DIOPHANTINE GEOMETRY OVER GROUPS IV:

AN ITERATIVE PROCEDURE FOR VALIDATION OF A SENTENCE

Z. Sela^{1,2}

posted Nov. 26, 2000

This paper is the fourth 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 fourth paper we present an iterative procedure that validates the correctness of an AE sentence defined over a free group. The terminating procedure presented in this paper is the basis for our analysis of elementary sets defined over a free group presented in the next paper in the sequence.

In the first 3 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 this paper we apply the structural results obtained in the first two papers in the sequence, to analyze AE sentences. As we have seen in the first section of the second paper, given a true sentence of the form:

∀y ∃x Σ(x, y, a) = 1 ∧ Ψ(x, y, a) ≠ 1

it is impossible to find a finite collection of formulas of the form x = x(y, a) that prove the validity of the sentence. To analyze an AE sentence we associate with it 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 are 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.

1Hebrew University, Jerusalem 91904, Israel.

2Partially supported by an Israel academy of sciences fellowship.

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.

Since the iterative procedure is rather involved, we prefered to present it first in two special cases, which are conceptually and technically simpler, but they already demonstrate some of the principals used in the general case. In the first section we present an iterative procedure in the minimal rank case, i.e., for sentences 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 analyze the ”opposite” case, the case of maximal rank homomorphisms. In both cases all maps in question are shown to be ”geometric”, which enables one to use ”purely” geometric concepts in analyzing the varieties and the resolutions constructed along the procedure. In the third section we construct tools needed for generalizing the iterative procedure. These involve resolutions inherited by a Diophantine set from a resolution of a variety. The fourth section finitely presents the iterative procedure for validation of a sentence in the general case.

§1. The Minimal Rank Case

Our ”trial and error” procedure for quantifier elimination is constructed iteratively. As we will see in the next paper the existence of formal solutions, graded formal limit groups, and graded configuration limit groups are the basic tools needed for constructing the procedure. This paper is devoted for the remaining tools needed for constructing our iterative procedure for quantifier elimination. In order to develop the needed tools and demonstrate our approach to quantifier elimination, we present a simplified version of the iterative procedure in this section.

Given a truth sentence of the form:

∀y ∃x Σ(x, y, a) = 1 ∧ Ψ(x, y, a) ≠ 1

the iterative procedure presented in this section constructs a finite sequence of limit groups (defined in terms of the coefficients a, the variables y, and some additional auxiliary variables), some well-structured resolutions of these limit groups, and for each well-structured resolution a set of formal solutions defined over a covering closure of the resolution. Using the resolutions, closures and formal solutions constructed by the procedure, we will be able to divide the domain of y’s into a finite collection of sets, so that on each set from this finite union, the validity of the sentence can be verified using a unique formula.

We start our warm-up procedure with the following rather straightforward proposition.

Proposition 1.1. Let F_{k} =< a_{1}, …, a_{k} > be a free group, and let:

∀y ∃x Σ(x, y, a) = 1 ∧ Ψ(x, y, a) ≠ 1

be a truth sentence defined over F_{k}. Let F_{y} =< y_{1}, …, y_{ℓ} > be the free group defined over the variables y_{1}, …, y_{ℓ}. Then there exists a formal solution x = x(y, a), and a finite set of restricted limit groups: Rlim_{1}(y, a), …, Rlim_{m}(y, a) for which:

(i) the words corresponding to the equations in the system Σ(x(y, a), y, a) represent the trivial word in the free group F_{k} ∗ F_{y}.

(ii) for every index i, Rlim_{i}(y, a) is a proper quotient of the free group F_{k} ∗ F_{y}.

(iii) let B_{1}(y), …, B_{m}(y) be the basic sets corresponding to the restricted limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a). If y ∉ B_{1}(y) ∪ … ∪ B_{m}(y), and ψ_{j}(x, y, a) is a word corresponding to one of the equations in the system Ψ(x, y, a), then: ψ_{j}(x(y, a), y, a) ≠ 1.

Proof: Let ψ_{1}(x, y, a) = 1, …, ψ_{r}(x, y, a) = 1 be the defining equations of the system Ψ(x, y, a) = 1. By theorem 1.2 of [Se2] there exists a formal solution x = x(y, a) for which condition (i) holds, and there exists some specialization y_{0} of the variables y, so that for every index j, 1 ≤ j ≤ r, ψ_{j}(x(y_{0}, a), y_{0}, a) ≠ 1. Let Rlim_{1}(y, a), …, Rlim_{m}(y, a) be the union of the maximal restricted limit groups corresponding to the equations: ψ_{j}(x(y, a), y, a) = 1, for 1 ≤ j ≤ r. Since there exists a specialization y_{0} for which for every index j, 1 ≤ j ≤ r, ψ_{j}(x(y_{0}, a), y_{0}, a) ≠ 1, each restricted limit group Rlim_{i}(y, a) is a proper quotient of the free group F_{k}∗F_{y}, and we get part (ii). Let B_{1}(y), …, B_{m}(y) be the basic sets corresponding to the maximal restricted limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a). By the construction of these maximal restricted limit groups, if y ∉ B_{1}(y) ∪ … ∪ B_{m}(y) then ψ_{j}(x(y, a), y, a) ≠ 1 for every 1 ≤ j ≤ r, so we get part (iii) of the proposition.

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Proposition 1.1 gives a formal solution that proves the validity of the given sentence on a co-basic set (F_{k})^{ℓ} \ (B_{1}(y)∪…∪B_{m}(y)), hence, the rest of the procedure needs to construct formal solutions that prove the validity of the predicate on the remaining basic sets B_{1}(y), …, B_{m}(y). Since our treatment of the restricted limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a) which define the basic sets B_{1}(y), …, B_{m}(y) is independent, we will continue with one of these limit groups, which for brevity we denote Rlim(y, a). Note that Rlim(y, a) is a proper quotient of the free group F_{k} ∗ F_{y}.

Let Rlim(y, a) be a restricted limit group. By proposition 1.10 of [Se2] with Rlim(y, a) one can associate its strict Makanin-Razborov diagram. Let Res_{1}(y, a), …, Res_{s}(y, a) be the resolutions in this strict diagram. Each resolution in the diagram is a strict Makanin-Razborov resolution of either the restricted limit group Rlim(y, a) or of a proper quotient of it, and every specialization of the restricted limit group Rlim(y, a) factors through one of the resolutions Res_{1}, (y, a), …, Res_{s}(y, a). Hence, with the strict Makanin-Razborov diagram of Rlim(y, a) we can associate a tuple of couples (Rlim_{i}(y, a), Res_{i}(y, a)), where Rlim_{i}(y, a) is either Rlim(y, a) or a proper quotient of it, and Res_{i}(y, a) is a strict Makanin-Razborov resolution of Rlim_{i}(y, a). Since every strict Makanin-Razborov resolution is, in particular, a well structured resolution (definition 1.11 in [Se2]), with each tuple (Rlim_{i}(y, a), Res_{i}(y, a)) we can associate its completed resolution Comp(Res)_{i}(z, y, a) and a completed limit group Comp(Rlim)_{i}(z, y, a). By lemma 1.13 of [Se2], every specialization y_{0} that factors through the resolution Res_{i}(y, a), factors through the completed resolution Comp(Res)_{i}(z, y, a) and through the completed limit group Comp(Rlim)_{i}(z, y, a). Therefore, if D_{i}(y) denotes the Diophantine set corresponding to the completed limit group Comp(Rlim)_{i}(z, y, a), and B(y) is the basic set corresponding to the original limit group Rlim(y, a), then: B(y) = D_{1}(y) ∪ … ∪ D_{s}(y). Since the continuation of the procedure is conducted independently for the tuples (Rlim_{i}(y, a), Res_{i}(y, a)), we will continue with one of them, which we denote (Rlim(y, a), Res(y, a)) for brevity, and we denote its completed resolution by Comp(Res)(y, a) and its completed limit group by Comp(Rlim)(y, a). Before we look for formal solutions defined over closures of the resolution Res(y, a), we need to define specializations of a completed limit group in shortest form.

Definition 1.2. Let Rlim(y, a) be a restricted limit group, let Res(y, a) be a wellstructured resolution of Rlim(y, a), let Comp(Res)(z, y, a) be its completed resolution, and let Comp(Rlim)(z, y, a) be its completed limit group. Given a specialization (z, y, a) that factors through the completed resolution Comp(Res)(z, y, a) we define the length of z, length(z), to be the ordered tuple consisting of lengths of the elements z_{i}, where z_{i} are the variables that appear in the i-th level of the completed resolution Comp(Res)(z, y, a), ordered from top to bottom. We say that a specialization (z_{0}, y_{0}, a) of the completed limit group Comp(Rlim)(z, y, a) is in shortest form, if given the specialization (y_{0}, a) of the restricted limit group Rlim(y, a) that factors through the resolution Res(y, a), z_{0} has the shortest possible length in the lexicographical order defined on the lengths of all the specializations (z, y_{0}, a) that factor through the completed resolution Comp(Res)(z, y, a).

Clearly, for any specialization (y_{0}, a) of the restricted limit group Rlim(y, a) that factors through the resolution Res(y, a), there exists a specialization (z_{0}, y_{0}, a) of the completed limit group Comp(Rlim)(z, y, a) that factors through the completed resolution Comp(Res)(z, y, a) which is in shortest form. Hence, when we consider specializations of completed resolutions corresponding to specializations of a given restricted limit group, we may restrict ourselves to specializations in shortest form.

Proposition 1.3. With the assumptions of proposition 1.1 and the notation above, there exists a covering closure Cl(Res)_{1}(s, z, y, a), …, Cl(Res)_{q}(s, z, y, a) of the resolution Res(y, a), and for each index n, 1 ≤ n ≤ q there exists a formal solution x_{n}(s, z, y, a), and a finite set of restricted limit groups QRlim_{1}^{n}(s, z, y, a), …, QRlim_{m(n)}^{n}(s, z, y, a), for which:

(i) for each index n, 1 ≤ n ≤ q, the words corresponding to the equations in the system Σ(x_{n}(s, z, y, a), y, a) represent the trivial word in the closure Cl(Res)_{n}(s, z, y, a).

(ii) for every index n, the restricted limit groups QRlim_{1}^{n}(s, z, y, a), …, QRlim_{m(n)}^{n}(s, z, y, a)

are proper quotients of the n-th closure Cl(Res)_{n}(s, z, y, a). (iii) for every index n, 1 ≤ n ≤ q, let D^{n}(y) be the Diophantine set corresponding to the closure Cl(Res)_{n}(s, z, y, a), and let D_{1}^{n}(y), …, D_{m(n)}^{n}(y) be the Diophantine sets corresponding to the restricted limit groups QRlim_{1}^{n}(s, z, y, a), …, QRlim_{m(n)}^{n}(s, z, y, a). Let ψ_{j}(x, y, a) be a word corresponding to one of the equations in the system Ψ(x, y, a). If for some index n, 1 ≤ n ≤ q, y_{0} ∈ D^{n}(y) and there exists some specialization (s_{0}, z_{0}, y_{0}, a) that factors through the closure Cl(Res)_{n}(s, z, y, a), and y_{0} ∉ D_{1}^{n}(y) ∪ … ∪ D_{m(n)}^{n}(y), then there exists a shortest form specialization (s_{1}, z_{1}, y_{0}, a) that factors through the closure Cl(Res)_{n}(s_{1}, z_{1}, y_{0}, a) so that for every index j, 1 ≤ j ≤ r: ψ_{j}(x_{n}(s_{1}, z_{1}, y_{0}, a), y_{0}, a) ≠ 1.

Proof: Let ψ_{1}(x, y, a) = 1, …, ψ_{r}(x, y, a) = 1 be the defining equations of the system Ψ(x, y, a) = 1. By theorem 1.17 of [Se2], there exists a covering closure Cl(Res)_{1}(s, z, y, a), …, Cl(Res)_{q}(s, z, y, a) of the resolution Res(z, y, a), and formal solutions x_{1}(s, z, y, a), …, x_{q}(s, z, y, a) for which (i) holds, and for each index n, 1 ≤ n ≤ q, there exists some specialization (s_{0}^{n}, z_{0}^{n}, y_{0}^{n}, a) of the variables (s, z, y, a) that factors through the n-th closure Cl(Res)_{n}(s, z, y, a), so that for every index j, 1 ≤ j ≤ r, ψ_{j}(x_{n}(s_{0}^{n}, z_{0}^{n}, y_{0}^{n}, a), y_{0}^{n}, a) ≠ 1.

For each index n, 1 ≤ n ≤ q, let QRlim_{1}^{n}(s, z, y, a), …, QRlim_{m(n)}^{n}(s, z, y, a) be the maximal restricted limit groups, which are the maximal quotients of the n-th closure Cl(Res)_{n}(s, z, y, a) containing all shortest form specializations of the closure Cl(Res)(s, z, y, a) that satisfy ψ_{j}(x_{n}(s, z, y, a), y, a) = 1 for some index j, 1 ≤ j ≤ r. Since for any specialization (y_{0}, a) that factors through the resolution Res(y, a) there exists a shortest form specialization (z_{0}, y_{0}, a) that factors through the completed resolution Comp(Res)(z, y, a), part (iii) of the proposition holds for the entire collection of restricted limit groups QRlim_{1}^{n}(s, z, y, a), …, QRlim_{m(n)}^{n}(s, z, y, a) for all possible n, 1 ≤ n ≤ q. Since for every n, 1 ≤ n ≤ q, there exists some specialization (s_{0}^{n}, z_{0}^{n}, y_{0}^{n}, a) that factors through the n-th closure Cl(Res)_{n}(s, z, y, a), for which for every index j, 1 ≤ j ≤ r, ψ_{j}(x_{n}(s_{0}^{n}, z_{0}^{n}, y_{0}^{n}, a), y_{0}^{n}, a) ≠ 1, QRlim_{i}^{n}(s, z, y, a) is a proper quotient of the n-th closure Cl(Rlim)(s, z, y, a) for every possible n, 1 ≤ n ≤ q, and i, 1 ≤ i ≤ m(n).

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The formal solutions constructed by propositions 1.1 and 1.3 prove the validity of the given sentence on the co-Diophantine set (F_{k})^{ℓ} \ (D_{1}^{1}(y)∪…∪D_{m(q)}^{q}(y)), hence, the rest of the procedure needs to construct formal solutions that prove the validity of the predicate on the remaining Diophantine sets D_{1}^{1}(y), …, D_{m(q)}^{q}(y). Note that for each index n, 1 ≤ n ≤ q, the Diophantine sets D_{1}^{n}(y), …, D_{m(n)}^{n}(y) are defined using the tuple of variables (s, z, y, a) of the closure Cl(Res)_{n}(s, z, y, a) of the strict Makanin-Razborov resolution Res(y, a) of the restricted limit group Rlim(y, a). This appearance of additional variables emerging from closures of resolutions, is a bi-product of the use of formal solutions, is unavoidable due to theorem 1.17 of [Se2], and is going to be an ”obstacle” throughout our iterative procedure. Naturally, these additional variables make the iterative procedure more technically involved.

Because of the technical complications, we prefer to present the iterative procedure assuming that all the restricted limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a) obtained in the initial step of the procedure (proposition 1.1) are of minimal rank. i.e., we will assume that for every restricted homomorphism from Rlim_{i}(y, a) onto a free group F_{k} ∗ F, F must be the trivial group. This assumption simplifies the technicalities considerably, and allows better demonstration of the geometric considerations that will eventually guarantee the termination of the procedure. In the next sections of this paper, we gradually modify the procedure to the general case. Since the continuation of the procedure is conducted independently for each tuple (Rlim_{i}(y, a), Res_{i}(y, a)), and a proper quotient QRlim_{t}^{n}(s, z, y, a) of some closure Cl(Res)_{n}(s, z, y, a) of the resolution Res_{i}(y, a), we will continue with one tuple which we denote (Rlim(y, a), Res(y, a)), and a proper quotient QRlim(s, z, y, a) of a closure Cl(Res)(s, z, y, a) of the resolution Res(y, a), and omit the relevant indices.

The iterative procedure we present produces a sequence of well-structured resolutions and their completions. We will associate a complexity with each of the resolutions produced by the procedure, show that the complexity decreases with each step of the procedure, a decrease that eventually forces the iterative procedure to terminate. As already pointed out, for presentation purposes we will start by restricting ourselves to the minimal rank case. We start describing the procedure by presenting the first step of the procedure, and define the resolutions it produces and their complexity. Next, we present the general step of the procedure, and prove that the complexity is strictly decreasing. By the way we define the complexity, the termination of the procedure is a straightforward consequence of the decrease in complexity.

The first step of the iterative procedure

Let z be the variables defining the completed resolution Comp(Res)(z, y, a) and let z_{base} be the union of the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, a) except the top level one. We will call this union of variables the basis of the completed resolution Comp(Res)(z, y, a). Following the construction of the strict Makanin-Razborov diagram ([Se2],1.10), we use the entire collection of shortest form specializations that factor through the quotient limit group QRlim(s, z, y, a) to construct the corresponding (canonical) strict graded Makanin-Razborov diagram of the limit group QRlim(s, z, y, a), viewed as a graded limit group with respect to the parameters z_{base}. Let GRes_{1}(s, z, y, z_{base}, a), …, GRes_{m}(s, z, y, z_{base}, a) be the strict graded Makanin-Razborov resolutions that appear in the strict graded Makanin-Razborov diagram of the (graded) limit group QRlim(s, z, y, z_{base}, a) with respect to the parameters z_{base}, where each graded resolution is terminating in either a rigid or a solid graded limit group (with respect to the defining parameters z_{base}). Note that by definition, a homomorphism factors through a graded resolution if it can be expressed as a composition of graded modular automorphisms of the different levels, the canonical epimorphisms between consecutive levels of the graded resolution, and a rigid (solid) homomorphism from the terminal rigid (solid) graded limit group of the resolution to the free group F_{k}.

We will treat the graded resolutions GRes_{i}(s, z, y, z_{base}, a) in parallel, so for the continuation we will restrict ourselves to one of them which we denote GRes(s, z, y, z_{base}, a) for brevity. Let GRlim(s, z, y, z_{base}, a) be the graded limit group corresponding to the graded resolution GRes(s, z, y, z_{base}, a). Let Glim_{j}(s, z, y, z_{base}, a) be a graded limit group that appears in the j-th level of the graded resolution GRes(s, z, y, z_{base}, a). Naturally, there exists a canonical map τ_{j} : Rlim(y, a) → Glim_{j}(s, z, y, z_{base}, a). Let Λ_{j} be the graded quadratic decomposition of Glim_{j}(s, z, y, z_{base}, a), i.e., the graded cyclic decomposition of Glim_{j}(s, z, y, z_{base}, a) obtained from the principal graded JSJ decomposition of Glim_{j}(s, z, y, z_{base}, a) by collapsing all the edges connecting two non-QH subgroups. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Rlim(y, 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 Glim_{j}(s, z, y, z_{base}, 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 1.4. With the notation above:

(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: Part (i) follows from the construction of the decomposition Γ_{j}(Q) and the associated collection of s.c.c. Γ_{j}(S). If a non-separating s.c.c. on the surface S is mapped to the trivial element by τ_{j}, then the limit group Rlim(y, a) admits an epimorphism onto the free group F_{k} ∗ Z, where Z is the infinite cyclic group, a contradiction to Rlim(y, a) being of minimal rank, and we get part (ii).

Let S_{1} be the connected subsurface of S \Γ_{j}(S) that is mapped non-trivially into Q′ by the homomorphism τ_{j}. Let Q_{1} be the fundamental group of S_{1}. By part (i), the homomorphism τ_{j} maps the boundary components of S_{1} to either the trivial element or to non-trivial elliptic elements in Λ_{j}. By part (ii) no s.c.c. on S_{1} which is a non-separating s.c.c. in S is mapped to the trivial element by the homomorphism τ_{j}. Clearly, genus(S_{1}) ≤ genus(S) and |χ(S_{1})| ≤ |χ(S)|.

If all the boundary components of the subsurface S_{1} are mapped to the trivial element by the homomorphism τ_{j}, and S_{1} itself is mapped non-trivially into Glim_{j}(s, z, y, z_{base}, a), then Rlim(y, a) admits an epimorphism onto a free group F_{k} ∗ F for some non-trivial free group F, a contradiction to Rlim(y, a) being of minimal rank. Hence, some of the boundary components of S_{1} are mapped by τ_{j} to non-trivial elliptic elements in the cyclic decomposition Λ_{j} of Glim_{j}(s, z, y, z_{base}, a). Let S_{2} be the punctured surface obtained from the punctured surface S_{1} by adjoing a disks along each of the boundary components of S_{1} that is mapped to the trivial element by τ_{j}. Let Q_{2} be the fundamental group of S_{2}. By the maximality of the collection Γ_{j}(S), no s.c.c. on the surface S_{2} is mapped to the trivial element in τ_{j}, and by the way the punctured surface S_{2} was constructed: genus(S_{2}) ≤ genus(S_{1}) ≤ genus(S) and |χ(S_{2})| ≤ |χ(S_{1})| ≤ |χ(S)|. To complete the proof of part (iii) of lemma 1.4, we use the following basic fact which is used often in the next sections.

lemma 1.5. Let M be a punctured surface with χ(M) ≤ −2, and let B be a f.g. subgroup of infinite index in π_{1}(M). Then either:

(i) B contains no conjugate of a boundary element in M.

(ii) B = < c_{1} > ∗ … ∗ < c_{ℓ} > ∗ B′. Each of the elements c_{i} is conjugate to some boundary element in M, and every element b ∈ B which is conjugate to a boundary element in π_{1}(M), is conjugate to one of the elements c_{j}.

