH, non-abelian vertex groups or into edge groups in the graded abelian JSJ decomposition of the limit group MP Lim(u, p̂, p′, a). This implies that for large enough u, 1 ≤ f ≤ u, and large enough ℓ, for the specializations (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ}, the sizes of the corresponding specializations of the boundary elements of QH vertex groups in Θ_{ND} (after appropriate conjugation) are much smaller the sizes of a corresponding specialization of a (fixed) generating set of at least one of the QH vertex groups in Θ_{ND}, which clearly contradicts the specializations (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} being almost shortest. Hence, Rcomp(Γ_{W P HG_{ND}}) < Rcomp(Θ_{ND}).

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By proposition 5.28 the reduced complexity of Γ_{W P HG_{ND}} is strictly smaller than the reduced complexity of the non-divisible abelian decomposition Θ_{ND}, which is bounded by the reduced complexity of the graded abelian decomposition Θ of the stable limit group W P HG. Since every divisible edge group in Θ_{ND} is elliptic in Γ_{W P HG_{ND}}, the graded abelian decomposition Γ_{W P HG_{ND}} can be naturally extended to a graded abelian decomposition of the stable limit group W P HG, by adding edges corresponding to the divisible Bass-Serre generators. We denote this graded abelian decomposition of the stable limit group W P HG, Θ^{2}. Since, Rcomp(Γ_{W P HG_{ND}}) < Rcomp(Θ_{ND}), by proposition 5.28, Rcomp(Θ^{2}) < Rcomp(Θ).

With the graded abelian decomposition Θ^{2} we associate its non-divisible subgraph (of groups) according to definition 5.23, which we denote Θ_{ND}^{2}. According to definition 5.23, with the graph of groups Θ_{ND}^{2} we also associate a collection of divisible edge groups and abelian vertex groups, a collection that includes all the divisible edge groups and abelian vertex groups in the (original) graded abelian decomposition Θ. With Θ_{ND}^{2} we associate its fundamental group, which we call the (second) non-divisible subgroup, and denote W P HG_{ND}^{2}. By construction, the non-divisible subgroup, W P HG_{ND}^{2}, is a subgroup of the non-divisible subgroup W P HG_{ND} associated with the graded abelian decomposition Θ. According to definition 5.23, along the construction of the (second) non-divisible subgraph and subgroup, we pass to a (stable) subsequence of our given sequence of penetrated sculpted resolutions, P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}}, and their associated test sequences. We continue the argument with this subsequence, and continue to denote it, P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}}.

Suppose that the number of (conjugacy classes of) divisible subgroups associated with the obtained decomposition Θ^{2} is strictly bigger than the number of divisible subgroups associated with the graded abelian decomposition of the stable limit group W P HG, Θ, we have started with. Then we replace the graded abelian decomposition Θ, by the multi-graded abelian decomposition of the stable limit group W P HG with respect to the parameter subgroup < h_{1}, w, p > and the obtained collection of divisible edge groups, which we denote Θ_{2}. By construction, the number of divisible edge groups associated with Θ_{2} is strictly bigger than the number of divisible edge groups associated with Θ. In this case, of larger number of divisible edge groups, we start the entire procedure with the graph of groups Θ_{2} instead of the graph of groups Θ. With Θ_{2} we associate a non-divisible subgraph, and a non-divisible group, and a stable rooted limit group, for which the corresponding specializations are R-AP W H_{1}-covered (theorem 5.26), hence, allows to analyze the action of the associated non-divisible subgroup on the corresponding real tree. Note that in case the number of divisible subgroups is larger, the number of Bass-Serre generators corresponding to the edges that are in Θ_{2} but not in its associated non-divisible subgraph, is strictly bigger than the number of Bass-Serre generators corresponding to such edges in the original graph of groups Θ.