Proof: Let G = π_{1}(M), and let φ ∈ Aut(G) be a pseudo-Anosov automorphism. By the work of Thurston (cf. [Ca-Bl]), the sequence of actions of G on its Cayley graph via the automorphisms {φ^{n}} converges into a stable action of G on a real tree Y. This action has one orbit of an IET component, and finitely many orbits of point stabilizers, where each orbit of a point stabilizer corresponds to a boundary component of the punctured surface S (cf. ([Se3],3)).

B < G and B is f.p. so we may apply the classification of stable actions of groups on real trees [Be-Fe], to analyze the action of the subgroup B on the real tree Y. The action of B on Y contains no axial component, and the stabilizer of any segment in the discrete part of the action is trivial. Any point stabilizer in the action of B on Y, is a subgroup of a point stabilizer in the action of Q on Y, so every point stabilizer in the action of B on Y can be conjugated into a boundary component of the punctures surface S. Furthermore, every element b ∈ B that can be conjugated into a boundary component of M, fixes a point in the real tree Y. Suppose that the action of B on the real tree Y contains an IET component. Let B_{1} < B be the subgroup corresponds to such an IET component, and let M_{1} be the punctured surface corresponding to B_{1}. We look at the fundamental group of the double of the surface M, i.e., the closed surface obtained by gluing two copies of M along their corresponding boundaries. The fundamental group of the double of the punctured surface M_{1} is a subgroup of the fundamental group of the double of the punctured surface M. Since they are both closed surfaces, the double of M_{1} finitely covers the double of M, so B_{1} is a subgroup of finite index in G = π_{1}(M), but B_{1} < B, and B is of infinite index in B, a contradiction. Hence, the action of B on the real tree Y contains no IET components. Therefore, by the classification of stable actions of groups on real trees [Be-Fe], admits a resolution of the action of B on Y contains only discrete parts, in which every non-degenerate segment has trivial stabilizer, orbits of Levitt (=thin) components, and orbits of point stabilizers, where each point stabilizer is cyclic and can be conjugated into a boundary component of the punctured surface S. Therefore, by [Be-Fe] the graph of groups corresponding to such a resolution of the action of B on Y is a free product of B of the form: B = < c_{1} > ∗ … ∗ < c_{ℓ} > ∗ F_{1} ∗ … ∗ F_{m} where the F_{i}’s are free groups, each of the cyclic subgroups < c_{i} > corresponds to an orbit of a point stabilizer in the action of B on the real tree Y, so every cyclic subgroup < c_{i} > can be conjugated into a boundary component of the punctured surface S, and every element in B which can be conjugated into a boundary component of M can be conjugated into one of the cyclic subgroups < c_{i} >, and we get part (ii) of the lemma.

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To complete the proof of lemma 1.4, note that if τ_{j}(π_{1}(S_{2})) = τ_{j}(Q_{2}) is of infinite index in the QH subgroup Q′, then lemma 1.5 implies that Rlim(y, a) admits an epimorphism onto a free group F_{k} ∗ F for some non-trivial free group F, a contradiction to Rlim(y, a) being of minimal rank. Hence, τ_{j}(Q_{2}) is of finite index in the QH subgroup Q′. Hence, genus(S′) ≤ genus(S_{2}) ≤ genus(S). Now, since τ_{j}(Q_{2}) is a finite index subgroup of Q′, the double of the punctured surface S_{2} along its boundary components is mapped non-trivially onto a finite cover of the double of the punctured surface S′. Hence: |χ(S′)| ≤ |χ(S_{2})| ≤ |χ(S)|, and we get part (iii) of the lemma.

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Part (iii) of lemma 1.4 bounds the topological complexity of those QH subgroups Q′ that appear in the principal graded abelian JSJ decompositions associated with the various levels of the graded resolution GRes(s, z, y, z_{base}, a) into which a QH subgroup Q that appears in the cyclic JSJ decomposition of Rlim(y, a) is mapped non-trivially. To show that lemma 1.4 can be applied to bound the topological complexity of all the QH subgroups Q′ that appear in the principal graded abelian decompositions associated with the various levels of GRes(s, z, y, z_{base}, a) we need the following proposition.

Proposition 1.6. Let Glim_{j}(s, z, y, z_{base}, a) be the graded limit group that appears in the j-th level of the graded resolution GRes(s, z, y, z_{base}, a), and let Q′ be a QHsubgroup that appears in the principal graded JSJ decomposition associated with Glim_{j}(s, z, y, z_{base}, a) and S′ be its corresponding punctured surface. Let τ_{j} be the natural map: τ_{j} : Rlim(y, a) → Glim_{j}(s, z, y, z_{base}, a). Then there exists a QH subgroup Q in the cyclic JSJ decomposition of Rlim(y, 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: Let Λ_{Q′} be the cyclic decomposition obtained from the principal graded JSJ decomposition of Glim_{j}(s, z, y, z_{base}, a) by collapsing all the edges that are not connected to the QH subgroup Q′. The cyclic decomposition Λ_{Q′} contains a (distinguished) vertex connected to Q′ and stabilized by a vertex group containing the subgroup < z_{base}, a >, and perhaps few other vertices connected to disjoint boundary components of the QH subgroup Q′. For each QH subgroup Q in the cyclic JSJ decomposition of Rlim(y, a), let G_{Q} be the subgroup of Comp(Rlim)(z, y, a) which is the fundamental group of the subgraph of groups containing two vertices, the vertex stabilized by the subgroup < z_{base}, a > and the vertex stabilized by the subgroup Q. There is a natural map ζ : Comp(Rlim)(z, y, a) → Glim_{j}(s, z, y, z_{base}, a). For each QH subgroup Q in the cyclic JSJ decomposition of Rlim(y, a), the image group ζ(G_{Q}) inherits a decomposition Δ_{Q} from the cyclic decomposition Λ_{Q′} of Glim_{j}(s, z, y, z_{base}, a).

Let S be the punctured surface corresponding to Q, and suppose that no subsurface of S is mapped non-trivially into a conjugate of Q. Since the subgroup ζ(< z_{base}, a >) is contained in a vertex group in Δ_{Q}, and the fundamental group of any subsurface of S is not mapped non-trivially into Q, Δ_{Q} is either a trivial graph of groups, or it gives rise to a free decomposition of G_{Q} in which the subgroup ζ(< z_{base}, a >) is contained in one of the factors. If the decomposition Δ_{Q} gives rise to a free decomposition of the group G_{Q} in which the subgroup ζ(< z_{base}, a >) is contained in one of the factors, then the restricted limit group Rlim(y, a) can be mapped onto a limit group H_{Q} which obtains a free product in which the subgroup F_{k} < H_{Q} is contained in one of the factors. Hence, in case Δ_{Q} gives rise to a free product of G_{Q}, the restricted limit group Rlim(y, a) admits an epimorphism onto a free group F_{k} ∗ F for some non-trivial free group F, in contradiction to Rlim(y, a) being of minimal rank. Therefore, for every QH subgroup Q in the JSJ decomposition of Rlim(y, a), the decomposition Δ_{Q} is the trivial decomposition of G_{Q}, so the homomorphism τ_{j} : Rlim(y, a) → Glim_{j}(s, z, y, z_{base}, a) maps the restricted limit group Rlim(y, a) into the vertex group stabilized by the subgroup < z_{base}, a > in the cyclic decomposition Λ_{Q′}. Since the quotient limit group QRlim(s, z, y, a) was obtained as the limit of a sequence of shortest form specializations {(s_{n}, z_{n}, y_{n}, a)}, if Rlim(y, a) is mapped to the vertex stabilized by < z_{base}, a > in Λ_{Q′}, the completed limit group Comp(Rlim)(z, y, a) is mapped to the vertex group stabilized by < z_{base}, a >, so the entire graded limit group Glim_{j}(s, z, y, z_{base}, a) stabilizes that vertex, and the decomposition Λ_{Q′} is trivial, which contradicts the way is was constructed. Therefore, there exists some QH subgroup Q in the cyclic JSJ decomposition of Rlim(y, a) with corresponding punctured surface S, so that a subsurface S_{1} of the punctured surface S is mapped non-trivially by τ_{j} into some conjugate of Q′. By lemma 1.4 the fundamental group of S_{1} is mapped into a finite index subgroup of a conjugate of Q′. In particular: genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|.

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Definition 1.7. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Rlim(y, 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}, 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 JSJ decomposition of Rlim(y, a), then every QH subgroup Q′ in any level of the graded resolution GRes(s, z, y, z_{base}, 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 GRes(s, z, y, z_{base}, a) to be smaller than that of the resolution Res(y, a) once one is able to ”isolate” the surviving surfaces. This is the purpose of the following theorem.

Theorem 1.8. Let Q_{1}, …, Q_{r} be the surviving QH subgroups in the JSJ decomposition of Rlim(y, a). Then the graded resolution GRes(s, z, y, z_{base}, 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, a) with respect to the parameter subgroup < z_{base}, Q_{1}, …, Q_{r} >, which we denote GRes(s, z, y, (z_{base}, Q_{1}, …, Q_{r}), a). The second part is a one step resolution that maps the rigid (solid) terminal graded limit group of GRes(s, z, y, (z_{base}, Q_{1}, …, Q_{r}), a) to the rigid (solid) terminal graded limit group of the resolution GRes(s, z, y, z_{base}, 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 GRes(s, z, y, z_{base}, 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 GRes(s, z, y, (z_{base}, 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}, 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}, Q_{1}, …, Q_{r} > in the first part, and by < z_{base} > in the second part.

Proof: Let Q_{1}, …, Q_{r} be the surviving surfaces in the graded JSJ decomposition of Rlim(y, a). By definition, for each surviving surface Q_{i} there exists a QH subgroup Q′_{i} that appears in one of the principal graded JSJ decompositions associated with the various levels of the graded resolution GRes(s, z, y, z_{base}, a), so that Q_{i} is mapped isomorphically onto Q′_{i}. For each Q_{i} we pick Q′_{i} to be the QH subgroup that appears in the highest possible level principal graded abelian decomposition so that Q_{i} is mapped isomorphically onto Q′_{i}. Up to a change of order, we may assume that the QH subgroups Q′_{1}, …, Q′_{r} are ordered according to the level of their appearance in the graded resolution GRes(s, z, y, z_{base}, a), from top to bottom. Note that it may be that some of the QH subgroups Q′_{1}, …, Q′_{r} are conjugate in the corresponding graded limit groups.

To replace the graded resolution GRes(s, z, y, z_{base}, a) with a graded resolution that have all its surviving surfaces in the bottom level we do the following. If a QH subgroup Q′ appears in a principal graded decomposition associated with a level of the graded resolution GRes(s, z, y, z_{base}, a) that lies above the level in which Q′_{i} appears, and the fundamental group of a subsurface of (the surviving surface) S_{i}, the punctured surface corresponding to the surviving QH subgroup Q_{i}, is mapped non-trivially into Q′_{i}, then we change the graded decomposition in which the QH subgroup appears by collapsing the vertex stabilized by Q′ into the (distinguished) vertex stabilized by < z_{base}, a >. Other than that, we leave the levels from the one that contains the QH subgroup Q′_{1} and above unchanged. Let Q′_{1}, …, Q′_{i_{m_{1}}} be the representatives of the conjugacy classes of those QH subgroups Q′_{1}, …, Q′_{r} that appear in the principal graded JSJ decomposition associated with the same level in which Q′_{1} appears. We continue the modified graded resolution by mapping each of the QH subgroups Q′_{i} from the set Q′_{1}, …, Q′_{i_{m_{1}}} to the image of the corresponding surviving surface Q_{i} in the subgroup < z_{base}, a > under the quotient map of the completed resolution Comp(Res)(z, y, a).

We continue the modification in the same way, going through the levels of the graded resolution GRes(s, z, y, z_{base}, a). If a QH subgroup Q′ appears in a principal graded decomposition associated with a level of the graded resolution GRes(s, z, y, z_{base}, a) that lies below the level in which Q′_{i} appears, where Q′_{i} is associated with one of the QH subgroups Q_{1}, …, Q_{i_{m_{1}}} , and the fundamental group of a subsurface of (the surviving surface) S_{i}, the punctured surface corresponding to the surviving QH subgroup Q_{i}, is mapped non-trivially into Q′_{i}, then we change the graded decomposition in which the QH subgroup appears by collapsing the vertex stabilized by Q′ into the (distinguished) vertex stabilized by < z_{base}, a >, and replacing the vertex stabilized by < z_{base}, a > by a maximal limit group obtained as a limit of specializations of the vertex group containing < z_{base}, a > and the QH subgroup Q′_{i}, specializations in which the QH subgroup Q_{i} is identified with its image in the subgroup < z_{base}, a >. Let Q′_{i_{m_{1}+1}}, …, Q′_{i_{m_{2}}} be the representatives of the conjugacy classes of those QH subgroups Q′_{1}, …, Q′_{r} that appear in the principal graded JSJ decomposition associated with the level below the one in which Q′_{1} appears. We continue the modified graded resolution by mapping each of the QH subgroups Q′_{i} from the set Q′_{i_{m_{1}+1}}, …, Q′_{i_{m_{2}}} to the image of the corresponding surviving surface Q_{i} in the subgroup < z_{base}, a > under the quotient map of the completed resolution Comp(Res)(z, y, a), and continue in the same way through the next levels.

Since according to ([Se1],5) there are only finitely many maximal limit groups corresponding to a given set of specializations, the modification described above gives rise to finitely many modified resolutions. Every shortest form specialization (s_{0}, z_{0}, y_{0}, a) with respect to the closure Cl(Res)(s, z, y, a) that factors through the graded resolution GRes(s, z, y, z_{base}, a) factors through at least one of the modified resolutions constructed by the above procedure. Let Q′_{1}, …, Q′_{r′} be representatives for the conjugacy classes of the QH subgroups Q′_{1}, …, Q′_{r}. Note that each of the surviving subgroups Q_{1}, …, Q_{r} is mapped to a conjugate of one of the subgroups Q′_{1}, …, Q′_{r′}, and in each other level it is mapped to the vertex group containing the subgroup < z_{base}, a >. Therefore, in each of the modified graded resolutions obtained by the above procedure, we can push all the QH subgroups Q′_{1}, …, Q′_{r′} to the bottom level, and the obtained (finitely many) graded resolutions satisfy the properties listed in the statement of the theorem.

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We continue the analysis of the limit group QRlim(s, z, y, a) by reducing the set of defining parameters sequentially. Recall that in order to obtain the graded resolution GRes(s, z, y, z_{base}, a) we used the variables z_{base} as parameters, where z_{base} were the subset of the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, 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 that appear in all levels of the completed resolution Comp(Res)(z, y, a) except for the two highest levels. Let T_{1}(s, z, y, a) be the terminal rigid or solid graded limit group in the graded resolution GRes(s, z, y, z_{base}, a) with respect to the parameters z_{base}. From the collection of rigid (solid) specializations of T_{1}(s, z, y, z_{base}, a), that are obtained (using our shortening procedure) from shortest form specializations that factor through the graded resolution GRes(s, z, y, z_{base}, a), we construct the graded strict Makanin-Razborov diagram of T_{1}(s, z, y, a), viewed as a graded limit group with respect to the parameters z_{base}^{2}. Let GRes_{1}(s, z, y, z_{base}^{2}, a), …, GRes_{m}(s, z, y, z_{base}^{2}, a) be the strict graded Makanin-Razborov resolutions that appear in the strict graded Makanin-Razborov diagram of the (graded) limit group T_{1}(s, z, y, z_{base}^{2}, a) with respect to the parameters z_{base}^{2}, where each graded resolution is terminating in either a rigid or a solid graded limit group (with respect to the defining parameters z_{basis}^{2}).

We will treat the graded resolutions GRes_{i}(s, z, y, z_{base}^{2}, a) in parallel, so for the continuation we will restrict ourselves to one of them which we denote GRes(s, z, y, z_{base}^{2}, a) for brevity. Let Glim(s, z, y, z_{base}^{2}, a) be the graded limit group corresponding to the graded resolution GRes(s, z, y, z_{base}^{2}, a). If the subgroup Glim(s, z, y, z_{base}^{2}, a) is a proper quotient of the subgroup T_{1}(s, z, y, a) we need to modify the graded resolution GRes(s, z, y, z_{base}, a) so that it becomes a strict graded Makanin-Razborov resolution terminating with the limit group Glim(s, z, y, z_{base}^{2}, a). Note that by modifying the graded resolution GRes(s, z, y, z_{base}, a), we may need to replace the quotient limit group QRlim(s, z, y, a) or one of the groups Glim_{j}(s, z, y, z_{base}, a) that appear in the graded resolution GRes(s, z, y, z_{base}, a) by a proper quotient of itself.

Let Glim_{j}(s, z, y, z_{base}^{2}, a) be a graded limit group that appears in the j-th level of the graded resolution GRes(s, z, y, z_{base}^{2}, a). Let Zlim(z_{base}, a) be the limit group generated by < z_{base}, a > in the completed limit group Comp(Rlim)(z, y, a). Naturally, there exists a canonical map τ_{j} : Zlim(z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, a). Let Λ_{j} be the graded quadratic decomposition of Glim_{j}(s, z, y, z_{base}^{2}, a), i.e., the graded cyclic decomposition of Glim_{j}(s, z, y, z_{base}^{2}, a) obtained from the principal graded JSJ decomposition of Glim_{j}(s, z, y, z_{base}^{2}, a) by collapsing all the edges connecting two non-QH subgroups. Let Q be a quadratically hanging subgroup in the principal graded JSJ decomposition of Zlim(z_{base}, 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 Glim_{j}(s, z, y, z_{base}^{2}, 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 1.9. With the notation above:

(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.

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The precise statement of proposition 1.6 is not valid for the graded resolution GRes(s, z, y, z_{base}^{2}, a). If Q′ is a QH subgroup that appears in an abelian decomposition associated with one of the levels of the graded resolution GRes(s, z, y, z_{base}^{2}, a), then it is not true that there exists some QH subgroup Q in the principal graded JSJ decomposition of Zlim(z_{base}, 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 1.10. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Zlim(z_{base}, 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}, 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 GRes(s, z, y, z_{base}^{2}, a) we need to ”isolate” the surviving QH subgroups. This can be done in a similar way to theorem 1.8.

Theorem 1.11. Let Q_{1}, …, Q_{r} be the surviving QH subgroups in the JSJ decomposition of Zlim(z_{base}, a). Then the graded resolution GRes(s, z, y, z_{base}^{2}, 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 parameters < z_{base}^{2}, Q_{1}, …, Q_{r} >, which we denote GRes(s, z, y, (z_{base}^{2}, Q_{1}, …, Q_{r}), a). The second part is a one step resolution that maps the rigid (solid) terminal graded limit group of GRes(s, z, y, (z_{base}^{2}, Q_{1}, …, Q_{r}), a) to the rigid (solid) terminal graded limit group of the resolution GRes(s, z, y, z_{base}^{2}, 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 GRes(s, z, y, z_{base}^{2}, 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 GRes(s, z, y, (z_{base}^{2}, 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}, 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}, Q_{1}, …, Q_{r} > in the first part, and by z_{base}^{2} in the second part.

Proof: identical to the proof of theorem 1.8.

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Given lemma 1.9 and theorem 1.11, to complete the analysis of the structure of the graded resolution GRes(s, z, y, z_{base}^{2}, a) we still need an appropriate analogue of proposition 1.6, i.e., we need to associate every QH subgroup that appears in one of the principal graded JSJ decompositions of the different levels of the graded resolution GRes(s, z, y, z_{base}^{2}, a) with a QH subgroup of either the graded JSJ decomposition of Zlim(z_{base}, a) or a QH subgroup that appears in the principal graded JSJ decomposition of the terminal subgroup T_{1}(s, z, y, z_{base}, a) of the graded resolution GRes(s, z, y, z_{base}, a) in case T_{1}(s, z, y, z_{base}, a) is solid. We divide the final analysis of GRes(s, z, y, z_{base}^{2}, a) into two cases depending on T_{1}(s, z, y, a) being rigid or solid.

Proposition 1.12. With the notation above, suppose that the terminal subgroup T_{1}(s, z, y, z_{base}, a) of the graded resolution GRes(s, z, y, z_{base}, a) is rigid. Let Q′ be a QH subgroup that appears in the j-th level principal graded JSJ decomposition of the graded resolution GRes(s, z, y, z_{base}^{2}, a), and let S′ be its corresponding surface. Then:

(i) there exists a QH subgroup Q in the JSJ decomposition of Zlim(z_{base}, a), with an associated surface S, so that the map τ_{j} : Zlim(z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, 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}, a), and Q is mapped to the vertex stabilized by < z_{base}^{2}, a > by the maps τ_{j} : Zlim(z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, a) in all the levels above the bottom one.

Proof: If under the map τ_{j} : Zlim(z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, a), non of the QH subgroups Q in the principal JSJ decomposition of Zlim(z_{base}, a) is mapped non-trivially into (a conjugate of) Q′, then the entire image of Zlim(z_{base}, a), τ_{j}(Zlim(z_{base}, a)), is contained in a proper subgraph of groups of the principal graded JSJ decomposition of Glim_{j}(s, z, y, z_{base}^{2}, a), a subgraph that does not contain the vertex stabilized by the QH subgroup Q′. Hence, if no QH subgroup Q is mapped non-trivially by τ_{j} into the QH subgroup Q′, the graded limit groups that appear along the graded resolution GRes(s, z, y, z_{base}^{2}, a) from Glim_{j}(s, z, y, z_{base}^{2}, a) and above: Glim_{j}(s, z, y, z_{base}^{2}, a), Glim_{j−1}(s, z, y, z_{base}^{2}, a), …, Glim_{1}(s, z, y, z_{base}^{2}, a), are all flexible quotients of the rigid limit group T_{1}(s, z, y, z_{base}, a). But the graded resolution GRes(s, z, y, z_{base}^{2}, a) was constructed from a collection of rigid specializations of T_{1}(s, z, y, z_{base}^{2}, a), a contradiction.

Therefore, there exists some QH subgroup Q in the principal JSJ decomposition of Zlim(z_{base}, a) that is mapped non-trivially into Q′ and we get part (i). parts (ii) and (iii) follow from (i) by applying lemma 1.9 and theorem 1.11 respectively.

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Proposition 1.13. With the notation above, suppose that the terminal subgroup T_{1}(s, z, y, z_{base}, a) of the graded resolution GRes(s, z, y, z_{base}, a) is solid. Let Q′ be a QH subgroup that appears in the j-th level principal graded JSJ decomposition of the graded resolution GRes(s, z, y, z_{base}^{2}, 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}, a), so that the map τ_{j} : Zlim(z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, 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}, a), so that the map η_{j} : T_{1}(s, z, y, z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, 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}, 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 GRes(s, z, y, z_{base}^{2}, a), and Q is mapped to the vertex stabilized by < z_{base}^{2}, a > by the maps τ_{j} : Rlim(z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, a) in all the levels above the bottom one.

Proof: The proof is essentially similar to the proof of proposition 1.12. If under the map τ_{j} : Zlim(z_{base}, a) → Glim_{j}(s, z, y, z_{base}^{2}, a), non of the QH subgroups Q in the principal JSJ decomposition of Zlim(z_{base}, a), is mapped non-trivially into (a conjugate of) Q′, then the entire image of Zlim(z_{base}, a), τ_{j}(Zlim(z_{base}, a)), is contained in a proper connected subgraph of groups of the principal graded JSJ decomposition of Glim_{j}(s, z, y, z_{base}^{2}, a), a subgraph that does not contain the vertex stabilized by the QH subgroup Q′. Let Θ_{j} be the graph of groups obtained from the principal graded JSJ decomposition of Glim_{j}(s, z, y, z_{base}^{2}, a) by collapsing the maximal connected subgraph of groups of the graded JSJ decomposition that does not contain the QH vertex group Q, and into which τ_{j} maps Zlim(z_{base}, a), into a point. If the quotient map: η_{j} : T_{1}(s, z, y, z_{base}, a) → Glim_{j}(s, z, y, z_{base}, a) maps a non-QH vertex group in the principal graded JSJ decomposition of T_{1}(s, z, y, z_{base}, a) into a non-elliptic or a QH subgroup in the graph of groups Θ_{j}, then the graded limit groups that appear along the graded resolution GRes(s, z, y, z_{base}^{2}, a) from Glim_{j}(s, z, y, z_{base}^{2}, a) and above: Glim_{j}(s, z, y, z_{base}^{2}, a), Glim_{j−1}(s, z, y, z_{base}^{2}, a), …, Glim_{1}(s, z, y, z_{base}^{2}, a), are all flexible quotients of the solid limit group T_{1}(s, z, y, z_{base}, a). But the graded resolution GRes(s, z, y, z_{base}^{2}, a) was constructed from a collection of solid specializations of T_{1}(s, z, y, z_{base}^{2}, a), a contradiction. Hence, every non-QH vertex group in the principal graded JSJ decomposition of T_{1}(s, z, y, z_{base}^{2}, a) is mapped by the quotient map η_{j} into a non-QH vertex group in Θ_{j}. Since the restricted limit group we have started with, Rlim(y, a), is assumed to be of minimal rank, if, in addition, no QH vertex group in the principal graded JSJ decomposition of T_{1}(s, z, y, z_{base}, a) is mapped non-trivially into the QH subgroup Q′ of Glim_{j}(s, z, y, z_{base}^{2}, a), then the entire group T_{1}(s, z, y, z_{base}^{2}, a) is mapped into the vertex group that contains the subgroup η_{j}(Zlim(z_{base}, a)) in the graph of groups Θ_{j}, a contradiction to η_{j} being a quotient map.

Therefore, there exists some QH subgroup Q either in the principal JSJ decomposition of Zlim(z_{base}, a) or in the principal graded JSJ decomposition of T_{1}(s, z, y, z_{base}^{2}, a) that is mapped non-trivially into Q′ and we get part (i). parts (ii) and (iii) follow from (i) by applying (the proof of) lemma 1.9 and theorem 1.11 respectively.

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We continue the analysis of the limit group QRlim(s, z, y, a) by further reducing the set of defining parameters sequentially. Recall that in order to obtain the graded resolution GRes(s, z, y, z_{base}^{2}, a) we first used the variables z_{base} as parameters, where z_{base} were the subset of the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, a) except those defining the highest level, and then used the variables z_{base}^{2} as defining parameters, where z_{base}^{2} form the subset of the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, a) except for the two highest levels. To continue the analysis of the limit group QRlim(s, z, y, a), we set the variables z_{base}^{ℓ} to be the subset of the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, a) except for the ℓ highest levels. Let T_{2}(s, z, y, a) be the terminal rigid or solid graded limit group in the graded resolution GRes(s, z, y, z_{base}^{2}, a) with respect to the parameters z_{base}^{2}. We continue the graded resolution GRes(s, z, y, z_{base}^{2}, a) by viewing T_{2}(s, z, y, a) as a graded limit group with respect to the defining parameters z_{base}^{3}, the obtained terminal rigid or solid graded limit group as a graded limit group with respect to the defining parameters z_{base}^{4} and so on, until we exclude all the z variables that appear in the different levels of the completed resolution Comp(Res)(z, y, a). Clearly, lemma 1.9 and theorem 1.11 remain valid for all the steps of the procedure. Note that the final resolution we obtain is an ungraded resolution of the limit group QRlim(s, z, y, a), or of some quotient of it. Also, note that the described procedure produces (canonically) finitely many such (ungraded) resolutions of the limit group QRlim(s, z, y, a).

We will denote an (ungraded) resolution obtained by our procedure QRes(s, z, y, a).

Proposition 1.14. With the notation above the resolution QRes(s, z, y, a) is a well-structured resolution.

Proof: Since the restricted limit group is assumed to be of minimal rank, so is the quotient limit group QRlim(s, z, y, a). By the definition of a well-structured resolution ([Se2],1.11), if the quotient limit group QRlim(s, z, y, a) is of minimal rank then any strict resolution of it is a well-structured resolution. By construction a quotient resolution QRes(s, z, y, a) is a strict resolution of QRlim(s, z, y, a) or of a quotient of it, hence, QRes(s, z, y, a) is a well-structured resolution.

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At this point we are finally ready to define the complexity of a resolution in the minimal rank case, and to show the complexity of a resolution QRes(s, z, y, a) obtained by the above procedure, Cmplx(QRes(s, z, y, a)), is strictly smaller than the complexity of the original resolution Cmplx(Res(y, a)). This sequential reduction in complexity in each of the next steps of the iterative procedure will finally lead to its termination.

Definition 1.15. Let Rlim(t, a) be a limit group of minimal rank and let Res(t, a) be a well-structured resolution of Rlim(t, a). Let Q_{1}, …, Q_{m} be the QH subgroups that appear in the resolution Res(t, a) and let S_{1}, …, S_{m} be the corresponding punctured surfaces. To each punctured surface S_{j} we may associate an ordered couple (genus(S_{j}), |chi(S_{j})|). We will assume that the QH subgroups Q_{1}, …, Q_{m} are ordered according to the lexicographically (decreasing) order of the ordered couples associated with their corresponding surfaces. Let A_{1}, …, A_{n} be the (pegged) abelian groups appear as vertex groups in the completed resolution Comp(Res)(u, t, a). Let eA_{j} < A_{j} be the subgroup generated by the edge groups connected to the vertex stabilized by the abelian group A_{j} in Comp(Res)(u, t, a). We set the abelian rank of the resolution Res(t, a), denoted Abrk(Res(t, a)), to be:

Abrk(Res(t, a)) = Σ(rk(A_{j}) − rk(eA_{j}))

We set the complexity of the resolution Res(t, a), denoted Cmplx(Res(t, a)), to be:

Cmplx(Res(t, a)) = ((genus(S_{1}), |chi(S_{1})|), …, (genus(S_{m}), |chi(S_{m})|), Abrk(Res(t, a))).

On the set of resolutions of minimal rank limit groups we can define a partial order. Let Rlim(t_{1}, a) and Rlim(t_{2}, a) be two minimal rank limit groups with well-structured resolutions Res(t_{1}, a), Res(t_{2}, a) in correspondence. We say that Cmplx(Res(t_{1}, a)) = Cmplx(Res(t_{2}, a)) if the tuples defining the two complexities are identical. We say that Cmplx(Res(t_{1}, a)) < Cmplx(Res(t_{2}, a)) if:

(1) the tuple ((genus(S_{1}^{1}), |chi(S_{1}^{1})|), …, (genus(S_{m_{1}}^{1}), |chi(S_{m_{1}}^{1}|)) is smaller in the lexicographical order than the tuple ((genus(S_{1}^{2}), |chi(S_{1}^{2})|), …, (genus(S_{m_{2}}^{2}), |chi(S_{m_{2}}^{2}|)).

(2) if the above tuples are equal then Abrk(Res(t_{1}, a)) < Abrk(Res(t_{2}, a)).

In order to ensure the termination of the iterative procedure we need the complexities of the resolutions obtained in its different steps to decrease. Indeed, the complexity of a (well-structured) resolution QRes(s, z, y, a), obtained after the first step of our procedure, is strictly smaller than the complexity of the resolution Res(y, a) we started with.

Theorem 1.16. The complexity of the resolution QRes(s, z, y, a) is strictly smaller than the complexity of the resolution Res(y, a), i.e.,

Cmplx(QRes(s, z, y, a)) < Cmplx(Res(y, a)).

Proof: Let Q′ be a QH subgroup that appears in a principal graded abelian decomposition associated with one of the levels of the quotient resolution QRes(s, z, y, a), and let S′ be its associated punctured surface. By propositions 1.6, 1.12 and 1.13 there exists some QH subgroup Q with associated punctured surface S that appears in a cyclic JSJ decomposition associated with one of the levels of the original resolution Res(y, a), so that the fundamental group of a subsurface of S is mapped into a finite index subgroup of S′, and in particular: genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|. Furthermore, if genus(S′) = genus(S) and |χ(S′)| = |χ(S)| then Q is by definition a surviving surface, so by theorems 1.8 and 1.11 the QH subgroup Q is mapped isomorphically onto Q′, Q is mapped to the (distinguished) vertex group containing F_{k} in all levels above the one in which Q′ appears, and in all the levels below the one containing Q′, Q is identified with its image in the completed resolution Comp(Res)(z, y, a).

Hence, for any surviving QH subgroup Q with associated punctured surface S, corresponds a unique QH subgroup Q′ in the resolution QRes(s, z, y, a) with associated punctured surface S′, so that S is homeomorphic to S′. If Q′ is a QH subgroup in the quotient resolution QRes(s, z, y, a) with an associated punctured surface S′, and no surviving surface Q is mapped onto a conjugate of Q′, then for any QH subgroup Q with an associated surface S in Res(y, a), for which the fundamental group of some subsurface of S is mapped into a finite index subgroup of a conjugate of Q′, genus(S′) ≤ genus(S), |χ(S′)| ≤ |χ(S)|a and either genus(S′) < genus(S) or |χ(S′)| < |χ(S)|. Therefore, if Q_{1}, …, Q_{m} are the QH subgroups that appear in the original resolution Res(y, a) and S_{1}, …, S_{m} are their associated punctured surfaces, and Q′_{1}, …, Q′_{m′} are the QH subgroups that appear in the quotient resolution QRes(s, z, y, a) and S′_{1}, …, S′_{m′} are their associated punctured surfaces, then the tuple ((genus(S_{1}), |chi(S_{1})|), …, (genus(S_{m}), |chi(S_{m}|)) is less than or equal in the lexicographical order than the tuple ((genus(S′_{1}), |chi(S′_{1})|), …, (genus(S′_{m′}), |chi(S′_{m′}|)), and the two tuples are equal if and only if all the QH subgroups Q_{1}, …, Q_{m} are surviving QH subgroups. Since the quotient limit group QRlim(s, z, y, a) is a proper quotient of the closure Cl(Res)(s, z, y, a), if all the QH subgroups Q_{1}, …, Q_{m} are surviving, then necessarily: Abrk(QRes(s, z, y, a)) < Abrk(Res(y, a)), and we finally get that: Cmplx(QRes(s, z, y, a)) < Cmplx(Res(y, a)).

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Having constructed the (finitely many) well-structured resolutions QRes(s, z, y, a) and their corresponding limit groups QRlim(s, z, y, a), to prove the validity of our given sentence we only need to prove its validity on the set of specializations of the variables y for which there exist specializations for the variables s and z so that the specialization (s, z, y, a) factors through one of the resolutions QRes(s, z, y, a).

The final part of the first step of the procedure for the validation of a sentence (in the minimal rank case), is the construction of formal solutions defined over a covering closure of the resolution QRes(s, z, y, a) in a similar way to the ones described in proposition 1.3.

Proposition 1.17. Let QRes(s, z, y, a) be one of the (well-structured) resolutions constructed in the first step of our iterative procedure for validation of a sentence, and let QRlim(s, z, y, a) be its corresponding limit group. For brevity we will denote the resolution by QRes(t, y, a) and the corresponding limit group by QRlim(t, y, a). There exists a covering closure Cl(QRes)_{1}(ŝ, ẑ, t, y, a), …, Cl(QRes)_{q}(ŝ, ẑ, t, y, a) of the resolution QRes(t, y, a), and for each index n, 1 ≤ n ≤ q, there exists a formal solution x_{n}(ŝ, ẑ, t, y, a), and a finite set of restricted limit groups QRlim_{1}^{n}(ŝ, ẑ, t, y, a), …, QRlim_{m(n)}^{n}(ŝ, ẑ, t, y, a) for which:

(i) for every index n, 1 ≤ n ≤ q, the words corresponding to the equations in the system Σ(x_{n}(ŝ, ẑ, t, y, a), y, a) represent the trivial word in the closure Cl(Res)_{n}(ŝ, ẑ, t, y, a).

(ii) for every index n, the restricted limit groups QRlim_{1}^{n}(ŝ, ẑ, t, y, a), …, QRlim_{m(n)}^{n}(ŝ, ẑ, t, y, a)

are proper quotients of the n-th closure Cl(Res)_{n}(ŝ, ẑ, t, y, a). (iii) for every index n, 1 ≤ n ≤ q, let D^{n}(y) be the Diophantine set corresponding

to the closure Cl(QRes)_{n}(ŝ, ẑ, t, y, a), and let D_{1}^{n}(y), …, D_{m(n)}^{n}(y) be the

Diophantine sets corresponding to the restricted limit groups QRlim_{1}^{n}(ŝ, ẑ, t, y, a), …, QRlim_{m(n)}^{n}(ŝ, ẑ, t, y, a). Let ψ_{j}(x, y, a) be a word corresponding to one of the equations in the system Ψ(x, y, a). If for some index n, 1 ≤ n ≤ q, y_{0} ∈ D^{n}(y) and there exists some specialization (ŝ_{0}, ẑ_{0}, t_{0}, y_{0}, a) that factors through the closure Cl(Res)_{n}(ŝ, ẑ, t, y, a), and y_{0} ∉ D_{1}^{n}(y) ∪ … ∪ D_{m(n)}^{n}(y), then there exists a shortest form specialization (ŝ_{1}, ẑ_{1}, t_{0}, y_{0}, a) that factors through the closure Cl(Res)_{n}(ŝ, ẑ, t, y, a) so that for every index j, 1 ≤ j ≤ r: ψ_{j}(x_{n}(ŝ_{1}, ẑ_{1}, t_{0}, y_{0}, a), y_{0}, a) ≠ 1.

Proof: identical with the proof of proposition 1.3.

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The restricted limit groups QRlim_{i}^{n}(s, z, t, y, a) are the input for the second step of the iterative procedure for validation of a sentence in the minimal rank case.

The general step of the iterative procedure

In a similar way to the first step of the procedure, the input to the general (m-th) step of the iterative procedure consists of finitely many well-structured resolutions QRes(t_{m}, y, a) with corresponding limit groups QRlim(t_{m}, y, a), and to each resolution QRes(t_{m}, y, a) there are finitely many associated limit groups of the form QRlim(s_{m}, z_{m}, t_{m}, y, a), where each limit group QRlim(s_{m}, z_{m}, t_{m}, y, a) is a proper quotient of a closure of the resolution QRes(t_{m}, y, a), Cl(QRes)(s_{m}, z_{m}, t_{m}, y, a).

Applying the same procedure used to construct the quotient resolutions QRes(s, z, y, a) in the first step, with each given proper quotient QRlim(s_{m}, z_{m}, t_{m}, y, a) of a closure of the resolution QRes(t_{m}, y, a), Cl(QRes)(s_{m}, z_{m}, t_{m}, y, a), we can associate finitely many (ungraded) resolutions {Res_{j}(s_{m}, z_{m}, t_{m}, y, a)} so that:

(1) every specialization of QRlim(s_{m}, z_{m}, t_{m}, y, a) which is a shortest form specialization of the closure Cl(QRes)(s_{m}, z_{m}, t_{m}, y, a), factors through at least one of the resolutions {Res_{j}(s_{m}, z_{m}, t_{m}, y, a)}.

(2) by a similar argument to the one used to prove theorem 1.16, for every possible index j:

Cmplx(Res_{j}(s_{m}, z_{m}, t_{m}, y, a)) < Cmplx(QRes(t_{m}, y, a)).

The final part of the general step of the procedure for validation of a sentence in the minimal rank case, is the construction of formal solutions in a similar way to the one described in propositions 1.3 and 1.17.

Proposition 1.18. Let Res(s_{m}, z_{m}, t_{m}, y, a) be one of the resolutions constructed above, and let Rlim(s_{m}, z_{m}, t_{m}, y, a) be its corresponding limit group. For brevity we will denote the resolution by Res(t_{m+1}, y, a) and the corresponding limit group by Rlim(t_{m+1}, y, a). There exists a covering closure Cl(Res)_{1}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), …, Cl(Res)_{q}(s_{m+1}, z_{m+1}, t_{m+1}, y, a) of the resolution Res(t_{m+1}, y, a), and for each index 1 ≤ n ≤ q there exists a formal solution x_{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), and a finite set of restricted limit groups QRlim_{1}^{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), …, QRlim_{u(n)}^{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a) for which:

(i) for each index n, 1 ≤ n ≤ q, the words corresponding to the equations in the system Σ(x_{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), y, a) represent the trivial word in the closure Cl(Res)_{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a).

(ii) for every index n, the restricted limit groups QRlim_{1}^{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), …, QRlim_{u(n)}^{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a)

are proper quotients of the n-th closure Cl(Res)_{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a). (iii) for every index n, 1 ≤ n ≤ q, let D^{n}(y) be the Diophantine set corresponding

to the closure Cl(QRes)_{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), and let D_{1}^{n}(y), …, D_{u(n)}^{n}(y)

be the Diophantine sets corresponding to the restricted limit groups QRlim_{1}^{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), …, QRlim_{u(n)}^{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a). Let ψ_{j}(x, y, a) be a word corresponding to one of the equations in the system Ψ(x, y, a). If for some index n, 1 ≤ n ≤ q, y_{0} ∈ D^{n}(y) and there exists some specialization (s_{m+1}_{0}, z_{m+1}_{0}, t_{m+1}_{0}, y_{0}, a) that factors through the closure Cl(Res)_{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a), and y_{0} ∉ D_{1}^{n}(y)∪…∪D_{u(n)}^{n}(y), then there exists a shortest form specialization (s_{m+1}_{1}, z_{m+1}_{1}, t_{m+1}_{0}, y_{0}, a) that factors through the closure Cl(Res)_{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a) so that for every index j, 1 ≤ j ≤ r: ψ_{j}(x_{n}(s_{m+1}_{1}, z_{m+1}_{1}, t_{m+1}_{0}, y_{0}, a), y_{0}, a) ≠ 1.

Proof: Identical with the proof of proposition 1.3.

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The resolutions Res(t_{m+1}, y, a), their corresponding limit groups QRlim(t_{m+1}, y, a), and the restricted limit groups QRlim_{i}^{n}(s_{m+1}, z_{m+1}, t_{m+1}, y, a) which are proper quotients of their corresponding closures, are the input for the next step of the iterative procedure for validation of a sentence in the minimal rank case.

The successive reduction in the complexity of the restricted limit groups constructed in the different steps of the iterative procedure for validation of a sentence in the minimal rank case, finally leads to its termination at a finite step. This termination makes it useful for the purposes of quantifier elimination.

Theorem 1.19. If the initial restricted limit group Rlim(y, a) is of minimal rank, then the iterative procedure for validation of a sentence terminates after finitely many steps.

Proof: Let Res(t_{m}, y, a) be a resolution obtained in the m-th step of the iterative procedure, and let Res(t_{m+1}, y, a) be a resolution obtained in the m + 1-th step from the resolution Res(t_{m}, y, a). By applying theorem 1.16 to the m + 1-th step of the iterative procedure: Cmplx(Res(t_{m+1}, y, a)) < Cmplx(Res(t_{m}, y, a)). Hence, if our iterative procedure does not terminate in a finite time there exists an infinite sequence of strictly decreasing finite tuples of (decreasing) positive integers. But the lexicographical order on finite tuples of (decreasing) positive integers guarantees that every sequence of strictly decreasing finite tuples of (decreasing) positive integers terminates, a contradiction.

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§2. Taut Homomorphisms of Maximal Rank

In the previous section we used an iterative procedure for validation of a sentence. Given a truth sentence of the form:

∀y ∃x Σ(x, y, a) = 1 ∧ Ψ(x, y, a) ≠ 1

the procedure uses a sequence of formal solutions, whose existence is guaranteed by theorem 1.17 in [Se2], to iteratively construct resolutions containing the decreasing sets of the ”remaining” y’s. To each resolution we have assigned its complexity, and the iterative procedure guarantees a decrease in the complexity of the resolutions constructed in consecutive steps of the procedure. This complexity decrease forces the iterative procedure to terminate in a finite time.