Suppose that the number of (conjugacy classes) of divisible edge groups and abelian vertex groups in Θ^{2} is identical to their number in Θ. For each couple of indices u and f, where 1 ≤ f ≤ u, and a specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} of the pen-

etrated sculpted resolution P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}}, we associate with the specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} a specialization (q̂_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} which is an almost shortest specialization in the same class of the specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} with respect to the modular groups associated with the graded abelian decomposition Θ^{2}, where the order on the modular groups according to which the specializations are almost shortest, starts with the modular groups associated with the non-divisible subgraph Θ_{ND}^{2}, and then the modular groups associated with the divisible edge groups in Θ^{2}. For each u and f, 1 ≤ f ≤ u, we replace the subgroup < q_{i_{j_{f}}} , h_{1}, w, p, a > of the pen-

etrated sculpted resolution P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}}, by a subgroup < q̂_{i_{j_{f}}} , h_{1}, w, p, a >, where the subgroup < q̂_{i_{j_{f}}} , h_{1}, w, p, a > is obtained as a limit group from a subsequence of specializations (q̂_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} that are almost shortest with respect to the action of the modular groups associated with Θ^{2}, according to their prescribed order. To the subgroup < q̂_{i_{j_{f}}} , h_{1}, w, p, a > of the (corresponding clo-

sure of the) penetrated sculpted resolution P enSCRes_{i_{j_{f}}}^{n_{mu} ,d_{mu}} we add (conjugating) elements that demonstrate that the subgroups < q_{i_{j_{f}}} , h_{1}, w, p, a > and < q̂_{i_{j_{f}}} , h_{1}, w, p, a > correspond to elements that are in the same modular blocks with respect to the graded abelian decomposition Θ^{2}, i.e., elements that conjugate subgroups of < q_{i_{j_{f}}} , h_{1}, w, p, a > that correspond to non-QH vertex groups in Θ^{2} to the corresponding subgroups in < q̂_{i_{j_{f}}} , h_{1}, w, p, a >. With the non-divisible subgroup Θ_{ND}^{2} and the given sequence of penetrated sculpted resolutions, and their associated test sequences, we further associate a stable rooted limit group according to definition 5.24, which we denote, RootW P HG_{ND}^{2}. Note that in defining the (second) rooted limit group we may need to pass a subsequence of our given sequence of penetrated sculpted resolutions and their associated test sequences.

In theorem 5.25 we have shown that there exists some global constant R, and for any positive integer u, an index ℓ(u), so that for every u, and f, 1 ≤ f ≤ u, and every ℓ > ℓ(u), the specialization of the (first) rooted stable limit group RootW P HG_{ND} associated with the specialization (q_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ} is R-AP W H_{1}-covered. A modified argument shows that a similar statement holds for the sequences of specializations of the (second) rooted stable limit group RootW P HG_{ND}^{2} associated with a specialization: {(q̂_{i_{j_{f}}} , h_{1}, w, p, a)_{ℓ}}, which are almost shortest with respect to the modular groups associated with Θ^{2} according to their prescribed order.

Theorem 5.29. There exists a (global) constant R, for which given any positive integer u, and a test sequence that factors through the penetrated sculpted resolution P enSCRes_{i_{mu}}^{n_{mu} ,d_{mu}} (r_{i_{mu}} , h_{1}, w, p, a) and associated with it, there exists an integer ℓ(u), so that the sets of specializations of the (second) stable rooted limit group RootW P HG_{ND}^{2} corresponding to the specializations:

(q_{i_{j_{1}}} , h_{1}, w, p, a)_{ℓ}, …, (q_{i_{ju}} , h_{1}, w, p, a)_{ℓ}

are R-AP W H_{1}-covered for all ℓ > ℓ(u).

Proof: The argument we use is similar to the one used to prove theorem 5.26. Suppose there is no such global constant R. Then for every positive integer t, there exists an index u(t), an index f(t), 1 ≤ f(t) ≤ u(t), and a test sequence that factors

through the penetrated sculpted resolution P enSCRes_{i_{j_{f(t)}}}^{n_{m_{u(t)}} ,d_{m_{u(t)}}} (r_{i_{j_{f(t)}}} , h_{1}, w, p, a), for which the sequence of specializations of the (second) stable rooted limit group RootW P HG_{ND}^{2} corresponding to the sequence of specializations: {(q_{i_{j_{f(t)}}} , h_{1}, w, p, a)_{ℓ}}_{ℓ=1}^{∞} are not t-AP W H_{1}-covered. For every t, we can pass to a subsequence that converges into an action of the stable rooted limit group RootW P HG_{ND}^{2} on a limit tree Y_{t}, and since each specialization in this sequence is not t-AP W H_{1}-covered, the action of RootW P HG_{ND}^{2} on the limit tree Y_{t} is not t-AP W H_{1}-covered.