The approach presented in the previous section for validation of a sentence, is the basis for our ”trial and error” procedure for quantifier elimination presented in the next paper. However, the procedure described in the previous section is bounded to the minimal rank case. In this section we generalize the notion of complexity to general resolutions, and start modifying the iterative procedure presented in the previous section to guarantee the decrease in the complexity of resolutions constructed by the procedure, which finally implies the termination in a finite time of the iterative procedure for validation of a general sentence, omitting the minimal rank assumption.

Since some of the intermediate claims made in the previous section remain true in the general case, we will continue using the notation used in the previous section. Before starting with the procedure itself, we need to introduce some further notions associated with restricted limit groups and their resolutions. We start with the natural notion of the rank of a resolution Res(y, a), which we denote rk(Res(y, a)).

Definition 2.1. Let Res(y, a) be a resolution of a restricted limit group Rlim(y, a). The rank of the resolution Res(y, a), denoted rk(Res(y, a)), is the maximal integer n for which there exists an epimorphism h : Rlim(y, a) → F_{k} ∗ F_{n} that factors through the resolution Res(y, a). Note that rk(Res(y, a)) is precisely the sum of the ranks of the free factors dropped along the resolution Res(y, a).

By definition, if h : Rlim(y, a) → F_{k} ∗F_{n} is an epimorphism that factors through the resolution Res(y, a), then n ≤ rk(Res(y, a)). Hence, the rank of a resolution is a basic measure of its complexity that was not needed in the minimal rank case. In section 1 of [Se2], we have defined the completion of a well-structured resolution. To get a terminating iterative procedure in the minimal rank case, analyzing well-structured resolutions and their completions is sufficient, as was shown in the previous section. However, to get a terminating iterative procedure in the general case, we need to restrict ourselves to a special class of well-structured resolutions which we call well-separated resolutions.

Definition 2.2. Let Res(y, a) be a well-structured resolution of a restricted limit group Rlim(y, a). Suppose that the resolution Res(y, a) is given by a decreasing sequence of restricted limit groups:

Rlim(y, a) = Rlim_{0}(y, a) → Rlim_{1}(y, a) → … → Rlim_{ℓ}(y, a) =< a, f > ∗H^{ℓ}

where < a, f > and H^{ℓ} are free groups. For each index i, 0 ≤ i ≤ ℓ − 1 let η_{i} : Rlim_{i}(y, a) → Rlim_{i+1}(y, a) be the canonical quotient map.

With each restricted limit group Rlim_{i}(y, a) that appears along the well-structured resolution Res(y, a), there is an associated free decomposition:

Rlim_{i}(y, a) = R_{1}^{i} ∗ … ∗ R_{q(i)}^{i} ∗ F_{rk(i)}^{i} ∗ H^{i}

where F_{rk(i)}^{i} is a (possibly trivial) free group of rank rk(i) and H^{i} is a (possibly trivial) free group. With each factor R_{j}^{i} there is an associated (restricted) (possibly trivial) abelian decomposition (graph of groups), and a corresponding restricted modular group.

Let R_{j}^{i} be a factor in the free decomposition of Rlim_{i}(y, a) which is not a surface or an abelian factor, and let Λ_{j}^{i} be the abelian decomposition associated with the factor R_{j}^{i} in the resolution Res(y, a). Let Δ_{j}^{i} be the cyclic decomposition of R_{j}^{i} obtained from Λ_{j}^{i} by collapsing all the edge groups connecting two non-QH vertex groups. In Δ_{j}^{i} each non-QH vertex group is connected only to QH vertex groups.

Let Q_{1}, …, Q_{s} be the QH vertex groups in the cyclic decomposition Δ_{j}^{i}, and for each QH subgroup Q_{h}, let S_{h} be its corresponding surface, and let bd_{h,1}, …, bd_{h,c} be its boundary components. Let η_{i}(Circum(Q_{h})) = V_{1}∗…∗V_{m(h)}∗H_{h} be the maximal (most refined) free decomposition inherited by η_{i}(Circum(Q_{h})) from the given free decomposition of Rlim_{i+1}(y, a) in which each of the images of the boundary elements η_{i}(bd_{h,n}) can be conjugated into one of the free factors V_{r}, and for each factor V_{r} there exists at least one image of a boundary element η_{i}(bd_{h,n}) that can be conjugated into it.

The free product inherited by the image η_{i}(Q_{h}) lifts to a decomposition of the surface S_{h}. Let Γ_{h}(Q_{h}) be a maximal decomposition of S_{h} which is a lift of the free decomposition inherited by η_{i}(Q_{h}). Let Θ_{j}^{i} be a graph of groups obtained from the graph of groups Δ_{j}^{i} by further cutting the surfaces S_{1}, …, S_{s} corresponding to the QH subgroups Q_{1}, …, Q_{s} along the collections of s.c.c of the decompositions Γ_{h}(Q_{h}) in correspondence.

With each connected component of Θ_{j}^{i} we associate a subgroup of Rlim(y, a). If a connected component of Θ_{j}^{i} contains a non-QH vertex group, then the group W_{u} associated with it is its fundamental group. If a connected component of Θ_{j}^{i} contains no non-QH vertex groups, then the group associated with it is the subgroup of Rlim(y, a) generated by its fundamental group and the Bass-Serre generators connecting it to the other parts of the corresponding ambient surface in the graph of groups Θ_{j}^{i}. Let W_{1}, …, W_{m} be the subgroups associated with the various connected components in the graph of groups Θ_{j}^{i}. We say that the well-structured resolution Res(y, a) is a well-separated resolution if the following conditions hold:

(i) for each subgroup associated with a connected component W_{u}, the image η_{i}(W_{u}) is a factor in a free decomposition of Rlim_{i+1}(y, a), which is a free product of some associated factors {R_{j′}^{i+1}}, a factor of the free group F_{rk(i+1)}^{i+1}, and a factor of the free group H^{i+1}, in case W_{u} contains a non-QH vertex group, and a factor of the free group H^{i+1} otherwise. Furthermore:

η_{i}(R_{j}^{i}) = η_{i}(< W_{1}, …, W_{m} >) = η_{i}(W_{1}) ∗ … ∗ η_{i}(W_{m}).

(ii) let W_{u} be a fundamental group of one of the connected components of Θ_{j}^{i}, and suppose this component contains a non-QH vertex group. Suppose further, that the components corresponding to W_{u} in Θ_{j}^{i} contains m_{d} ≥ 2 subsurfaces of the punctured surface S_{c_{d}} corresponding to the QH-subgroup Q_{c_{d}} of Λ_{j}^{i}, for some 1 ≤ d ≤ r_{u} ≤ s. Then the image of the component W_{u} under the quotient map η_{i} admits the free decomposition:

η_{i}(W_{u}) = M ∗ η_{i}(t_{b_{1}}) ∗…∗ η_{i}(t_{b_{m_{1}−1}}) ∗…∗ η_{i}(t_{b_{m_{1}+...+m_{ru−1}−ru+2}}) ∗…∗ η_{i}(t_{b_{m_{1}+...+mr_{u} −ru}})

where the elements t_{b} are Bass-Serre generators in the decomposition Λ_{j}^{i} connecting the corresponding components of each of the QH subgroups Q_{c_{d}} for 1 ≤ d ≤ r_{u}.

(iii) let S′ be a punctured surface obtained from a punctured surface in the decomposition Θ_{j}^{i} that is connected to a non-QH vertex group in the decomposition Θ_{j}^{i}, by adjoining disks to the boundary components that are mapped to the trivial element in η_{i}. Then no s.c.c. on S′ is mapped to the trivial element by the map η_{i}.

(iv) let Q be one of the QH subgroups in the graph of groups Λ_{j}^{i}, and let S_{Q} be the punctured surface associated with Q. Let Ŝ_{1}, …, Ŝ_{q} be all the punctured surfaces in Θ_{j}^{i} which are subsurfaces of the surface S_{Q}, so that each of the subsurfaces Ŝ_{b} is connected to a non-QH vertex group in the graph of groups Θ_{j}^{i}, and each subsurface of S_{Q} that is connected to a non-QH vertex group in Θ_{j}^{i} is one of the surfaces Ŝ_{1}, …, Ŝ_{q}. Let Ŝ be the punctured surface Ŝ = S_{Q}\{Ŝ_{1}∪…∪Ŝ_{q}} and let S′ be the closed surface obtained from Ŝ by adding disks to all its boundary components. Then η_{i}(circum(Ŝ)) = η_{i}(π_{1}(S′))∗F, where η_{i}(S′) is the maximal possible rank free quotient of π_{1}(S′), and F is a (possibly trivial) free factor generated by the image under the map η_{i} of Bass-Serre generators connecting Ŝ to the components Ŝ_{1}, …, Ŝ_{Q}, and η_{i}(circum(Ŝ)) is a free factor of the free group H^{i+1} of the restricted limit group Rlim_{i+1}.

(v) let R_{j}^{i}, a factor in the given free decomposition of Rlim_{i}(y, a), be a closed surface group. Then η_{i}(R_{j}^{i}) is a free group, which is the maximal possible rank free quotient of the surface group R_{j}^{i}, and η_{i}(R_{j}^{i}) is a free factor of H^{i+1}.

By lemma 1.13 of [Se2] if Res(y, a) is a well-structured resolution, and Comp(Res)(z, y, a) is its completion, then if we replace each of the completed limit groups Comp(Rlim)_{i}(z, y, a) by Comp(Rlim)_{i}(z, y, a) ∗ H^{i}, where H^{i} is the corresponding free factor that appears in the given free decomposition of Rlim_{i}(y, a), then the modified completed resolution is a well-structured resolution. By construction, the same is true for well-separated resolutions.

Lemma 2.3. Let Res(y, a) be a well-separated resolution, and Comp(Res)(z, y, a) be its completion. If we replace each of the completed limit groups Comp(Rlim)_{i}(z, y, a) by Comp(Rlim)_{i}(z, y, a) ∗ H^{i}, where H^{i} is the corresponding free factor that appears in the given free decomposition of Rlim_{i}(y, a), then the modified completed resolution is a well-separated resolution.

In the minimal rank case, we have analyzed the entire collection of homomorphisms that factor through a given well-structured resolution in each step of our iterative procedure. We start the analysis of general sentences by imposing a further restriction on the homomorphisms associated with a given well-separated resolution. Later on, we show that there exists a canonical Makanin-Razborov diagram, called the taut Makanin-Razborov diagram, containing only well-separated resolutions, so that every specialization of the limit group Rlim(y, a) satisfies the additional (taut) condition with respect to at least one of the well-separated resolutions Res(y, a) from the taut Makanin-Razborov diagram.

Definition 2.4. Let Res(y, a) be a well-separated resolution of a restricted limit group Rlim(y, a), and let Comp(Res)(z, y, a) be its completion. Let h : Rlim(y, a) → F_{k} be a homomorphism that factors through the resolution Res(y, a). Since the homomorphism h factors through the resolution Res(y, a), it can be extended to a homomorphism ĥ : Comp(Rlim)(z, y, a) → F_{k} that factors through the completed resolution Comp(Res)(z, y, a). We say that the homomorphism h : Rlim(y, a) → F_{k} is taut with respect to the well-separated resolution Res(y, a) if the following conditions hold:

(i) the image under the homomorphism ĥ of an abelian edge group in any of the abelian decompositions associated with the various levels of the completed resolution Comp(Res)(z, y, a) is non-trivial.

(ii) the image under the homomorphism ĥ of any QH subgroup Q in any of the abelian decompositions associated with the various levels of the completed resolution Comp(Res)(z, y, a) is non-abelian.

(iii) let Q be a QH subgroup in the abelian decomposition Λ_{j}^{i} associated with a factor R_{j}^{i} of the restricted limit group Rlim_{i}(y, a), let S_{Q} be its corresponding punctured surface. Let Ŝ be a connected punctured surface of the surface S_{Q} connected to some non-QH vertex group in the decomposition Θ_{j}^{i} associated with the factor R_{j}^{i} (see definition 2.2). Let S′ be the punctured surface obtained from Ŝ by adjoining disks to the boundary components of Ŝ that are mapped trivially by the quotient map η_{i}. Then no s.c.c. on the punctured surface S′ is mapped to the trivial element by the homomorphism ĥ ∘ η_{i}.

Proposition 2.5. Given a limit group Rlim(y, a) there exists a (canonical) taut Makanin-Razborov diagram, in which every resolution is well-separated, and every homomorphism h : Rlim(y, a) → F_{k} is a taut homomorphism with respect to at least one of the Makanin-Razborov resolutions in the (canonical) taut Makanin-Razborov diagram.

Proof: We start with the restricted limit group Rlim(y, a). If Rlim(y, a) is freely decomposable we continue with each factor separately. hence, w.l.o.g. we assume that Rlim(y, a) is freely indecomposable. Hence, we may associate with Rlim(y, a) its principal cyclic JSJ decomposition which we denote Λ_{0}, and the collection of homomorphisms h : Rlim(y, a) → F_{k}. First we look at degenerate homomorphisms h : Rlim(y, a) → F_{k}, i.e., those homomorphisms that either map an edge group in Λ_{0} into the trivial element in F_{k}, or a non-abelian vertex group in Λ_{0} into a cyclic (or trivial) subgroup in F_{k}. By the arguments used in the construction of the Makanin-Razborov diagram of a limit group ([Se1],5), the collection of degenerate homomorphisms h : Rlim(y, a) → F_{k} factor through finitely many maximal limit groups DRlim_{1}(y, a), …, DRlim_{s}(y, a), where each of the restricted limit groups DRlim_{i}(y, a) is a proper quotient of the original limit group Rlim(y, a). Hence, to analyze the entire set of homomorphisms h : Rlim(y, a), we can restrict ourselves to the set of all the non-degenerate homomorphisms h : Rlim(y, a) → F_{k}, and later analyze the collections of homomorphisms from the proper quotients of Rlim(y, a): DRlim_{1}(y, a), …, DRlim_{s}(y, a) to the free group F_{k}, in the same way we have analyzed the collection of all homomorphisms from Rlim(y, a) to F_{k}.

Given the graph of groups Λ_{0}, up to automorphisms of the (punctured) surfaces associated with its QH subgroups, there are only finitely many possibilities to decompose these (punctured) surfaces along collections of disjoint non-homotopic s.c.c. , hence, up to automorphisms of these (punctured) surfaces there are only finitely many possibilities for the graph Θ_{1}^{0} (see definition 2.2). With each possibility for the graph Θ_{1}^{0} we associate the collection of non-degenerate homomorphisms: h : Rlim(y, a) → F_{k} which are taut with respect to it. Note that this collection may be empty, in which case we exclude the corresponding possibility for the graphs Θ_{1}^{0} from our list of possibilities. We continue with each of the (finitely many) possibilities for the graph Θ_{1}^{0} in parallel. Given the graph Θ_{1}^{0}, we apply the shortening procedure for all the non-degenerate homomorphisms: Rlim(y, a) → F_{k} which are taut with respect to the graph Θ_{1}^{0}. From the collection of shortened homomorphisms we construct a (canonical) collection of finitely many maximal shortening quotients of Rlim(y, a), which we denote: Rlim_{1}^{1}(y, a), …, Rlim_{q_{1}}^{1}(y, a). By ([Se1],5) every shortening quotient Rlim_{d}^{1}(y, a) is a proper quotient of the restricted limit group Rlim(y, a).

We continue with each of the maximal shortening quotients Rlim_{d}^{1}(y, a) and its associated collection of homomorphisms h_{1} : Rlim_{d}^{1}(y, a) → F_{k} in parallel. If Rlim(y, a) is freely decomposable we continue with each of its factors in parallel, so w.l.o.g. we may assume that Rlim(y, a) is freely indecomposable. Hence, we may associate with Rlim(y, a) a cyclic JSJ decomposition which we denote Λ_{d}^{1}. With Rlim_{d}^{1}(y, a) we canonically associate its canonical finite collection of proper quotients obtained from the set of degenerate homomorphisms: h_{1} : Rlim_{d}^{1}(y, a) → F_{k}. We further continue the analysis of all homomorphisms h_{1} : Rlim(y, a)_{d}^{1}(y, a) → F_{k} that factor through one of the degenerate quotients, and the set of non-degenerate homomorphisms h_{1} : Rlim_{d}^{1}(y, a) → F_{k} in parallel. To analyze the set of nondegenerate homomorphisms: h_{1} : Rlim_{d}^{1}(y, a) → F_{k} we again look for the finitely many possibilities (up to automorphisms of the relevant surfaces) for the graph Θ_{1}^{1} associate with Rlim_{d}^{1}, with each possibility for the graph Θ_{1}^{1} we associate the collection of non-degenerate homomorphisms: h_{1} : Rlim_{d}^{1}(y, a) → F_{k} which are taut with respect to Θ_{1}^{1}, and from this collection we obtain a canonical finite collection of maximal shortening (proper) quotients, and continue to the next level.

Since at each step in our construction we obtain (canonical) collections of proper quotients of limit groups obtained in previous steps, the ascending chain condition for decreasing sequences of limit groups proved in ([Se1],5), guarantees that the first part for the construction of the taut Makanin-Razborov diagram terminates. Once it terminates, we obtained a (canonical) finite set of resolutions Res(y, a) of the restricted limit group Rlim(y, a), and with each resolution Res(y, a) a collection of homomorphisms that factor through the resolution Res(y, a) and are non-degenerated with respect to the cyclic decompositions associated with all its levels. by construction, the union of the collections of homomorphisms associated with the resolutions Res(y, a) constructed by the above procedure includes every homomorphism: h : Rlim(y, a) → F_{k}.

Since the resolutions we get are obtained by successive application of the shortening procedure, if a resolution in our final collection is a strict resolution, then it is necessarily a well-separated resolution, so we can take it to be a resolution in the taut Makanin-Razborov diagram. If a resolution Res(y, a) is not a strict resolution, we modify it as follows. We replace each non-abelian, non-QH vertex group associated with one of the levels of the resolution Res(y, a) to be the corresponding limit group obtained from the set of homomorphisms associated with the resolution Res(y, a), i.e., from the collection of homomorphisms that factor and that are non-degenerate and taut with respect to the resolution Res(y, a). The modified resolution MRes(y, a) is a strict resolution by construction, and the collection of homomorphisms that factor and are non-degenerate and taut with respect to the resolution Res(y, a) also factor through the modified resolution MRes(y, a) and they are non-degenerate and taut with respect to it. Furthermore, since the resolution Res(y, a) was constructed by successive applications of the shortening procedure, and since the modified resolution MRes(y, a) is constructed from it to be a strict resolution, the modified resolution MRes(y, a) is a well-separated resolution. Therefore, we can set the (canonical) taut Makanin-Razborov diagram to be the collection of strict resolutions among the resolutions obtained by by our procedure, together with the modified resolutions obtained from the non-strict resolutions constructed through the procedure. Every resolution in this collection is well-separated, and every homomorphism is taut with respect to at least one of the resolutions in the collection.

□

In each of the steps of the iterative procedure, we analyze all the taut homomorphisms that factor through the resolutions constructed in the previous step of the procedure. By proposition 2.5, every homomorphism is taut with respect to at least one of the resolutions in the taut Makanin-Razborov diagram, hence, we are able to use the taut Makanin-Razborov diagram to further impose the taut restriction on the homomorphisms and the well-separated restriction on the resolutions we study at each step of the procedure.

Let:

∀y ∃x Σ(x, y, a) = 1 ∧ Ψ(x, y, a) ≠ 1

be a truth sentence defined over F_{k}. Let F_{y} =< y_{1}, …, y_{ℓ} > be the free group defined over the variables y_{1}, …, y_{ℓ}. By proposition 1.1 there exists a formal solution x = x(y, a), and a finite set of restricted limit groups: Rlim_{1}(y, a), …, Rlim_{m}(y, a) for which:

(i) the words corresponding to the equations in the system Σ(x(y, a), y, a) represent the trivial word in the free group F_{k} ∗ F_{y}.

(ii) for every index i, Rlim_{i}(y, a) is a proper quotient of the free group F_{k} ∗ F_{y}. (iii) let B_{1}(y), …, B_{m}(y) be the basic sets corresponding to the restricted limit

groups Rlim_{1}(y, a), …, Rlim_{m}(y, a). If y ∉ B_{1}(y) ∪ … ∪ B_{m}(y), and ψ_{j}(x, y, a) is a word corresponding to one of the equations in the system Ψ(x, y, a), then: ψ_{j}(x(y, a), y, a) ≠ 1.

Proposition 1.1 gives a formal solution that proves the validity of the given sentence on a co-basic set (F_{k})^{ℓ} \ (B_{1}(y)∪…∪B_{m}(y)), hence, the rest of the procedure needs to construct formal solutions that prove the validity of the sentence on the remaining basic sets B_{1}(y), …, B_{m}(y). Since our treatment of the restricted limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a) which define the basic sets B_{1}(y), …, B_{m}(y) is independent, we will continue with one of these limit groups, which for brevity we denote Rlim(y, a). Note that Rlim(y, a) is a proper quotient of the free group F_{k} ∗ F_{y}. Let Rlim(y, a) be a restricted limit group. By proposition 2.5 with Rlim(y, a) one can associate the (canonical) taut Makanin-Razborov diagram. Let Res_{1}(y, a), …, Res_{r}(y, a) be the resolutions in this taut diagram. Each resolution in the diagram is a well-separated Makanin-Razborov resolution of either the restricted limit group Rlim(y, a) or of a proper quotient of it, and every specialization of the restricted limit group Rlim(y, a) is taut with respect to at least one of the resolutions Res_{1}, (y, a), …, Res_{r}(y, a). Since every resolution in the taut Makanin-Razborov diagram is, in particular, a well-separated resolution, with each resolution Res_{i}(y, a) we can associate its completed resolution Comp(Res)_{i}(z, y, a) with a completed limit group Comp(Rlim)_{i}(z, y, a). By lemma 1.13 of [Se2] for every specialization (y_{0}, a) that factors through the resolution Res_{i}(y, a), there exists some specialization (z_{0}, y_{0}, a) that factors through the completed resolution Comp(Res)_{i}(z, y, a). Therefore, if D_{i}(y) denotes the Diophantine set corresponding to the completed limit group Comp(Rlim)_{i}(z, y, a), and B(y) is the basic set corresponding to the original limit group Rlim(y, a), then: B(y) = D_{1}(y) ∪ … ∪ D_{r}(y).