Since for each index t the action of RootW P HG_{ND}^{2} on the tree Y_{t} is not t-AP W H_{1} covered, from the sequence of actions of the limit group W P HG_{ND}^{2} on the trees Y_{t}, it is possible to extract a subsequence (still denoted Y_{t}), that converges into a (faithful) action of the stable rooted limit group RootW P HG_{ND}^{2} on a limit tree Y. By construction, the action of the limit group RootW P HG_{ND}^{2} on the limit tree Y satisfies the properties of the action obtained in proving theorem 1.7 of [Se3], i.e., the orbit of the subtree Y_{AP W H_{1}} in the limit tree Y misses either a non-degenerate segment or a germ of a branching point in Y.

The continuation of our argument is a modification of the argument used to prove theorem 5.25. By the proof of theorem 1.2 of [Se3], from the action of the limit group RootW P HG_{ND}^{2} on the real tree Y, RootW P HG_{ND}^{2} inherits an abelian decomposition in which all the divisible edge groups of RootW P HG_{ND}^{2} are elliptic. Hence, its subgroup, W P HG_{ND}^{2}, inherits an abelian decomposition Λ_{W P HG_{ND}^{2}}, a decomposition in which the image of the subgroup AP W H_{1} =< h_{1}, w, p, a >, η_{2}(AP W H_{1}) is contained in a proper subgraph of groups Λ′, and every divisible edge group is elliptic. Let Δ_{W P HG_{ND}^{2}} be the graph of groups obtained from Λ_{W P HG_{ND}^{2}} by collapsing Λ′ to a vertex. In Δ_{W P HG_{ND}^{2}} the subgroup AP W H_{1} is contained in a vertex stabilizer, each divisible subgroup is elliptic, each edge group is a non-trivial abelian subgroup of W P HG_{ND}^{2}, and each vertex group that does not contain a conjugate of AP W H_{1} corresponds to either a point stabilizer, an IET component, or an axial component in Y.

The abelian decomposition Δ_{W P HG_{ND}^{2}}^{2} is a multi-graded abelian decomposition of W P HG_{ND}^{2} with respect to the parameter subgroup < h_{1}, w, p > and the divisible edge groups in the graph of groups Θ_{ND}, hence, every edge group and every non-QH vertex group in Θ_{ND} is elliptic in Δ_{W P HG_{ND}^{2}}. The sequence of abelian decompositions of the subgroup W P HG_{ND}^{2} = W P HG_{ND}, that were used to construct the multi-graded compatible abelian JSJ decomposition Γ_{W P HG_{ND}}, are identical to the corresponding sequence of abelian decompositions inherited by the subgroup W P HG_{ND}^{2} from its action on the real tree Y, since the modular groups associated with the graded compatible decomposition do not change the actions of the corresponding subgroups of W P HG_{ND}. Hence, the multigraded abelian decomposition Δ_{W P HG_{ND}^{2}} has to be compatible with the multigraded compatible abelian JSJ decomposition Γ_{W P HG_{ND}}, so it is compatible with the multi-graded abelian decomposition Θ^{2} of the stable limit group W P HG. Therefore, for large enough t and large enough ℓ, the specializations {(q̂_{i_{j_{f(t)}}} , h_{1}, w, p, a)_{ℓ}} are not almost shortest with respect to the action of the modular group associated with the graded abelian decomposition Θ^{2}, which clearly contradicts our assumptions.

□

We continue with the sequences of specializations {(q̂_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}} as we continued with the specializations {(q_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}}. We sequentially ”uncover” fractions p′ of the defining parameters {h_{1}, w, p} until we obtain a perturbed subsequence of specializations. With the perturbed subsequence we associate limit groups MLim(u, p̂, a) and MP Lim(u, p̂, p′, a), where u are variables representing fluctuations and p̂ represent the parts of the defining parameters which are not yet ”uncovered”. With the limit group MP Lim(u, p̂, p′, a) there is associated a natural embedding η_{2} : RootW P HG_{ND}^{2} → MP Lim(u, p̂, p′, a) which maps the divisible subgroup of the non-divisible subgroup W P HG_{ND}^{2} into conjugates of the distinguished vertex group in the graded abelian decomposition of MP Lim(u, p̂, p′, a) (graded with respect to the subgroup < p̂, p′, a >). By theorem 2.7 of [Se3], the graded limit group MP Lim(u, p̂, p′, a) can not be rigid (with respect to the parameter subgroup < p, p′ >), hence, its graded abelian JSJ decomposition has to be non-trivial.