As in the minimal rank case, the continuation of the iterative procedure in the general case is conducted independently for the different limit groups and each of their resolutions. Hence, for the rest of the procedure, we will denote the restricted limit group in question Rlim(y, a), and its resolution Res(y, a) omitting their indices. By theorem 1.17 of [Se2] from the validity of our given sentence, we get the existence of a covering closure Cl(Res)_{1}(s, z, y, a), …, Cl(Res)_{q}(s, z, y, a) of the resolution Res(y, a), and for each index n, 1 ≤ n ≤ q, there exists a formal solution x_{n}(s, z, y, a) for which:

(i) the words corresponding to the equations in the system Σ(x_{n}(s, z, y, a), y, a) represent the trivial word in the closure Cl(Res)_{n}(s, z, y, a).

(ii) there exists some specialization (s_{0}^{n}, z_{0}^{n}, y_{0}^{n}) for which Ψ(x_{n}(s_{0}^{n}, z_{0}^{n}, y_{0}^{n}, a), y_{0}^{n}, a) ≠ 1.

The formal solutions described above, prove the validity of the given sentence on a certain co-Diophantine set. As we did in the minimal rank case, to analyze the ”remaining” set of y’s, we construct an iterative procedure that produces a sequence of well-separated resolutions and their completions. The iterative procedure for analyzing general resolutions is a modification of the procedure presented in the previous section. To get such a generalization, the analysis of the set of the remaining y’s in each step of the procedure needs to be modified, as well as the complexity presented for the minimal rank case. As in the minimal rank case, the decrease in the modified complexity will finally force the iterative procedure to terminate.

In each of the steps of the iterative procedure, we analyze all the taut homomorphisms that factor through the resolution Res(y, a) and the closure Cl(res)(s, z, y, a), and satisfy the equation ψ_{j}(x(s, z, y, a), y, a) = 1 for some index j. By proposition 2.5, every homomorphism is taut with respect to at least one of the resolutions in the taut Makanin-Razborov diagram, hence, we do not loose any homomorphism by imposing the taut condition. In this section we present a modification of the procedure for validation of a sentence in the minimal rank case to a procedure for validation of a sentence for taut maximal rank homomorphisms. This is still much simpler than the iterative procedure for validation of a sentence in the general case which is presented in the next sections. In this section we analyze all the epimorphisms h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} that are taut with respect to the resolution Res(y, a), where F_{rk(Res(y,a))} is the free group of rank rk(Res): F =< f_{1}, …, f_{rk(Res)} >, the epimorphisms are assumed to factor through the closure Cl(Res)(s, z, y, a), and satisfy the equality ψ_{j}(x(s, z, y, a), y, a) = 1 in the free group F_{k} ∗ F_{rk(Res(y,a))} for some index j. In the next sections we make use of this analysis to get an iterative procedure for validation of a general sentence of the form given above.

Clearly, we can extend a taut maximal rank homomorphism that factors through the closure Cl(Res)(s, z, y, a) to a taut homomorphism h : Cl(Res)(s, z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}. Since these homomorphisms are the maximal rank homomorphisms that factor through the resolution Res(y, a), the rank of the image under the homomorphism h of each quotient of the limit group Rlim(y, a) that appears along the resolution is completely determined. In particular, whenever a free factor which is a free group is dropped along the resolution Res(y, a), its image under the homomorphism h is a free factor of the subgroup F_{rk(Res(y,a))}. For analyzing the entire collection of maximal rank homomorphisms we may further assume that the image of a free factor that drops along the resolution is a fixed free factor in the free group F. Hence, we may assume that the image of the corresponding z generators of a free factor that drops along the resolution are fixed generators f of the free group F. Furthermore, the image of each circumference of a QH subgroup that appears along the resolution Res(y, a) under the homomorphism h, is contained in a free factor F_{k} ∗F_{1} ∗F_{2} of the free group F_{k} ∗F_{rk(Res(y,a))}, where the factor F_{k} ∗F_{1} contains the image of the boundary components of the QH subgroup, and F_{2} is the (possibly trivial) free factor generated by some new generators corresponding to the QH subgroup and (some of) the Bass-Serre generators connecting its boundary components. The rank of the free factor F_{2} depends on the resolution Res(y, a), and is fixed for all maximal rank homomorphisms. Hence, we may assume that for each QH subgroup, the factor F_{2} is a fixed free factor of the free group F_{rk(Res(y,a))}.

Fixing the image of the ”dropped” free factors along the resolution Res(y, a), we can modify the procedure given in section 1 for analyzing the quotients of a minimal rank resolution, to analyze the quotients obtained as limits of taut maximal rank homomorphisms. First, the entire set of taut maximal rank homomorphisms (that fix the image of ”dropped” free factors) from the closure Cl(Res)(s, z, y, a) onto the free group F_{k} ∗ F, h : Cl(Res)(s, z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}, that satisfy ψ_{j}(x(h(s), h(z), h(y), a), h(y), a) = 1 (in the free group F_{k} ∗ F_{rk(Res(y,a))}) for some index j, is contained in a finite set of maximal limit groups (over the free group F_{k}∗F_{rk(Res(y,a))}) QRlim_{1}(s, z, y, a), …, QRlim_{m}(s, z, y, a). By the construction of the formal solution x(s, z, y, a) each of the quotient limit groups QRlim_{t}(s, z, y, a) is a proper quotient of the closure Cl(Res)(s, z, y, a). Since our analysis of the quotient groups QRlim_{1}(s, z, y, a), …, QRlim_{m}(s, z, y, a) is conducted in parallel, we will continue with only one of them which we denote QRlim(s, z, y, a).

At this point we can modify the procedure given in section 1 to analyze the well-separated resolutions containing all the taut maximal rank specializations that factor through the quotient group QRlim(s, z, y, a). We will keep the notation and notions used for minimal rank resolutions in the previous section. Suppose the free decomposition associated with the top level of the resolution Res(y, a), i.e., with the restricted limit group Rlim_{0}(y, a) = Rlim(y, a) is: Rlim(y, a) = L_{1}∗…∗L_{s}. Then the closure Cl(Res)(s, z, y, a) admits a natural free decomposition Cl(Res)(s, z, y, a) = L̂_{1} ∗ … ∗ L̂_{s} and the canonical embedding ν_{0} : Rlim(y, a) → Cl(Res)(s, z, y, a) sends the factor L_{i} to the factor L̂_{i}. By our assumptions on the maximal rank homomorphisms h : Cl(Res)(s, z, y, a) → F_{k} ∗F_{rk(Res(y,a))}, any maximal limit group QRlim(s, z, y, a) admits a free decomposition QRlim(s, z, y, a) = M_{1}∗…∗M_{s} where some of the factors M_{i} may be trivial, and the canonical quotient map Cl(Res)(s, z, y, a) → QRlim(s, z, y, a) sends each of the factors L̂_{i} onto the factor M_{i}. Hence, in the continuation of the analysis of the structure of the quotient limit group QRlim(s, z, y, a) and its resolutions, we can analyze the factors L̂_{i} and their quotients in parallel, so w.l.o.g., we may assume that the free decomposition associated with the restricted limit group Rlim(y, a) in the resolution Res(y, a) is trivial.

Let z be the variables defining the completed resolution Comp(Res)(z, y, a) and let z_{base} be the union of the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, a) but not in the top level. These variables are called the basis of the completed resolution Comp(Res)(z, y, a). With the top level of the well-separated resolution Res(y, a) we have associated a graph of groups Θ_{1}^{0}. Let W_{1}, …, W_{μ} be the fundamental groups of those connected components of Θ_{1}^{0} that contain a non-QH vertex group. Since the resolution Res(y, a) is well-separated,

η_{0}(Rlim(y, a)) = Rlim_{1}(y, a) = η_{0}(W_{1}) ∗ … ∗ η_{0}(W_{μ}) ∗ H^{1}

where each factor η_{0}(W_{d}) is a free product of some of the factors R_{j}^{1} and a factor of the free group F_{rk(1)}^{1}. Since the resolution Res(y, a) is well-separated, it is in particular a well-structured resolution, so the free decomposition of Rlim_{1}(y, a) is inherited by all the restricted limit groups that lie below Rlim_{1}(y, a) in the resolution Res(y, a) and, hence, by the part of the completed resolution Comp(Res)(z, y, a) that contains all its levels except the top one. Therefore, the part of the completed limit group Comp(Rlim)_{1}(z, y, a) admits the free decomposition: Comp(Rlim)_{1}(z, y, a) = U_{1} ∗ … ∗ U_{μ} where η_{0}(W_{d}) < U_{d}, and F_{k} < U_{1}. We will denote the z generators that lie in the factor U_{d} by z_{base}^{d}. Since the homomorphism in question are taut maximal rank ones, the subgroup < z_{base}, a > < QRlim(s, z, y, a) admits the free decomposition: < z_{base}, a >=< z_{base}^{1}, a > ∗ < z_{base}^{2} > ∗ … ∗ < z_{base}^{μ} >.

In the sequel we will set the subgroup P < QRlim(s, z, y, a) to be P =< z_{base}^{1}, a > and for each index d, 2 ≤ d ≤ μ, we set R_{d} < QRlim(s, z, y, a) to be R_{d} =< z_{base}^{d} >. Following the construction of the taut Makanin-Razborov diagram (proposition 2.5), we construct the (canonical) taut multi-graded Makanin-Razborov diagram of the limit group QRlim(s, z, y, a), containing all the taut homomorphisms h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} that factor through the closure Cl(Res)(s, z, y, a) and the quotient limit group QRlim(s, z, y, a), viewed as a multi-graded limit group with respect to the subgroups P, R_{2}, …, R_{μ}. Let MGRes_{1}(s, z, y, r, p, a), …, MGRes_{m}(s, z, y, r, p, a) be the well-separated multigraded Makanin-Razborov resolutions that appear in the taut multi-graded Makanin-Razborov diagram of the (multi-graded) limit group QRlim(s, z, y, r, p, a) with respect to the subgroups P, R_{2}, …, R_{μ}, where each multi-graded resolution is terminating in either a rigid or a solid multi-graded limit group (with respect to the defining subgroups P, R_{2}, …, R_{μ}). With each multi-graded resolution MGRes_{i}(s, z, y, r, p, a) we associate its collection of taut maximal rank epimorphisms h : QRlim(s, z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} that are taut with respect to that given resolution and factor through the closure Cl(Res)(s, z, y, a). Note that by proposition 2.5 every taut maximal rank epimorphism h : Rlim(y, a) → F_{k} ∗F_{rk(Res(y,a))}, that factors through the closure Cl(Res)(s, z, y, a) and the quotient limit group QRlim(s, z, y, a), is taut with respect to one of the multi-graded resolutions: MGRes_{1}(s, z, y, r, p, a), …, MGRes_{m}(s, z, y, r, p, a).

We will treat the multi-graded resolutions MGRes_{i}(s, z, y, r, p, a) in parallel, so for the continuation we will restrict ourselves to one of them which we denote MGRes(s, z, y, r, p, a) for brevity. Let MGlim(s, z, y, r, p, a) be the multi-graded limit group corresponding to the multi-graded resolution MGRes(s, z, y, r, p, a). For each j, let MGlim_{j}(s, z, y, r, p, a) be the multi-graded limit group that appears in the j-th level of the multi-graded resolution MGRes(s, z, y, r, p, a). To analyze the complexity of the multi-graded resolution MGRes(s, z, y, r, p, a) we start with the top level multi-graded limit group MGlim_{1}(s, z, y, r, p, a), and go down through the levels of the multi-graded resolution. MGlim_{1}(s, z, y, r, p, a) admits a canonical free decomposition: MGlim_{1}(s, z, y, r, p, a) = L_{0}^{1} ∗ L_{1}^{1} ∗ … ∗ L_{n(1)}^{1}, which is the most refined free decomposition in which AP =< z_{base}^{1}, a > < L_{0}^{1} and each of the subgroups R_{d} =< z_{base}^{d} > can be conjugated into one of the factors L_{i}^{1}. Let Λ_{i} be the multi-graded principal JSJ decomposition of L_{0}^{i} with respect to the subgroups P, R_{2}, …, R_{μ}. Naturally, there exists a canonical map τ_{1} : Rlim(y, a) → MGlim_{1}(s, z, y, r, p, a).

Rlim(y, a) was assumed freely indecomposable so it admits a principal cyclic JSJ decomposition. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Rlim(y, a) and let S_{Q} be the corresponding (punctured) surface. The boundary elements of Q are mapped by τ_{1} to non-trivial elements that can be conjugated to the various factors L_{0}^{1}, …, L_{n(1)}^{1} in MGlim_{1}(s, z, y, r, p, a). τ_{1}(Q) is a subgroup of MGlim_{1}(s, z, y, r, p, a), so it inherits a (possibly trivial) free decomposition from the canonical free decomposition MGlim_{1}(s, z, y, r, p, a) = L_{0}^{1} ∗…∗L_{n(1)}^{1}. This free decomposition naturally lifts to a decomposition of the QH subgroup Q and its corresponding surface S_{Q}.

Theorem 2.6. With the notation above there exists a (maximal) collection of nonhomotopic, non-boundary parallel s.c.c. ζ(S_{Q}) on the surface S_{Q}, that separates S_{Q} into (punctured) surfaces S_{1}^{1}, …, S_{m(1)}^{1} with fundamental groups Q_{1}^{1}, …, Q_{m(1)}^{1} with the following properties:

(i) every s.c.c. from the defining collection ζ(S_{Q}) is in the kernel of the map τ_{1}.

(ii) the fundamental group of each of the subsurfaces S_{c}^{1} which is connected to a non-QH subgroup in the cyclic JSJ decomposition of Rlim(y, a) is mapped by τ_{1} into a non-trivial subgroup of a conjugate of one of the factors L_{i}^{1} of MGlim_{1}(s, z, y, r, p, a) that either contains the subgroup AP or a conjugate of one of the subgroups R_{d}.

(iii) Let S_{c}^{1} be a punctured subsurface of S_{Q} which is not connected to a non-QH subgroup in the cyclic JSJ decomposition of Rlim(y, a) and let Ŝ_{c}^{1} be the closed surface obtained from S_{c}^{1} by adding disks to its boundary components. Given a taut maximal rank homomorphism h : Cl(Res)(s, z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} which is taut with respect to the graded resolution MGRes(s, z, y, r, p, a), the subgroup circum(Q_{c}^{1}) of Q inherits an h-rank, rk_{h}(circum(Q_{c}^{1})), which is the rank of h(circum(Q_{c}^{1})) in the free group F_{k} ∗ F_{rk(Res(y,a))}. Since all the boundary components of Q_{c}^{1} are mapped by the maximal rank homomorphism h to the identity in F_{k} ∗F_{rk(Res(y,a))}, we can define the maximal rank of Q_{c}^{1}, maxrk(Q_{i}^{1}), as the maximal rank of a free image of the closed surface Ŝ_{c}^{1}. Then: rk_{h}(circum(Q_{c}^{1})) = maxrk(Ŝ_{c}^{1}) + t_{BS}, where t_{BS} is the number of Bass-Serre generator connected to the punctured surface S_{c}^{1} in the decomposition of S_{Q} corresponding to the collection of s.c.c. ζ(S_{Q}).

(iv) let c, 1 ≤ c ≤ m(1), be an index for which the punctured surface S_{c}^{1} is not connected to any non-QH vertex group in the cyclic JSJ decomposition of Rlim(y, a). Then τ_{1}(circum(Q_{c}^{1})) is a factor in a free decomposition of MGlim_{1}(s, z, y, r, p, a) in which all the subgroups P, R_{2}, …, R_{μ} can be conjugated into other factors, and τ_{1}(circum(Q_{c}^{1})) = τ_{1}(Q_{c}^{1}) ∗ F where F is a free group generated freely by the image of the Bass-Serre generators connecting Q_{c}^{1} to the other components. Furthermore, if S_{c_{1}}^{1}, …, S_{c_{v}}^{1} are all the punctured surfaces that are not connected to any non-QH vertex groups in the cyclic JSJ decomposition of Rlim(y, a) then:

MGlim_{1}(s, z, y, r, p, a) = M ∗ τ_{1}(circum(Q_{c_{1}}^{1})) ∗ … ∗ τ_{1}(circum(Q_{c_{v}}^{1}))

and all the subgroups P, R_{2}, …, R_{μ} can be conjugated into the factor M.

(v) let Q_{1}, …, Q_{s} be the QH subgroups in the cyclic JSJ decomposition of Rlim(y, a). Then:

MGlim_{1}(s, z, y, r, p, a) = M^{1} ∗ F_{BS} ∗ τ_{1}(circum(Q_{c_{1}}^{1})) ∗ … ∗ τ_{1}(circum(Q_{c_{v}}^{1}))

where all the subgroups P, R_{2}, …, R_{μ} can be conjugated into the factor M, and F_{BS} is a free group freely generated by the image under the map τ_{1} of some of the Bass-Serre generators in the cyclic JSJ decomposition of Rlim(y, a), τ_{1}(t_{λ}), where these Bass-Serre generators connect the various components corresponding to the subgroups W_{1}, …, W_{μ} in the decomposition Θ_{1}^{0} to subsurfaces S_{c}^{1} that are connected to a non-QH vertex group in the decomposition corresponding to the collection of curves ζ(S_{Q_{1}}), …, ζ(S_{Q_{s}}).

(vi) let h : Rlim(y, a) → F_{k}∗F_{rk(Res(y,a))} be a taut maximal rank homomorphism with respect to the resolution Res(y, a), and let ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗F_{rk(Res(y,a))} be the corresponding (completed) homomorphism, and suppose that ĥ factors through the closure Cl(Res)(s, z, y, a) and the multigraded limit group MGlim_{1}(s, z, y, r, p, a). Then with the notation of part (v):

ĥ(MGlim_{1}(s, z, y, r, p, a)) = F_{k}∗F_{rk(Res(y,a))} = ĥ(M^{1}) ∗ ĥ(F_{BS}) ∗ ĥ(τ_{1}(circum(Q_{c_{1}}^{1}))) ∗ … ∗ ĥ(τ_{1}(circum(Q_{c_{v}}^{1})))

and ĥ maps the factor F_{BS} to its image ĥ(F_{BS}).

Proof: (i) follows from the construction of the decomposition ζ(S_{Q}). The boundary components of the surface S_{Q} are mapped by τ_{1} to elements that can be conjugated into either P or one of the subgroups R_{d}. Since the homomorphisms ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} from which the multi-graded limit group MGlim_{1}(s, z, y, r, p, a) is constructed, correspond to taut homomorphisms with respect to the resolution Res(y, a): h : Rlim(y, a) → F_{k} ∗F_{rk(Res(y,a))}, all the boundary components of the surface S_{Q} are mapped by τ_{1} into non-trivial elements in MGlim_{1}(s, z, y, r, p, a). Since in the free decomposition: MGlim_{1}(s, z, y, r, p, a) = L_{0}^{1} ∗ … ∗ L_{n(1)}^{1} the subgroups P, R_{2}, …, R_{μ} can be conjugated into the various factors L_{0}^{1}, …, L_{n(1)}^{1}, each of the images under the map τ_{1} of the boundary components of the surface S_{Q} can be conjugated into one of the factors L_{0}^{1}, …, L_{n(1)}^{1}. Let S_{c}^{1} be a one of the (punctured) surfaces separated by the collection ζ(S_{Q}), and suppose that some of the boundary components of S_{c}^{1} are the boundary components of S_{Q}. Let Ŝ_{c}^{1} be the punctured surface obtained by adjoining disks to the those boundary components of S_{c}^{1} which are not boundary components of the ambient surface S_{Q}. All the boundary components of Ŝ_{c}^{1} are boundary components of S_{Q}, so they can all be conjugated to the various factors L_{0}^{1}, …, L_{n(1)}^{1}. Hence, if the image τ_{1}(π_{1}(Ŝ_{c}^{1})) can not be conjugated into a factor L_{i}^{1} it inherits a non-trivial free decomposition from the decomposition: MGlim_{1}(s, z, y, r, p, a) = L_{0}^{1} ∗ … ∗ L_{n(1)}^{1}, a decomposition in which all the boundary components of S_{Q}^{1} are mapped to one of the factors. Since the collection of s.c.c. ζ(S_{Q}) is assumed to be a maximal collection of s.c.c. inherited by the ambient surface S_{Q} from the free decomposition of the image τ_{1}(S_{Q}), the free decomposition inherited by τ_{1}(π_{1}(Ŝ_{c}^{1})) from the free decomposition of MGlim_{1}(s, z, y, r, p, a) has to be trivial. Therefore, τ_{1}(π_{1}(Ŝ_{c}^{1})) = τ_{1}(π_{1}(S_{c}^{1})) can be conjugated into one of the factors L_{i}^{1}, and we get part (ii) of the theorem.

Let S_{c}^{1} be one of the punctured surfaces that does not contain any of the boundary components of the ambient surface S_{Q}. Since the resolution Res(y, a) is wellseparated, and since the homomorphisms h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} are taut maximal rank homomorphisms with respect to the resolution Res(y, a), the rank of the image h(circum(Q_{c}^{1})) has to be a maximal possible rank free factor of the free group F_{k} ∗ F_{rk(Res(y,a))}. Clearly, the maximal rank of a free quotient of circum(Q_{c}^{1})) has the rank maxrk(Ŝ_{c}^{1}) + t_{BS} and we get part (iii). Furthermore, if S_{c_{1}}^{1}, …, S_{c_{v}}^{1} are the punctured surfaces separated by the collection of curves ζ(S_{Q_{1}}), …, ζ(S_{Q_{s}}) that do not contain any boundary components of the ambient surfaces S_{Q_{1}}, …, S_{Q_{s}}, then for every homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(res(y,a))} which is taut and maximal rank with respect to the resolution Res(y, a):

F_{k} ∗ F_{rk(Res(y,a))} = h(Rlim(y, a)) = ĥ(QRlim(s, z, y, a)) = F_{h} ∗ h(circum(Q_{c_{1}}^{1})) ∗ … ∗ h(circum(Q_{c_{v}}^{1})) where h(P), h(R_{2}), …, h(R_{μ}) can all be conjugated into the factor F_{h}.

By composing each taut maximal rank homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} with an automorphism of the free group F_{k} ∗ F_{rk(Res(y,a))}, w.l.o.g. we may assume that the factors: F_{h}, h(circum(Q_{c_{1}}^{1})), …, h(circum(Q_{c_{v}}^{1})) are all fixed factors of the free group F_{k} ∗ F_{rk(Res(y,a))}: A, B_{1}, …, B_{v}. Let T be the Bass-Serre tree corresponding to the free decomposition: F_{k} ∗ F_{rk(Res(y,a))} = A ∗ B_{1} ∗ … ∗ B_{v}. Naturally, with every taut maximal rank homomorphism: ĥ : QRlim(s, z, y, a) → F_{k} ∗ F_{Res(y,a))} we can associate an action λĥ : QRlim(s, z, y, a) × T → T. Now, the multi-graded limit group MGlim_{1}(s, z, y, r, p, a) is constructed from a converging sequence of taut maximal rank homomorphisms: {ĥ_{n} : QRlim(s, z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}}. This sequence of homomorphisms corresponds to a sequence of actions {λĥ_{n}} of the quotient limit group QRlim(s, z, y, a) on the Bass-Serre tree T. Since the sequence of homomorphisms {ĥ_{n}} converge into the multi-graded limit group MGlim_{1}(s, z, y, r, p, a), a converging sequence of actions {λ_{h_{nm}} } converge in the Gromov-Hausdorff topology into a faithful action of the multi-graded limit group MGlim_{1}(s, z, y, r, p, a) on some real tree Y. Since we have assumed that the factor F_{h} and the factors: h(circum(Q_{c_{1}}^{1})), …, h(circum(Q_{c_{v}}^{1})) are all fixed factors of the free group F_{k} ∗ F_{rk(Res(y,a))}, and these fixed free factors fix points in the Bass-Serre tree T, every converging sequence of actions λĥ_{nm} converge into a faithful action of the multi-graded limit group MGlim_{1}(s, z, y, r, p, a) on a discrete tree T′, where the stabilizer of each edge is trivial, and the subgroups: τ_{1}(circum(Q_{c_{1}}^{1})), …, τ_{1}(circum(Q_{c_{v}}^{1})) and the subgroups P, R_{2}, …, R_{μ} fix points in the discrete tree T′. Hence, by standard Bass-Serre theory, the multi-graded limit group MGlim_{1}(s, z, y, r, p, a) admits a free decomposition of the form:

MGlim_{1}(s, z, y, r, p, a) = M ∗ τ_{1}(circum(Q_{c_{1}}^{1})) ∗ … ∗ τ_{1}(circum(Q_{c_{v}}^{1}))

in which the subgroups P, R_{2}, …, R_{μ} can be conjugated into the factor M. If S_{c}^{1} is a component that do not contain a boundary component of its corresponding ambient surface S_{Q_{j}}, then for any taut maximal rank homomorphism: h : Rlim9y, a) → F_{k} ∗ F_{rk(res(y,a))}, we have: h(circum(Q_{c}^{1})) = h(Q_{c}^{1}) ∗ h(t_{1}) ∗ … ∗ h(t_{S_{c}^{1}}), where t_{1}, …, t_{S_{c}^{1}} are the Bass-Serre generators connecting the punctured surface S_{c}^{1} to the other components of the ambient surface S_{Q_{j}}. W.l.o.g. we may assume that the factors h(Q_{c}^{1}), h(t_{1}), …, h(t_{S_{c}^{1}}) are all fixed factors of the free group F_{k} ∗ F_{rk(Res(y,a))}, so by applying the same argument used to get the free decomposition of the multi-graded limit group MGlim_{1}(s, z, y, r, p, a) we get: τ_{1}(circum(Q_{c}^{1})) = τ_{1}(Q_{c}^{1}) ∗ τ_{1}(t_{1}) ∗ … ∗ τ_{1}(t_{S_{c}^{1}}), and we finally get part (iv) of the theorem.

Since the homomorphisms h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} are taut maximal rank with respect to the resolution Res(y, a), the image h(t_{λ}) of a Bass-Serre generator {t_{λ}} that connect certain non-QH vertex groups in the decomposition Θ_{1}^{0} to some of the subsurfaces S_{c}^{1} connected to them, is a cyclic factor in a free decomposition of the ambient free group F_{k} ∗ F_{rk(Res(y,a))}. Furthermore, for every taut maximal rank homomorphism h : Rlim(y, a) → F_{k}, h(Rlim(y, a)) admits the free decomposition:

F_{k} ∗ F_{rk(Res(y,a))} = h(Rlim(y, a)) = ĥ(QRlim(s, z, y, a)) = F_{h} ∗ h(circum(Q_{c_{1}}^{1})) ∗ … ∗ h(circum(Q_{c_{v}}^{1})) h(t_{λ_{1}})∗…∗h(t_{λ_{u_{1}}} ) where t_{λ_{1}}, …, t_{λ_{u−1}} are the corresponding Bass-Serre generators, and ĥ(P), ĥ(R_{2}), …, ĥ(R_{μ}) can all be conjugated into the factor F_{h}. Hence, assuming (w.l.o.g.) that the factors: F_{h}, h(circum(Q_{c_{1}}^{1})), …, h(circum(Q_{c_{v}}^{1})), h(t_{λ_{1}}), …, h(t_{λ_{u_{1}}} ) are fixed factors of the free group F_{k} ∗ F_{rk(Res(y,a))} for all taut maximal maximal rank homomorphisms h : Rlim(y, a) → F_{k} ∗F_{rk(Res(y,a))}, and applying the same construction used to prove part (iv) of the theorem, we obtain the free decomposition of the multigraded limit group MGlim_{1}(s, z, y, r, p, a) presented in part (v) of the theorem:

MGlim_{1}(s, z, y, r, p, a) = M^{1} ∗ F_{BS} ∗ τ_{1}(circum(Q_{c_{1}}^{1})) ∗ … ∗ τ_{1}(circum(Q_{c_{v}}^{1}))

where all the subgroups P, R_{2}, …, R_{μ} can be conjugated into the factor M, and F_{BS} is a free group freely generated by the image under the map τ_{1} of some of the Bass-Serre generators in the cyclic JSJ decomposition of Rlim(y, a), τ_{1}(t_{λ}), where these Bass-Serre generators connect the various components corresponding to the subgroups W_{1}, …, W_{μ} in the decomposition Θ_{1}^{0} to subsurfaces S_{c}^{1} that are connected to a non-QH vertex group in the decomposition corresponding to the collection of curves ζ(S_{Q_{1}}), …, ζ(S_{Q_{s}}). This concludes the proofs of parts (iv), (v) and (vi).

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At this point we can apply the relevant part of the argument used in the minimal rank case, for analyzing the complexity of the graded resolution MGRes(s, z, y, r, p, a). If the image of a QH subgroup Q_{i}^{1} of some QH subgroup Q_{t} in the principal cyclic JSJ decomposition of Rlim(y, a) is in the factor M^{1}, i.e., τ_{1}(Q_{i}^{1}) < M^{1}, then Q_{i}^{1} inherits a (possibly trivial) decomposition from the multi-graded principal JSJ decomposition of one of the factors in the most refined free decomposition of the factor M^{1} in which the subgroups P, R_{2}, …, R_{μ} can be conjugated into the different factors. Otherwise, Q_{i}^{1} inherits a decomposition from the cyclic JSJ decomposition of the factor τ_{1}(Q_{i}^{1}). We will denote these decompositions Γ_{i}^{1}(Q_{i}^{1}). Such a decomposition corresponds to some decomposition of the surface S_{i}^{1}, corresponding to the QH subgroup Q_{i}^{1}, along a (possibly trivial) collection of disjoint non-homotopic s.c.c. which we denote Γ_{i}^{1}(S_{i}^{1}). Note that by construction every s.c.c. from the defining collection of Γ_{i}^{1}(S) is mapped by τ_{1} to either a trivial or an elliptic element in the principal JSJ decomposition of either a factor of M^{1} or of τ_{1}(Q_{i}^{1}).

Lemma 2.7. With the notation above let Q′ be a quadratically hanging subgroup in the multi-graded principal JSJ decomposition of either a factor of M^{1} or the factor τ_{1}(Q_{i}^{1}). Suppose that τ_{1}(Q_{i}^{1}) is contained in a factor of M^{1}, but τ_{1}(Q_{i}^{1}) is not contained in the fundamental group of a subgraph of the multi-graded decomposition of the corresponding factor of M^{1} that does not contain the vertex stabilized by Q′, or that τ_{1}(Q_{i}^{1}) can not be conjugated into M^{1}. Then some subsurface of S_{i}^{1} is mapped into a finite index subgroup of Q′, genus(S′) ≤ genus(S_{i}^{1}) and |χ(S′)| ≤ |χ(S_{i}^{1})|.

Proof: Let S_{i}^{1} be the surface corresponding to Q_{i}^{1}, and suppose that Q_{i}^{1} is a subgroup of the QH subgroup Q in the cyclic JSJ decomposition of Rlim(y, a). Let S_{Q} be the surface associated with the QH subgroup Q, and let Ŝ_{i}^{1} be the surface obtained from the punctured surface S_{i}^{1} by adding disks to those of its boundary components which are not boundary components of the ambient surface S_{Q}.

If there exists a s.c.c. on the closed surface Ŝ_{i}^{1} which is mapped to the identity by the map τ_{1}, then this s.c.c. can be added to the collection ζ(S_{Q}), which contradicts the maximality of that collection ζ(S_{Q}). Hence, no s.c.c. on the surface Ŝ_{i}^{1} is mapped to the identity by the homomorphism τ_{1}. Since there is no s.c.c. on Ŝ_{i}^{1} that is mapped to the trivial element by the homomorphism τ_{1}, and since by the assumptions of the lemma Ŝ_{i}^{1} can not be mapped into the fundamental group of a subgraph that does not contain Q′ in the multi-graded decomposition of the corresponding factor of MGlim_{1}(s, z, y, r, p, a), lemma 1.5 implies that there must exists some subsurface of Ŝ_{i}^{1} separated by the collection of curves Γ(S_{i}^{1}), which we denote S̃, so that S̃ is mapped into a subgroup of finite index of Q′, and all the boundary components of S̃ are mapped into conjugates of boundary elements of S_{Q′}. Hence, the double of S̃ is mapped into a subgroup of finite index of the double of S′. Therefore: genus(S′) ≤ genus(S̃) ≤ genus(S_{Q}), and |χ(S′)| ≤ |χ(S̃)| ≤ |χ(S_{Q})|.

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We continue analyzing the levels of the multi-graded resolution MGRes(s, z, y, r, p, a) in the same way we have analyzed the first level, i.e., the multi-graded limit group MGlim_{1}(s, z, y, r, p, a). Clearly, theorem 2.6 and lemma 2.7 remain valid for the next levels of the multi-graded resolution MGRes(s, z, y, r, p, a). After finitely many steps we are left with either a rigid or solid multi-graded limit group with respect to (some of) the subgroups P, R_{1}, …, R_{μ}. Like in the minimal rank case, to analyze the multi-graded resolution MGRes(s, z, y, r, p, a) we need to look at surviving surfaces.

Definition 2.8. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Rlim(y, a), and let S be its corresponding (punctured) surface. The QH subgroup Q (and the corresponding surface S) is called surviving if along the multigraded resolution MGRes(s, z, y, r, p, a) there exists some quadratically hanging subgroup Q′ in the multi-graded principal JSJ decomposition of a factor of one of the multi-graded limit groups MGlim_{j}(s, z, y, r, p, a), with corresponding surface S′, so that τ_{1} 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 JSJ decomposition of Rlim(y, a), then every QH subgroup Q′ in the any level of the multi-graded resolution MGRes(s, z, y, r, 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 multigraded resolution MGRes(s, z, y, r, p, a) to be smaller than that of the resolution Res(y, a) if one is able to ”isolate” the surviving surfaces.

Theorem 2.9. Let Q_{1}, …, Q_{r} be the surviving QH subgroups in the JSJ decomposition of Rlim(y, a), i.e., those QH vertex groups that are mapped nontrivially into QH subgroups of the same genus and Euler characteristic in some level of the multi-graded resolution MGRes(s, z, y, r, p, a). Then in the taut Makanin-Razborov diagram of QRlim(s, z, y, a) associated with those specializations that are taut with respect to the closure Cl(Res)(s, z, y, a), the multi-graded resolution MGRes(s, z, y, r, p, a) can be replaced by finitely many multi-graded resolutions, each composed from two consecutive parts. The first part is a multi-graded resolution of QRlim(s, z, y, a) with respect to the subgroups < P, R_{2}, …, R_{μ}, Q_{1}, …, Q_{r} >, which we denote MGRes(s, z, y, (r, p, Q_{1}, …, Q_{r}), a). The second part is a one step resolution that maps the rigid (solid) terminal multi-graded limit group of MGRes(s, z, y, (r, p, Q_{1}, …, Q_{r}), a) to the rigid (solid) terminal multi-graded limit group of the resolution MGRes(s, z, y, r, p, a). The two consecutive parts of the multi-graded resolution have the following properties:

(1) the multi-graded decomposition corresponding to the second part of the resolution contains a vertex stabilized by the terminal rigid (solid) multi-graded limit group of the resolution MGRes(s, z, y, r, p, a) connected to r′ surviving QH subgroups Q_{i_{1}}, …, Q_{i_{r′}} for some r′ ≤ r and 1 ≤ i_{1} < … < i_{r′} ≤ r.

(2) if the terminal multi-graded limit group of the multi-graded resolution MGRes(s, z, y, (r, p, Q_{1}, …, Q_{r}), a) is rigid (solid) so is the terminal multi-graded limit group of the multi-graded resolution MGRes(s, z, y, r, p, a). Furthermore, if they are both solid, their principal multi-graded decompositions are in correspondence. i.e., the decomposition differ only in the stabilizer of the vertices stabilized by the subgroups P and R_{d}’s. These vertices are stabilized by the subgroups P and R_{d}’s and the subgroups Q_{i_{1}}, …, Q_{i_{r′}} in the first part, and by the subgroups P and R_{d}’s in the second.

Proof: Identical to the proof of theorem 1.8.

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Since the resolution Res(y, a) is well-separated, the groups associated with the various connected components of the graph of groups Θ_{1}^{0}, W_{1}, …, W_{μ}, are mapped to different factors of the second restricted limit group along the resolution Res(y, a), Rlim_{1}(y, a). In accordance with this free decomposition, the completed limit group Comp(Rlim)_{1}(z, y, a) admits a free decomposition: Comp(Rlim)_{1} = P ∗ R_{2} ∗ … ∗ R_{μ}. Furthermore, for every taut maximal rank homomorphism: h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))}, which is taut with respect to the resolution Res(y, a), and its associated homomorphism: ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}, we have:

h(Rlim(y, a)) = ĥ(Comp(Rlim)(z, y, a)) = ĥ(Comp(Rlim)_{1}(z, y, a)) ∗ ĥ(H^{1})

where H^{1} is the free group freely generated by free groups ”dropped” in the top level of the resolution Res(y, a) from the circumference of the various QH vertex groups that appear in this level. Let T_{1}(s, z, y, r, p, a) be the terminal rigid or solid multi-graded limit group of the multi-graded resolution MGRes(s, z, y, r, p, a). To continue the analysis of the resolutions associated with the quotient limit group QRlim(s, z, y, a), we need a ”linkage” between the free decomposition of the completed limit group Comp(Rlim)_{1}(z, y, a) and the free decomposition of T_{1}(s, z, y, r, p, a), a correspondence between (the ranks of) the free groups ”dropped” along the top level of the resolution Res(y, a) and along the various levels of the multi-graded resolution MGRes(s, z, y, r, p, a), and a correspondence between the image ĥ(Comp(Rlim)_{1}(z, y, a)) and the image h_{1}(T_{1}(s, z, y, r, p, a)), where h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} is a rigid or solid shortening of a maximal rank taut homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} which is taut with respect to both the closure Cl(Res)(s, z, y, a) and the multi-graded resolution MGRes(s, z, y, r, p, a).

Theorem 2.10. Let T_{1}(s, z, y, r, p, a) be the solid or rigid terminal limit group of the multi-graded resolution MGRes(s, z, y, r, p, a), let h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} be a homomorphism which is taut with respect to the resolution Res(y, a), let ĥ : Comp(Rlim)(z, y, a) → F_{k}∗F_{rk(Res(y,a))} be the corresponding homomorphism of the completed limit group, and suppose the homomorphism ĥ factors through the closure Cl(Res)(s, z, y, a) and is taut with respect to the multi-graded resolution MGRes(s, z, y, r, p, a). Let h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} be a rigid or solid homomorphism obtained from the homomorphism ĥ by the shortening procedure applied along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a). Then the multi-graded resolution MGRes(s, z, y, r, p, a) can be extended in finitely many ways, to give finitely many multi-graded resolutions with respect to the subgroups P, R_{2}, …, R_{μ}, which we still denote MGRes(s, z, y, r, p, a), so that the terminal rigid or solid multi-graded limit group (which we still denote) T_{1}(s, z, y, r, p, a) has the following properties:

(i) the summation of the ranks of the free groups dropped along the various levels of the multi-graded resolution MGRes(s, z, y, r, p, a) is identical to the rank of the free group dropped along the top level of the resolution Res(y, a).

(ii) by our assumptions the completed limit group Comp(Rlim)_{1}(z, y, a) admits the free decomposition: Comp(Rlim)_{1}(z, y, a) = P ∗ R_{2} ∗ … ∗ R_{μ}. Then the terminal solid or rigid multi-graded limit group T_{1}(s, z, y, r, p, a) admits a free decomposition: T_{1}(s, z, y, r, p, a) = N_{1} ∗ … ∗ N_{μ} ∗ F where F is a free group having a rank which is equal to the summation of the ranks of the free groups drooped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a), rk(F) = rk(H^{1}), P < N_{1} and for every d ≥ 2, R_{d} can be conjugated into N_{d}.

(iii) h_{1}(T_{1}(s, z, y, r, p, a)) = ĥ(Comp(Rlim)_{1}(z, y, a)) and with the notation of

part (ii): h_{1}(N_{1}) = ĥ(P), and for every d ≥ 2: h_{1}(N_{d}) = ĥ(R_{d}). (iv) the limit group Rlim(y, a) admits a natural map into each multi-graded limit

group along the multi-graded resolution MGRes(s, z, y, r, p, a). Given the map ν : Rlim(y, a) → T_{1}(s, z, y, r, p, a), the image of every QH subgroup Q in the top level of the resolution Res(y, a) under the map ν admits a (possibly trivial) free decomposition from the free decomposition T_{1}(s, z, y, r, p, a) = N_{1} ∗ … ∗ N_{μ} ∗ F. Let S_{Q} be the (punctured) surface corresponding to Q, and let ζ(S_{Q}) be the decomposition of the surface S corresponding to the free decomposition of ν(Q). Let S_{1}, …, S_{m} be the punctured subsurfaces of S_{Q} in the decomposition ζ(S_{Q}) that contain a boundary component of S_{Q}, let Ŝ_{Q} = S_{Q} \ (S_{1} ∩ … ∩ S_{m}), and let S_{Q}′ be the closed surface obtained from Ŝ_{Q} by adding disks to its boundary components. Then the maximal rank of a free quotient of π_{1}(S_{Q}′) plus the number of Bass-Serre generators connecting the surface Ŝ_{Q} to the other components of the surface S_{Q} is precisely the rank of the free group dropped from the QH subgroup Q in the top level of the resolution Res(y, a).