As we did with the sequences of specializations {(q_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}}, the (finite) sequential ”uncovering” process associates a sequence of (hierarchical) abelian decompositions with the rooted stable limit group RootW P HG_{ND}^{2}. Since the graded abelian decomposition of MP Lim(u, p̂, p′, a) is non-trivial, and η_{2} embeds the rooted stable limit group RootW P HG_{ND}^{2} into MP Lim(u, p̂, p′, a), the multi-graded compatible abelian decomposition of the rooted graded limit group RootW P HG_{ND}^{2} is non-trivial, and so is the multi-graded compatible abelian decomposition inherited by the (second) non-divisible subgroup W P HG_{ND}^{2}, which we denote, Γ_{W P HG_{ND}^{2}}^{2}

Since the sequences of specializations {(q̂_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}} were obtained by shortening the sequences of specializations {(q_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}} using the modular groups associated with the multi-graded compatible abelian JSJ decomposition Γ_{W P HG_{ND}}, the sequence of abelian decompositions associated with the sequences of specializations {(q̂_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}} starts with the sequence of specializations associated with the sequences {(q_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}}, possibly followed by additional abelian decompositions. Hence, the abelian decomposition Γ_{W P HG_{ND}^{2}}^{2} being compatible with the sequence of abelian decompositions associated with the sequences {(q̂_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}}, is multi-graded compatible with respect to the sequence of abelian decompositions associated with the sequences of specializations {(q_{i_{j(t)}}, h_{1}, w, p, a)_{ℓ}}. Therefore, every abelian edge group and every non-QH vertex group in Γ_{W P HG_{ND}} is elliptic in Γ_{W P HG_{ND}^{2}}^{2}. Hence, Rcomp(Γ_{W P HG_{ND}^{2}}^{2}) ≤ Rcomp(Γ_{W P HG_{ND}}), and by the same argument used to prove theorem 5.29 Rcomp(Γ_{W P HG_{ND}^{2}}^{2}) < Rcomp(Γ_{W P HG_{ND}}).

By iteratively repeating these constructions, we either strictly increase the number of (conjugacy classes of) divisible edge groups, or we obtain an (infinite) sequence of multi-graded (compatible) abelian decompositions of the non-divisible limit group W P HG: Γ_{W P HG_{ND}}, Γ_{W P HG_{ND}^{2}}^{2}, Γ_{W P HG_{ND}^{3}}^{3}, …, for which:

Rcomp(Γ_{W P HG_{ND}}) > Rcomp(Γ_{W P HG_{ND}^{2}}^{2}) > Rcomp(Λ_{W P HG_{ND}^{3}}^{3}) > …

but any sequence of graded abelian decompositions of the limit group W P HG_{ND} with strictly decreasing reduced complexity terminates in a finite time, and we get a contradiction. Hence, we sequentially increase the number of (conjugacy classes of) divisible edge groups in the graphs of groups Θ, Θ_{2}, Θ_{3}, . . .. But each time we add an additional divisible edge group, we associate at least one additional Bass-Serre generator corresponding to edges that are in the graphs of groups Θ_{i} but not in their associated non-divisible subgraphs. Since the number of Bass-Serre generators in Θ_{i} is (globally) bounded by the first betti number of the stable limit group W P HG, the whole process has to terminate. which finally contradicts the existence of an unbounded number of sculpted resolutions of the same width along an infinite path of the sieve procedure, which concludes the proof of theorem 5.17.

□

The combination of proposition 5.16 and theorem 5.17 clearly gives a contradiction, that finally proves the termination of the sieve procedure (theorem 5.14).