Proof: Since the resolution Res(y, a) is well-separated, and since the homomorphisms ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} are all taut maximal rank homomorphisms with respect to the completed resolution Comp(Res)(z, y, a), the summation of the ranks of the free groups that are dropped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a) is at most the rank of the free group dropped along the top level of the resolution Res(y, a). Let τ_{ℓ} : Rlim(y, a) → T_{1}(s, z, y, r, p, a) be the natural map. As we have pointed out, the formulation and proof of theorem 2.6 which are presented for the top level of the multi-graded resolution MGRes(s, z, y, r, p, a) hold for all its levels, and in particular for its terminal rigid or solid limit group T_{1}(s, z, y, r, p, a). By parts (iv) and (v) of theorem 2.6:

T_{1}(s, z, y, r, p, a) = M^{1} ∗ F_{BS} ∗ τ_{ℓ}(circum(Q_{c_{1}}^{ℓ})) ∗ … ∗ τ_{ℓ}(circum(Q_{c_{v}}^{ℓ}))

where all the subgroups P, R_{2}, …, R_{μ} can be conjugated into the factor M^{1}, for each index j, 1 ≤ j ≤ v, τ_{ℓ}(circum(Q_{c_{j}}^{ℓ})) is the free group generated by the Bass-Serre generators connecting S_{c_{j}}^{ℓ}, the surface corresponding to Q_{c_{j}}^{ℓ}, to the other parts of the ambient surface S_{Q_{i}} in the cyclic JSJ decomposition of Rlim(y, a) containing it, free product with the maximal possible free quotient of Q̂_{c_{j}}^{ℓ}, the fundamental group of the closed surface obtained by adding disks to the boundary components of the surface S_{c_{j}}^{ℓ}, F_{BS} is a free group freely generated by the image under the map τ_{ℓ} of some of the Bass-Serre generators in the cyclic JSJ decomposition of Rlim(y, a), τ_{ℓ}(t_{λ}), where these Bass-Serre generators connect the various components corresponding to the subgroups W_{1}, …, W_{μ} in the decomposition Θ_{1}^{0} to subsurfaces S_{c}^{ℓ} that are connected to a non-QH vertex group in the decomposition corresponding to the collection of curves ζ(S_{Q_{1}}), …, ζ(S_{Q_{s}}) associated with the multi-graded free decomposition of T_{1}(s, z, y, r, p, a). Furthermore, let h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} be a taut maximal rank homomorphism with respect to the resolution Res(y, a), let ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} be the corresponding (completed) homomorphism, and suppose that ĥ factors through the closure Cl(Res)(s, z, y, a) and the multi-graded resolution MGRes(s, z, y, r, p, a). Let h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} be the homomorphism obtained from the homomorphism ĥ by applying the shortening procedure along the various levels of the multi-graded resolution MGRes(s, z, y, r, p, a). Then by successively applying part (vi) of theorem 2.6:

h_{1}(T_{1}(s, z, y, r, p, a)) = F_{k}∗F_{rk(Res(y,a))} = h_{1}(M^{1}) ∗ h_{1}(F_{BS}) ∗ h_{1}(τ_{ℓ}(circum(Q_{c_{1}}^{ℓ}))) ∗ … ∗ h_{1}(τ_{ℓ}(circum(Q_{c_{v}}^{ℓ})))

and h_{1} maps the factor F_{BS} and the factors τ_{ℓ}(circum(Q_{c_{1}}^{ℓ})), …, τ_{ℓ}(circum(Q_{c_{v}}^{ℓ})) isomorphically onto their image. Now, from the above compatible free factorizations of T_{1}(s, z, y, r, p, a) and its image h_{1}(T_{1}(s, z, y, r, p, a), if the summation of the rank of the factor F_{BS} and the ranks of the factors τ_{ℓ}(circum(Q_{c_{1}}^{ℓ})), …, τ_{ℓ}(circum(Q_{c_{v}}^{ℓ})) is strictly smaller than the rank of the free group dropped along the top level of the resolution Res(y, a), then for every homomorphism: h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))}, the image of the factor M^{1}, h_{1}(M^{1}), admits a free decomposition of the form: h_{1}(M^{1}) = F_{h_{1}}∗ < v >, where h_{1}(P) < F_{h_{1}}, and the subgroups h_{1}(R_{2}), …, h_{1}(R_{μ}) can all be conjugated into F_{h_{1}} in the free group F_{k} ∗ F_{rk(Res(y,a))}, which contradicts our assumption that the homomorphisms h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} are either rigid or solid multi-graded homomorphisms, and we get part (i) of the theorem.

By composing each taut maximal rank homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))}, with an automorphism of the free group F_{k} ∗ F_{rk(Res(y,a))}, w.l.o.g. we may assume that the factors: h_{1}(M^{1}), h_{1}(circum(Q_{c_{1}}^{ℓ})), …, h_{1}(circum(Q_{c_{v}}^{ℓ})) are all fixed factors of the free group F_{k} ∗ F_{rk(Res(y,a))}, and similarly that the factors h_{1}(P), h_{1}(R_{2}), …, h_{1}(R_{μ}) are all fixed factors of the free group F_{k} ∗ F_{rk(Res(y,a))}:

B_{1}, …, B_{μ}. Let T be the Bass-Serre tree corresponding to the free decomposition: B_{1} ∗ … ∗ B_{μ}. Naturally, with every homomorphism: h_{1} : M^{1} → F_{k} ∗ F_{Res(y,a))} we can associate an action λ_{h_{1}} : M^{1} × T → T.

Now, the factor M^{1} is constructed from a converging sequence of homomorphisms: {h_{1}(n) : M^{1} → B_{1} ∗ … ∗ B_{μ}}. This sequence of homomorphisms corresponds to a sequence of actions {λ_{h_{1}(n)}} of the factor M^{1} on the Bass-Serre tree T. Since the sequence of homomorphisms {h_{1}(n)} converges into the factor M^{1}, a converging sequence of actions {λ_{h_{1}(n_{m})}} converge in the Gromov-Hausdorff topology into a faithful action of the factor M^{1} on some real tree Y. Since the resolution Res(y, a) is a well-separated resolutions, and the homomorphisms: h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} from which the multi-graded resolution MGRes(s, z, y, r, p, a) are all taut and maximal rank with respect to the resolution Res(y, a), and since by part (i) the summation of the ranks of the free groups dropped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a) is equivalent to the rank of the free group dropped in the top level of the resolution Res(y, a), the factor M^{1} does not admit a free decomposition in which the subgroups P, R_{2}, …, R_{μ} can all be conjugated into one of the factors. Hence, if all the subgroups P, R_{2}, …, R_{μ} fix the same point in the real tree Y, the real tree Y is composed from finitely many orbits of IET and axial components and finitely many orbits of segments with non-trivial (abelian) stabilizers. Therefore, we can successively apply the shortening procedure for the action of M^{1} on the real tree Y, and continue the multi-graded resolution MGRes(s, z, y, r, p, a) (canonically) in finitely many ways, until the action of each of the terminal multi-graded limit group, which we still denote T_{1}(s, z, y, r, p, a) acts on a discrete tree T′, where the stabilizer of each edge in T′ is trivial, and the subgroups: P, R_{2}, …, R_{μ} fix points in the discrete tree T′. Hence, by standard Bass-Serre theory, the multi-graded limit group T_{1}(s, z, y, r, p, a) admits a free decomposition of the form: T_{1}(s, z, y, r, p, a) = N_{1} ∗ … ∗ N_{μ} ∗ F, where F is a free group having a rank which is equal to the summation of the ranks of the free groups dropped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a), rk(F) = rk(H^{1}), P < N_{1} and for every d ≥ 2, R_{d} < N_{d}, and by the construction of the limit discrete tree T′ for every homomorphism h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} that factors through the obtained multi-graded resolution MGRes(s, z, y, r, p, a) and its corresponding homomorphism: ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}: h_{1}(T_{1}(s, z, y, r, p, a)) = ĥ(Comp(Rlim)_{1}(z, y, a)), h_{1}(N_{1}) = ĥ(P), and for every d ≥ 2: h_{1}(N_{d}) = ĥ(R_{d}), and we get parts (ii) and (iii) of the theorem.

To prove part (iv), note that if Q is a QH subgroup in the cyclic JSJ decomposition of Rlim(y, a), then the rank of the free group dropped from the image of the subgroup Q along the various levels of the multi-graded resolution MGRes(s, z, y, r, p, a), is at most the maximal rank of a free quotient of π_{1}(S_{Q}′) plus the number of Bass-Serre generators connecting S_{Q}′ to the other components of the ambient surface S_{Q}, which is bounded by the rank of the free group dropped from the QH subgroup Q in the top level of the well-separated resolution Res(y, a). By part (i), the summation of the ranks of the free groups dropped along various levels of the multi-graded resolution MGRes(s, z, y, r, p, a) is precisely the rank of the free group dropped in the top level of the resolution Res(y, a), hence, the rank of free group dropped from the image of the QH subgroup Q along the various levels of the multi-graded resolution MGRes(s, z, y, r, p, a) is precisely the rank of the maximal free quotient of S_{Q}′ plus the number of Bass-Serre generators that connect S_{Q}′ to the other parts of the ambient surface S_{Q}.

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Theorem 2.10 associates a free decomposition T_{1}(s, z, y, r, p, a) = N_{1} ∗ … ∗ N_{μ} ∗ F of the terminal rigid or solid limit group of the multi-graded resolution MGRes(s, z, y, r, p, a) with the free decomposition Comp(Rlim)_{1}(z, y, a) = P ∗ R_{2} ∗ … ∗ R_{μ}. However, the free decomposition associated with the restricted limit group Rlim_{1}(y, a) along the resolution Res(y, a), hence, the free decomposition of the completed limit group Comp(Rlim)_{1}(z, y, a) along the completed resolution Comp(Res)(z, y, a), may be finer than the above free decomposition. To continue the analysis of the resolutions of the quotient limit group QRlim(s, z, y, a) and their associated maximal rank taut homomorphisms, we need to show that the terminal multi-graded limit group T_{1}(s, z, y, r, p, a) admits a free decomposition compatible with the associated finer free decomposition of Rlim_{1}(y, a).

Proposition 2.11. Let Rlim_{1}(y, a) = L_{1} ∗ … ∗ L_{s} ∗ H^{1} be the free decomposition associated with Rlim_{1}(y, a) along the resolution Res(y, a), where H^{1} is the free group generated by the free groups dropped in the top level of the resolution Res(y, a), and let Comp(Rlim)_{1}(z, y, a) = L̂_{1} ∗ … ∗ L̂_{s} be the corresponding free decomposition of Comp(Rlim)_{1}(z, y, a). Let T_{1}(s, z, y, r, p, a) be the solid or rigid terminal limit group of the multi-graded resolution MGRes(s, z, y, r, p, a), let h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} be a maximal rank homomorphism which is taut with respect to the resolution Res(y, a), let ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} be the corresponding homomorphism of the completed limit group, and suppose the homomorphism ĥ factors through the closure Cl(Res)(s, z, y, a) and is taut with respect to the multi-graded resolution MGRes(s, z, y, r, p, a). Let h_{1} : T (s, z, y, r, p, a) → F_{k} ∗F_{rk(Res(y,a))} be a rigid or solid homomorphism obtained from the homomorphism ĥ by the shortening procedure applied along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a). Then the multi-graded resolution MGRes(s, z, y, r, p, a) can be extended in finitely many ways, to obtain finitely many multi-graded resolutions with respect to the subgroups P, R_{2}, …, R_{μ}, which we still denote MGRes(s, z, y, r, p, a), so that the terminal rigid or solid multi-graded limit group of each of these multi-graded resolutions, which we still denote T_{1}(s, z, y, r, p, a), satisfies:

(1) T_{1}(s, z, y, r, p, a) = D_{1}∗…∗D_{s}∗F where the natural map λ : Comp(Rlim)_{1}(z, y, a) → T_{1}(s, z, y, r, p, a) maps the factor L̂_{j} into the factor D_{j}, F is a free group and rk(F) = rk(H^{1}).

(2) for every index j, 1 ≤ j ≤ s, h_{1}(D_{j}) = ĥ(L̂_{j}).

Proof: The proof of theorem 2.11 is basically identical to the proof of parts (ii) and (iii) of theorem 2.10. By composing each taut maximal rank homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))}, with an automorphism of the free group F_{k} ∗ F_{rk(Res(y,a))}, w.l.o.g. we may assume that the factors: h_{1}(L̂_{1}), …, h_{1}(L̂_{s}) are all fixed factors of the free group F_{k} ∗ F_{rk(Res(y,a))}: B_{1}, …, B_{s} in correspondence. By part (i) of theorem 2.10, T_{1}(s, z, y, r, p, a) = M^{1} ∗ F where F is a free group for which rk(F) = rk(H^{1}). Let B = B_{1} ∗ … ∗ B_{s}, and let T be the Bass-Serre tree corresponding to the free decomposition: B = B_{1} ∗ … ∗ B_{s}. Naturally, with every homomorphism: h_{1} : M^{1} → F_{k} ∗ F_{Res(y,a))} we can associate an action λ_{h_{1}} : M^{1} × T → T.

The factor M^{1} is constructed from a converging sequence of homomorphisms: {h_{1}(n) : M^{1} → B_{1} ∗ … ∗ B_{s}}. This sequence of homomorphisms corresponds to a sequence of actions {λ_{h_{1}(n)}} of the factor M^{1} on the Bass-Serre tree T. Since the sequence of homomorphisms {h_{1}(n)} converges into the subgroup M^{1}, a converging sequence of actions {λ_{h_{1}(n_{m})}} converges in the Gromov-Hausdorff topology into a faithful action of the factor M^{1} on some real tree Y. Since the resolution Res(y, a) is a well-separated resolutions, and the homomorphisms: h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} from which the multi-graded resolution MGRes(s, z, y, r, p, a) are all taut and maximal rank with respect to the resolution Res(y, a), and since by part (i) of theorem 2.10 the summation of the ranks of the free groups dropped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a) is equivalent to the rank of the free group dropped in the top level of the resolution Res(y, a), the factor M^{1} does not admit a free decomposition in which the subgroups L̂_{1}, …, L̂_{s}, as well as the subgroups P, R_{2}, …, R_{μ} generated by L̂_{1}, …, L̂_{s}, are all subgroups of one of the factors. Hence, if all the subgroups L̂_{1}, …, L̂_{s} fix the same point in the real tree Y, the real tree Y is composed from finitely many orbits of IET and axial components and finitely many orbits of segments with non-trivial (abelian) stabilizers. Therefore, we can successively apply the shortening procedure for the action of M^{1} on the real tree Y, and continue the multi-graded resolution MGRes(s, z, y, r, p, a) (canonically) in finitely many ways, until the action of each of the terminal multi-graded limit group, which we still denote T_{1}(s, z, y, r, p, a) acts on a discrete tree T′, where the stabilizer of each edge in T′ is trivial, and the subgroups: λ(L̂_{1}), …, λ(L̂_{s}) fix points in the discrete tree T′. Hence, by standard Bass-Serre theory, the multi-graded limit group T_{1}(s, z, y, r, p, a) admits a free decomposition of the form: T_{1}(s, z, y, r, p, a) = D̂_{1}∗…∗D̂_{s}∗F, where, by theorem 2.10 F is a free group having a rank which is equal to the summation of the ranks of the free groups dropped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a), rk(F) = rk(H^{1}), for every index j, λ(L̂_{j}) < D_{j}, and by the construction of the limit discrete tree T′ for every homomorphism h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗F_{rk(Res(y,a))} that factors through the obtained multi-graded resolution MGRes(s, z, y, r, p, a) and its corresponding homomorphism: ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}: h_{1}(D_{j}) = ĥ(L̂_{j}).

□

In general, the restricted limit group Rlim_{1}(y, a) is associated with a free decomposition along the resolution Res(y, a), a free decomposition of the form: Rlim_{1}(y, a) = L_{1} ∗ … ∗ L_{s} ∗ H^{1}. From this free decomposition, the completed limit group, Comp(Rlim)_{1}(z, y, a), inherits a free decomposition Comp(Rlim)_{1}(z, y, a) = L̂_{1} ∗ … ∗ L̂_{s}, where the canonical map ν_{1} : Rlim_{1}(y, a) → Comp(Rlim)_{1}(z, y, a) maps each of the factors L_{j} into the corresponding factor L̂_{j}. Furthermore, given any taut maximal rank homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} that is taut with respect to the resolution Res(y, a), and its corresponding (completed) homomorphism ĥ : Comp(Rlim)(a, y, a) → F_{k} ∗F_{rk(Res(y,a))}, ĥ(Comp(Rlim)_{1}(z, y, a)) = ĥ(L̂_{1})∗… ĥ(L̂_{s}), and we have even assumed (w.l.o.g.) that for each factor L̂_{j}, ĥ(L̂_{j}) is a fixed factor of the free group F_{k} ∗ F_{rk(Res(y,a))}. Proposition 2.11 shows that the terminal (rigid or solid) multi-graded limit group T_{1}(s, z, y, r, p, a) of the multi-graded resolution MGRes(s, z, y, r, p, a) admits a free decomposition T_{1}(s, z, y, r, p, a) = D_{1} ∗ …, D_{s} ∗ F′ which is compatible with the free decomposition of Comp(Rlim)_{1}(z, y, a). i.e., if λ : Comp(Rlim)_{1}(z, y, a) → T_{1}(s, z, y, r, p, a) is the natural map, then for every index j, 1 ≤ j ≤ s, λ(L̂_{j}) < D_{j}. Furthermore, let h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} be a taut maximal rank homomorphism with respect to the resolution Res(y, a), let ĥ : Comp(Rlim)_{1}(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}, be the corresponding homomorphism, and suppose that the homomorphism ĥ is taut with respect to both the closure Cl(Res)(s, z, y, a) and the multi-graded resolution MGRes(s, z, y, r, p, a), and h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} is the rigid or solid homomorphism obtained from ĥ by applying the shortening procedure along the multi-graded resolution MGRes(s, z, y, r, p, a), then h_{1}(D_{j}) = ĥ(L̂_{j}). The compatibility of the free decompositions of Rlim_{1}(y, a), Comp(Rlim)_{1}(z, y, a) and T_{1}(s, z, y, r, p, a) proved in proposition 2.11, enables one to analyze the factors D_{j} of the terminal rigid or solid multi-graded limit group T_{1}(s, z, y, r, p, a) in parallel, hence, it reduces the continuation of the analysis of the resolutions associated with the quotient limit group QRlim(s, z, y, a) to the case in which the free decomposition associated with the restricted limit group Rlim_{1}(y, a) in the resolution Res(y, a) is the trivial one. This will be our assumption for the rest of this section.

We continue the analysis of the collection of taut maximal rank homomorphisms h : Cl(Res)(s, z, y, a) → F_{k} ∗ F that factor through the multi-graded resolution MGRes(s, z, y, r, p, a) as we did in the previous section for minimal rank resolutions, i.e., by sequentially reducing the subgroups P and R_{d}’s according to which we construct our multi-graded resolutions. Let z be the variables defining the completed resolution Comp(Res)(z, y, a), let z_{base} be the union of the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, a) but not in the top level, and let z_{scbase} be the union of all the variables z that appear in all levels of the completed resolution Comp(Res)(z, y, a) but not in the highest two levels. With the second level of the well-separated resolution Res(y, a) we have associated a graph of groups Θ_{1}^{1}. Let W_{1}^{1}, …, W_{μ_{1}}^{1} be the fundamental groups of those connected components of Θ_{1}^{1} that contain a non-QH vertex group. Since the resolution Res(y, a) is well-separated:

η_{1}(Rlim_{1}(y, a)) = Rlim_{2}(y, a) = η_{1}(W_{1}^{1}) ∗ … ∗ η_{1}(W_{μ_{1}}^{1}) ∗ H^{2}

where each factor W_{d}^{1} is a free product of some of the factors R_{j}^{2} and a factor of the free group F_{rk(2)}^{2}. Since the resolution Res(y, a) is well-separated, it is in particular a well-structured resolution, so the free decomposition of Rlim_{2}(y, a) is inherited by all the restricted limit groups that lie below Rlim_{2}(y, a) in the resolution Res(y, a) and, hence, by the part of the completed resolution Comp(Res)(z, y, a) that contains all its levels except the two highest ones. Therefore, this (bottom) part of the completed limit group, Comp(Rlim)_{2}(z, y, a), admits the free decomposition: Comp(Rlim)_{2}(z, y, a) = U_{1} ∗ … ∗ U_{μ_{1}} where η_{1}(W_{d}^{1}) = U_{d}, and F_{k} < U_{1}. We will denote the z generators that lie in the factor U_{d} by z_{scbase}^{d}. Since the homomorphisms in question are taut maximal rank ones, the subgroup < z_{scbase}, a > < T_{1}(s, z, y, r, p, a) admits the free decomposition: < z_{scbase}, a >=< z_{scbase}^{1}, a > ∗ < z_{scbase}^{2} > ∗ … ∗ < z_{scbase}^{μ_{1}} >.

In the sequel we will set the subgroup P^{1} < T_{1}(s, z, y, r, p, a) to be P^{1} =< z_{scbase}^{1}, a > and for each index d, 2 ≤ d ≤ μ_{1}, we set R_{d}^{1} < T_{1}(s, z, y, r, p, a) to be R_{d}^{1} =< z_{scbase}^{d} >. Following the construction of the taut Makanin-Razborov diagram (proposition 2.5), we construct sequentially a taut multi-graded Makanin-Razborov diagram of the limit group T_{1}(s, z, y, r, p, a) with respect to the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1}, containing all the rigid or solid homomorphisms h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} that are obtained by applying the shortening procedure along the multi-graded resolution MGRes(s, z, y, r, p, a) from homomorphisms ĥ : Comp(Rlim)(s, z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} that correspond to a taut maximal rank homomorphism with respect to the resolution Res(y, a): h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))}, so that the homomorphism ĥ factors through both the closure Cl(Res)(s, z, y, a) and the multi-graded resolution MGRes(s, z, y, r, p, a). We denote each of the obtained multi-graded resolutions in our sequential construction MGRes(s, z, y, r^{1}, p^{1}, a).

To analyze the complexity of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) we start with the top level multi-graded limit group T_{1}(s, z, y, r, p, a) = T_{1}(s, z, y, r^{1}, p^{1}, a). T_{1}(s, z, y, r^{1}, p^{1}, a) admits a canonical free decomposition: T_{1}(s, z, y, r^{1}, p^{1}, a) = D_{0}^{1} ∗ D_{1}^{1} ∗ … ∗ D_{n(1)}^{1}, which is the most refined free decomposition in which AP =< z_{scbase}^{1}, a > < D_{0}^{1} and each of the subgroups R_{d}^{1} =< z_{scbase}^{d} > can be conjugated into one of the factors D_{i}^{1}. Let Λ_{i}^{1} be the multi-graded decomposition associated with the factor D_{i}^{1} with respect to the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1}. Naturally, there exist canonical maps τ_{1} : Rlim_{1}(y, a) → T_{1}(s, z, y, r^{1}, p^{1}, a) and λ_{1} : Comp(Rlim)_{1}(z, y, a) → T_{1}(s, z, y, r^{1}, p^{1}, a). The image λ_{1}(Comp(Rlim)_{1}(z, y, a)) < T_{1}(s, z, y, r^{1}, p^{1}, a) admits a canonical free decomposition: λ_{1}(Comp(Rlim)_{1}(z, y, a)) = L_{0}^{1} ∗ L_{1}^{1} ∗ … ∗ L_{s(1)}^{1}, which is the most refined free decomposition in which AP =< z_{scbase}^{1}, a > < L_{0}^{1} and each of the subgroups R_{d}^{1} =< z_{scbase}^{d} > can be conjugated into one of the factors L_{i}^{1}.