□

Recall that T_{1}(p) ⊂ EAE(p) is defined to be the set of specializations of the defining parameters p, that have a witness with valid P S statement that terminates after the initial step of the procedure for the construction of the tree of stratified sets (cf. theorem 3.2). At this stage we have all the tools needed for showing that the set T_{1}(p) is in the Boolean algebra of AE sets. By construction, if p_{0} ∈ T_{1}(p) then there must exist a valid P S statement of the form: ((h_{1}^{2}, g_{1}^{1}), …, (h_{ν(ps)}^{2}, g_{ν(ps)}^{1}), h_{0}^{1}, w_{0}, p_{0}, a) that factors through one the of the P S resolutions P SHGHRes constructed with respect to all proof systems that terminate after the initial step of the procedure for the construction of the tree of stratified sets. By proposition 5.6, the sets T SP S(p) associated with the various P S resolutions P SHGHRes, i.e., the sets of specializations p_{0} of the defining parameters P =< p > for which there exists a test sequence of valid P S statements that factor through any of the P S resolutions P SHGHRes, is in the Boolean algebra of AE sets. By lemma 5.5 if there exists a valid P S statement that factors through a P S resolution P SHGHRes, then either there exists a test sequence of valid P S statements that factors through that P S resolution, or there must exist a valid P S statement that can be extended to a specialization that factors through one the of the collapse extra P S limit groups associated with the various P S resolutions P SHGHRes.

We continue with each of the collapse P S limit groups in parallel. Exactly as we did with each of the P S resolutions P SHGHRes, we associate (canonically) with each of the developing resolution constructed along the sieve procedure, a set T SP S(p), that contains all the specializations of the defining parameters p, that have a test sequence of valid P S statements that factor through the given developing resolution. By proposition 5.6 applied to the developing resolutions constructed along the sieve procedure, the sets T SP S(p) associated with these developing resolutions are all in the Boolean algebra of AE sets. By lemma 5.5 applied to the various developing resolutions, if there exists a valid P S statement that factors through a developing resolution, then either there exists a test sequence of valid P S statements that factors through that developing resolution, or there must exist a valid P S statement that factors through one of the anvils associated with that developing resolution. Hence, by iteratively applying lemma 5.5 and proposition 5.6 to the various developing resolutions constructed along the sieve procedure, if for some specialization p_{0} of the defining parameters p, there exists a valid P S statement that factors through one of the P S limit groups P SHGH we have started the sieve procedure with, then this specialization p_{0} is an element of at least one of the sets T SP S(p) associated with the various developing resolutions constructed along the sieve procedure.

The set T_{1}(p) is the union of the sets T SP S(p) associated with the various developing resolutions constructed along the sieve procedures corresponding to the finitely many possibilities of proof systems of depth 1 (i.e., proof systems that terminates after the initial step of the sieve procedure). Since each of the sets T SP S(p) is in the Boolean algebra of AE sets, and a Boolean algebra is closed under finite unions, we have finally completed the proof of theorem 3.2.

Theorem 3.2. Let T_{1}(p) ⊂ EAE(p) be the subset of all specializations p_{0} ∈ EAE(p) of the defining parameters p, that have witnesses with a proof system that terminates after the initial step of the procedure for the construction of the tree of stratified sets. Then T_{1}(p) is in the Boolean algebra of AE sets.

At this stage we are finally ready to show that the entire set EAE(p) is in the Boolean algebra of AE sets. The tree of stratified sets has a finite depth, which (by definition) bounds the depth of all possible proof systems associated with the tree of stratified sets. For each integer d, we set T_{d}(p) to be the set of specializations p_{0} of the defining parameters P =< p > for which there exists a valid P S statement for some proof system of depth d. Clearly: EAE(p) = T_{1}(p) ∪ … ∪ T_{d_{0}}(p) where d_{0} is the depth of tree of stratified sets. Since a Boolean algebra is closed under finite unions, to show that the set EAE(p) is in the Boolean algebra of AE sets, it is enough to show that each of the sets T_{d}(p) is in the Boolean algebra of AE sets. By the structure of the tree of stratified sets, and the global bounds on possible disjoint rigid and strictly solid families associated with each of the stratas in these tree, given a fixed depth d, there exist finitely many possible proof systems of depth d. Given a fixed proof system of depth d, which we denote P S, we collect all its associated valid P S statements in a canonical collection of finitely many P S limit groups which we denote: P S(HG)^{d}H. With each of the P S limit groups P S(HG)^{d}H we associate a sieve procedure, precisely as we did with P S limit groups P SHGH that contain all the valid proof statements of depth 1. With each of the developing resolutions constructed along these sieve procedures, we associate a set T SP S(p), that contain the set of specializations of the defining parameters p that have a test sequence of valid P S statements that factor through that developing resolution. By iteratively applying lemma 5.5 and proposition 5.6, the sets T SP S(p) associated with the various developing resolutions constructed along these sieve procedures are all in the Boolean algebra of AE sets, and if a specialization p_{0} has a valid P S statement of depth d, i.e., if p_{0} ∈ T_{d}(p), then p_{0} is contained in at least one of the sets T SP S(p) constructed along the sieve procedures associated with the P S limit groups P S(HG)^{d}H.

The set T_{d}(p) is the union of the sets T SP S(p) associated with the various developing resolutions constructed along the sieve procedures corresponding to the finitely many possibilities of proof systems of depth d. Since each of the sets T SP S(p) is in the Boolean algebra of AE sets, and a Boolean algebra is closed under finite unions, for each depth d, the set T_{d}(p) is in the Boolean algebra of AE sets. Since the sets EAE(p) is the union: EAE(p) = T_{1}(p) ∪ … ∪ T_{d_{0}}(p), where d_{0} is the depth of tree of stratified sets, the set EAE(p) itself is in the Boolean algebra of AE sets, and we have finally completed the proof of theorem 1.4.

Theorem 1.4. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let EAE(p) be a set defined by the predicate:

EAE(p) = ∃w ∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1.

Then EAE(p) is in the Boolean algebra of AE sets.

The proof of theorem 1.4 shows that a set EAE(p) defined using a conjunction of a system of equalities and a system of inequalities:

EAE(p) = ∃w ∀y ∃x Σ(x, y, w, p, a) = 1 ∧ Ψ(x, y, w, p, a) ≠ 1

is in the Boolean algebra of AE sets. The generalization of that proof to a set EAE(p) defined using a (finite) disjunction of conjunctions of a system of equalities and a system of inequalities:

EAE(p) = ∃ ∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …

… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1)

is rather straightforward. Indeed, the only change required is in the construction of the tree of stratified sets. In proving theorem 1.4, i.e., when the predicate defining the set EAE(p) is the conjunction of a system of equalities and a system of inequalities, we have constructed the tree of stratified sets iteratively, where in each step we have first collected all the formal solutions defined over closures of the resolutions of the remaining y’s from the previous step, and then applied the collections of formal solutions to the set of the remaining y’s from the previous step, and analyzed the set of y’s for which (at least) one of the inequalities from our given system is in fact an equality when we substitute the families of formal solutions we have collected, using an iterative procedure for the analysis of quotient resolutions.

When the set EAE(p) is defined using a (finite) disjunction of conjunctions of a system of equalities and a system of inequalities we work in parallel with each of the conjunctions in each step of the iterative procedure that constructs the tree of stratified sets. In each step of this iterative procedure, we do the following for the indices j, 1 ≤ j ≤ r, in parallel. We first collect all the formal solutions of the system of equalities Σ_{j}(x, y, w, p, a) defined over closures of the resolutions of the remaining y’s from the previous step, and then apply the collections of these formal solutions to the set of the remaining y’s from the previous step, and analyze the set of y’s for which (at least) one of the inequalities from the system Ψ_{j}(x, y, w, p, a) is in fact an equality when we substitute the families of formal solutions we have collected, using the iterative procedure for the analysis of quotient resolutions.

The termination of this modified procedure for the construction of the tree of stratified sets follows using the same argument used in proving theorem 1.4 (theorem 2.8). Given the tree of stratified sets, the analysis of the set EAE(p) is identical to the analysis described in proving theorem 1.4. This finally shows that the set EAE(p) is in the Boolean algebra of AE sets, also when EAE(p) is defined using a (finite) disjunction of conjunctions of a system of equalities and a system of inequalities, hence, concludes the proof of theorem 1.3 for general EAE sets.

Theorem 1.3. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let the AE set AE(w, p) be defined as:

AE(w, p) = ∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1).

Then the projection of the AE set AE(w, p), i.e., the set:

EAE(p) = ∃w ∀y ∃x (Σ_{1}(x, y, w, p, a) = 1 ∧ Ψ_{1}(x, y, w, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, w, p, a) = 1 ∧ Ψ_{r}(x, y, w, p, a) ≠ 1)

is in the Boolean algebra of AE sets.

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