Rlim_{1}(y, a) was assumed freely indecomposable so it admits a principal cyclic JSJ decomposition. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Rlim_{1}(y, a) and let S_{Q} be the corresponding (punctured) surface. The boundary elements of Q are mapped by τ_{1} to non-trivial elements that can be conjugated to vertex groups in the multi-graded decompositions associated with the various factors Λ_{0}^{1}, …, Λ_{n(1)}^{1}.

Theorem 2.12. With the notation above let ζ(S_{Q}) be a maximal collection of nonhomotopic, non-boundary parallel s.c.c. on the surface S_{Q}, so that each curve in the collection ζ(S_{Q}) is mapped to the trivial element in T_{1}(s, z, y, r^{1}, p^{1}, a) by the map τ_{1}. The collection ζ(S_{Q}) separates S_{Q} into (punctured) surfaces S_{1}^{1}, …, S_{m(1)}^{1} with fundamental groups Q_{1}^{1}, …, Q_{m(1)}^{1} with the following properties:

(i) the fundamental group of each of the subsurfaces S_{c}^{1} which is connected to a non-QH subgroup in the cyclic JSJ decomposition of Rlim_{1}(y, a) is mapped by τ_{1} into a non-trivial subgroup of a conjugate of one of the factors L_{i}^{1} of T_{1}(s, z, y, r^{1}, p^{1}, a) that either contains the subgroup AP^{1} or a conjugate of one of the subgroups R_{d}^{1}. Similarly, S_{c}^{1} is mapped by λ_{1} into a non-trivial subgroup of a conjugate of one of the factors D_{j}^{1} that either contains AP^{1} or a conjugate of one of the subgroups R_{d}^{1}.

(ii) Let S_{c}^{1} be a subsurface which is not connected to any non-QH vertex group in the cyclic JSJ decomposition of Rlim_{1}(y, a) and let Ŝ_{c}^{1} be the closed surface obtained from S_{c}^{1} by adding disks to its boundary components. Given a rigid or solid homomorphism h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} which is obtained from a taut maximal rank homomorphism with respect to the resolution Res(y, a): h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} which is also taut with respect to the multi-graded resolution MGRes(s, z, y, r, p, a) by sequentially applying the shortening procedure through the various levels of MGRes(s, z, y, r, p, a), the subgroup circum(Q_{c}^{1}) of Q inherits an h_{1}-rank, rk_{h_{1}}(circum(Q_{c}^{1})), which is the rank of h_{1}(circum(Q_{c}^{1})) in the free group F_{k} ∗ F_{rk(Res(y,a))} (note that rk_{h_{1}}(circum(Q_{c}^{1})) = rk_{h}(circum(Q_{c}^{1}))). Since all the boundary components of Q_{c}^{1} are mapped by the homomorphism h_{1} to the identity in F_{k} ∗ F_{rk(Res(y,a))}, we can define the maximal rank of Q_{c}^{1}, maxrk(Q_{c}^{1}), as the maximal rank of a free image of the closed surface Ŝ_{c}^{1}. Then: rk_{h_{1}}(circum(Q_{c}^{1})) = maxrk(Ŝ_{i}^{1}) + t_{BS}, where t_{BS} is the number of Bass-Serre generator connected to the punctured surface S_{c}^{1} in the decomposition of S_{Q} corresponding to the collection of s.c.c. ζ(S_{Q}).

(iii) there exists a graded taut Makanin-Razborov diagram of T_{1}(s, z, y, r^{1}, p^{1}, a) with respect to the parameter subgroup < z_{scbase}, a > containing the graded resolutions: GRes_{1}(s, z, y, z_{scbase}, a), …, GRes_{ℓ}(s, z, y, z_{scbase}, a) that satisfy the following. Let c, 1 ≤ c ≤ m(1), be an index for which the punctured surface S_{c}^{1} is not connected to any non-QH vertex group in the cyclic JSJ decomposition of Rlim_{1}(y, a). Then the terminal rigid or solid limit group T_{j}^{1}(s, z, y, z_{scbase}, a) of the graded resolution GRes_{j}(s, z, y, z_{scbase}, a) has the following properties:

(1) for each index j, 1 ≤ j ≤ ℓ, let λ_{j} : Comp(Rlim)_{1}(z, y, a) → T_{j}^{1}(s, z, y, z_{scbase}, a) be the canonical map. Then λ_{j}(circum(Q_{c}^{1})) is contained in a factor in a free decomposition of T_{j}^{1}(s, z, y, z_{scbase}, a) in which all the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} and all the subgroups λ_{j}(circum(Q_{c′}^{1}), can be conjugated into other factors.

(2) λ_{j}(circum(Q_{c}^{1})) = λ_{j}(Q_{c}^{1}) ∗ F_{Q_{c}^{1}} where F_{Q_{c}^{1}} is a free group generated by the Bass-Serre generators associated with the surface S_{c}^{1} in the decomposition ζ(S_{Q}) of the surface S_{Q}.

(3) let S_{c_{1}}^{1}, …, S_{c_{v}}^{1} be all the punctured surfaces that are not connected to any non-QH vertex groups in the cyclic JSJ decomposition of Rlim_{1}(y, a) then for each index j, 1 ≤ j ≤ ℓ, T_{j}^{1}(s, z, y, z_{scbase}, a) admits the free decomposition:

T_{j}^{1}(s, z, y, z_{scbase}, a) = M_{j} ∗ N_{1} ∗ F_{Q_{c_{1}}^{1}} ∗ … ∗ N_{v} ∗ F_{Q_{cv}^{1}}

where N_{t} is the trivial group if λ_{j}(Q_{c_{t}}^{1}) is the trivial group, all the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} can be conjugated into the factor M_{j}, and each of the subgroups λ_{j}(Q_{c_{t}}^{1}) can be conjugated into the factor N_{t}.

(iv) let Q_{1}, …, Q_{s} be the QH subgroups in the cyclic JSJ decomposition of Rlim_{1}(y, a), and S_{1}^{1}, …, S_{v}^{1} be the punctured subsurfaces of Q_{1}, …, Q_{s} that are not connected to non-QH subgroups in the decompositions ζ(S_{Q_{1}}), …, ζ(S_{Q_{c}}). Then:

T_{j}^{1}(s, z, y, z_{scbase}, a) = D_{0}^{1} ∗ D_{q(1)}^{1} ∗ F_{BS} ∗ N_{1} ∗ F_{Q_{c_{1}}^{1}} ∗ … ∗ N_{v} ∗ F_{Q_{cv}^{1}} where N_{t} is the trivial group if λ_{j}(Q_{c_{t}}^{1}) is the trivial group, all the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} can be conjugated into the various factors D_{0}^{1}, …, D_{q(1)}^{1}, λ_{j}(Q_{c_{t}}^{1}) can be conjugated into the factor N_{t}, and F_{BS} is a free group freely generated by the image under the map τ_{j} : Rlim_{1}(y, a) → T_{j}^{1}(s, z, y, z_{scbase}, a) of some of the Bass-Serre generators in the cyclic JSJ decomposition of Rlim_{1}(y, a), λ_{j}(t_{λ}), where these Bass-Serre generators connect the various components corresponding to the subgroups W_{1}^{1}, …, W_{μ_{1}}^{1} in the decomposition Θ_{1}^{1} to subsurfaces S_{c}^{1} that are connected to a non-QH vertex group in the decomposition corresponding to the collection of curves ζ(S_{Q_{1}}), …, ζ(S_{Q_{s}}).

(v) let λ_{j} : Comp(Rlim)_{1}(z, y, a) → T_{j}^{1}(s, z, y, r^{1}, p^{1}, a), and let λ_{j}(Comp(Rlim)_{1}(z, y, a)) = L_{0}^{1} ∗ … ∗ L_{s(1)}^{1} be the maximal (most refined) free decomposition in which AP^{1} < L_{0}^{1} and the subgroups R_{d}^{1} can be conjugated into the various factors L_{j}^{1}. Then one can reorder the factors in the free decomposition: λ_{j}(Comp(Rlim)_{1}(z, y, a)) = L_{0}^{1} ∗ … ∗ L_{s(1)}^{1}, so that each of the factors L_{j}^{1} can be conjugated into the j − th factor in the above free decomposition of T_{j}^{1}(s, z, y, r^{1}, p^{1}, a).

(vi) let h′_{j} : T_{j}^{1}(s, z, y, z_{scbase}, a) → F_{k} ∗ F_{rk(Res(y,a))} be a homomorphism obtained from the homomorphism h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} by applying the shortening procedure through the various levels of the graded resolution GRes_{j}(s, z, y, z_{scbase}, a). Then:

h′_{j}(T_{j}^{1}(s, z, y, z_{scbase}, a)) = h′_{j}(D_{0}^{1}) ∗ ∗ h′_{j}(D_{q(1)}^{1}) ∗ h′_{j}(F_{BS}) ∗ h′_{j}(L_{1}) ∗ h′_{j}(F_{Q_{c_{1}}^{1}} ) ∗ … ∗ h′_{j}(L_{v}) ∗ h′_{j}(F_{Q_{cv}^{1}} ).

Proof: The proof of part (i) is identical with the proof of part (ii) in theorem 2.6, and the proof of part (ii) is identical to the proof of part (iii) in theorem 2.6. To prove parts (iii)-(vi) we use a similar construction to the one used in proving parts parts (iv)-(vi) of theorem 2.6. Let S_{c_{1}}^{1}, …, S_{c_{v}}^{1} be the punctured surfaces separated by the collection of curves ζ(S_{Q_{1}}), …, ζ(S_{Q_{s}}) that do not contain any boundary components of the ambient surfaces S_{Q_{1}}, …, S_{Q_{s}}. Let λ_{1} : Comp(Rlim)_{1}(z, y, a) → T_{1}(s, z, y, r, p, a) be the natural map, and let the free decomposition: λ(Comp(Rlim)_{1}(z, y, a)) = L_{0}^{1} ∗ … ∗ L_{s(1)}^{1} be the maximal (most refined) free decomposition of λ_{1}(Comp(Rlim)_{1}(z, y, a)) in which the subgroups AP^{1} and R_{d}^{1} and be conjugated into the various factors. Let L_{0}^{1}, …, L_{q(1)}^{1} be the factors containing conjugates of at least one of the subgroups: AP^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1}. Then for every homomorphism h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} which is obtained from a taut and maximal rank homomorphism with respect to the resolution Res(y, a): h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))}

h_{1}(T_{1}(s, z, y, r, p, a) = ĥ(Comp(Rlim)_{1}(z, y, a)) = h_{1}(L_{0}^{1}) ∗ … ∗ h_{1}(L_{q(1)}^{1}) ∗ h_{1}(F_{BS}) ∗ h_{1}(circum(Q_{c_{1}}^{1})) ∗ … ∗ h_{1}(circum(Q_{c_{v}}^{1})) where F_{BS} is a free group freely generated by some of the Bass-Serre generators connecting various subgraphs in each of the connected components in the graph Θ_{1}^{1}.

By composing each taut maximal rank homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} with an automorphism of the free group F_{k} ∗ F_{rk(Res(y,a))}, w.l.o.g. we may assume that the factors: h_{1}(L_{0}^{1}), …, h_{1}(L_{q(1)}^{1}), h_{1}(F_{BS}), h_{1}(circum(Q_{c_{1}}^{1})), …, h_{1}(circum(Q_{c_{v}}^{1})) are all fixed factors of the free group F_{k} ∗F_{rk(Res(y,a))}: C_{1}, …, C_{q(1)}, E, B_{1}, …, B_{v}. Let T be the Bass-Serre tree corresponding to the free decomposition: V = C_{1}∗…∗ C_{q(1)}∗E∗B_{1}∗…∗B_{v}. Naturally, with every homomorphism: h_{1} : T_{1}(s, z, y, r, p, a) → V we can associate an action α_{h_{1}} : T_{1}(s, z, y, r, p, a) × T → T. Now, the multi-graded limit group T_{1}(s, z, y, r, p, a) is constructed from a converging sequence of homomorphisms: {(h_{1})_{n} : T_{1}(s, z, y, r, p, a) → V }. This sequence of homomorphisms corresponds to a sequence of actions {α_{(h_{1})_{n}}} of the rigid or solid limit group T_{1}(s, z, y, r, p, a) on the Bass-Serre tree T. Since the sequence of homomorphisms {(h_{1})_{n}} converge into the (rigid or solid) limit group T_{1}(s, z, y, r, p, a), a converging sequence of actions {α_{(h_{1})_{nm}} } converge in the Gromov-Hausdorff topology into a faithful action of the multi-graded limit group T_{1}(s, z, y, r, p, a) on some real tree Y. Since we have assumed that the factors: h_{1}(L_{0}^{1}), …, h_{1}(L_{q(1)}^{1}), h_{1}(F_{BS}), h_{1}(circum(Q_{c_{1}}^{1})), …, h_{1}(circum(Q_{c_{v}}^{1})) are all fixed factors of the free group F_{k}∗F_{rk(Res(y,a))}, and these fixed free factors fix points in the Bass-Serre tree T, every converging sequence of actions α_{(h_{1})_{nm}} converge into a faithful action of the multi-graded limit group T_{1}(s, z, y, r, p, a) on a real treeY for which the subgroups L_{0}^{1}, …, L_{q(1)}^{1}, F_{BS}, circum(Q_{c_{1}}^{1}), ldots, circum(Q_{c_{v}}^{1}) all fix points in Y. Since the resolution Res(y, a) is a well-separated resolutions, and the homomorphisms: h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} from which the multi-graded resolution MGRes(s, z, y, r, p, a) is constructed are all taut and maximal rank with respect to the resolution Res(y, a), and since by part (i) of theorem 2.10 the summation of the ranks of the free groups dropped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a) is equivalent to the rank of the free group dropped in the top level of the resolution Res(y, a), T_{1}(s, z, y, r, p, a) does not admit a free decomposition in which the subgroups L_{0}^{1}, …, L_{q(1)}^{1}, F_{BS}, circum(Q_{c_{1}}^{1}), ldots, circum(Q_{c_{v}}^{1}) are all subgroups of one of the factors. Hence, if all the subgroups L_{0}^{1}, …, L_{q(1)}^{1}, F_{BS}, circum(Q_{c_{1}}^{1}), ldots, circum(Q_{c_{v}}^{1}) fix the same point in the real tree Y, the real tree Y is composed from finitely many orbits of IET and axial components and finitely many orbits of segments with nontrivial (abelian) stabilizers. Therefore, we can successively apply the shortening procedure for the action of T_{1}(s, z, y, r, p, a) on the real tree Y, and continue the multigraded resolution MGRes(s, z, y, r, p, a) (canonically) in finitely many ways, until the action of each of the terminal multi-graded limit group, which we still denote T_{1}(s, z, y, r, p, a) acts on a discrete tree T′, where the stabilizer of each edge in T′ is trivial, and the subgroups: L_{0}^{1}, …, L_{q(1)}^{1}, F_{BS}, circum(Q_{c_{1}}^{1}), ldots, circum(Q_{c_{v}}^{1}) fix points in the discrete tree T′. Hence, by standard Bass-Serre theory, each of the terminal multi-graded limit groups, of the multi-graded resolutions MGRes_{j}(s, z, y, r, p, a), T_{j}^{1}(s, z, y, r, p, a), admits a free decomposition of the form:

T_{j}^{1}(s, z, y, z_{scbase}, a) = D_{0}^{1} ∗ D_{q(1)}^{1} ∗ F_{BS} ∗ N_{1} ∗ F_{Q_{c_{1}}^{1}} ∗ … ∗ N_{v} ∗ F_{Q_{cv}^{1}}

where N_{t} is the trivial group if λ_{j}(Q_{c_{t}}^{1}) is the trivial group, all the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} can be conjugated into the various factors D_{0}^{1}, …, D_{q(1)}^{1}, λ_{j}(Q_{c_{t}}^{1}) can be conjugated into the factor N_{t}, and F_{BS} is a free group freely generated by the image under the map τ_{j} : Rlim_{1}(y, a) → T_{j}^{1}(s, z, y, z_{scbase}, a) of some of the Bass-Serre generators in the cyclic JSJ decomposition of Rlim_{1}(y, a), λ_{j}(t_{λ}), where these Bass-Serre generators connect the various components corresponding to the subgroups W_{1}^{1}, …, W_{μ_{1}}^{1} in the decomposition Θ_{1}^{1} to subsurfaces S_{c}^{1} that are connected to a non-QH vertex group in the decomposition corresponding to the collection of curves ζ(S_{Q_{1}}), …, ζ(S_{Q_{s}}). Furthermore, one can reorder the factors in the free decomposition: λ_{j}(Comp(Rlim)_{1}(z, y, a)) = L_{0}^{1} ∗ … ∗ L_{s(1)}^{1}, so that each of the factors L_{j}^{1} can be conjugated into the j − th factor in the above free decomposition of T_{j}^{1}(s, z, y, r^{1}, p^{1}, a). Also, by the construction of the limit discrete tree T′, if h′_{j} : T_{j}^{1}(s, z, y, z_{scbase}, a) → F_{k} ∗ F_{rk(Res(y,a))} is a homomorphism obtained from the homomorphism h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} by applying the shortening procedure through the various levels of the graded resolution GRes_{j}(s, z, y, z_{scbase}, a). Then:

h′_{j}(T_{j}^{1}(s, z, y, z_{scbase}, a)) = h′_{j}(D_{0}^{1}) ∗ ∗ h′_{j}(D_{q(1)}^{1}) ∗ h′_{j}(F_{BS}) ∗ h′_{j}(L_{1}) ∗ h′_{j}(F_{Q_{c_{1}}^{1}} ) ∗ … ∗ h′_{j}(L_{v}) ∗ h′_{j}(F_{Q_{cv}^{1}} ).

and we finally get parts (iii)-(vi) of the theorem.

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We will continue constructing the multi-graded resolutions of QRlim(s, z, y, a) with each of the resolutions GRes_{j}(s, z, y, z_{scbase}, a) in parallel, so in the sequel we will denote the multi-graded resolution we continue with: GRes(s, z, y, z_{scbase}, a) = MGRes(s, z, y, r^{1}, p^{1}, a), omitting the index. Similarly, we will denote the terminal rigid or solid limit group of GRes(s, z, y, z_{scbase}, a) by T^{1}(s, z, y, z_{scbase}, a). Clearly, there exists a canonical map τ_{1} : Rlim_{1}(y, a) → T^{1}(s, z, y, z_{scbase}, a). If for some QH subgroup Q in the cyclic decomposition of Rlim_{1}(y, a), the decomposition ζ(S_{Q}) can be refined, we apply theorem 2.12 starting with the multi-graded limit group T^{1}(s, z, y, z_{scbase}, a) and the further refined decompositions of the QH subgroups Q_{1}, …, Q_{s} in the cyclic JSJ decomposition of Rlim_{1}(y, a). Hence, by repeatedly applying theorem 2.12 we can assume that T^{1}(s, z, y, z_{scbase}, a) = T^{1}(s, z, y, r^{1}, p^{1}, a) = L_{0}^{1} ∗ …, L_{n(1)}^{1} is the most refined free decomposition of T^{1}(s, z, y, r^{1}, p^{1}, a) in which AP^{1} < L_{0}^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} can be conjugated into the various factors L_{j}^{1}, and the decompositions ζ(S_{Q_{i}}), corresponding to the QH subgroups Q_{1}, …, Q_{s} in the cyclic JSJ decomposition of Rlim_{1}(y, a) and the map τ : Rlim_{1}(y, a) → T^{1}(s, z, y, r^{1}, p^{1}, a) can not be further refined. Each of the factors Ł_{j}^{1} admits a multi-graded decomposition which we denote Λ_{j}.

At this point we can apply the relevant part of the argument used in the minimal rank case, for analyzing the complexity of the graded resolution MGRes(s, z, y, r^{1}, p^{1}, a). If the image of a QH subgroup Q_{i}^{1} of some QH subgroup Q_{t} in the principal JSJ decomposition of Rlim_{1}(y, a) is in the factor L_{j}^{1}, i.e., τ^{1}(Q_{i}^{1}) < L_{j}^{1}, then Q_{i}^{1} inherits a (possibly trivial) decomposition from the multi-graded principal JSJ decomposition Λ_{j} of L_{j}^{1}. This decomposition corresponds to a decomposition of the surface S_{i}^{1}, corresponding to the QH subgroup Q_{i}^{1}, along a (possibly trivial) collection of disjoint non-homotopic s.c.c. which we denote Γ_{i}^{1}(S_{i}^{1}). Note that by construction every s.c.c. from the defining collection of Γ_{i}^{1}(S_{i}^{1}) is mapped by τ_{1} to either a trivial or an elliptic element in the principal JSJ decomposition of L_{j}^{1}

Lemma 2.13. With the notation above let Q′ be a quadratically hanging subgroup in the multi-graded principal JSJ decomposition of a factor D_{j}^{1} of T^{1}(s, z, y, r^{1}, p^{1}, a).

Suppose that τ_{1}(Q_{i}^{1}) is contained in a factor D_{j}^{1}, but τ_{1}(Q_{i}^{1}) is not contained in the fundamental group of a subgraph of the multi-graded decomposition Λ_{j} that does not contain the vertex stabilized by Q′. Then some subsurface of S_{i}^{1} is mapped into a finite index subgroup of Q′, genus(S′) ≤ genus(S_{i}^{1}) and |χ(S′)| ≤ |χ(S_{i}^{1})|.

Proof: Let Ŝ_{i}^{1} be the closed surface obtained from the surface S_{i}^{1} by adding disks to those boundary components of S_{i}^{1} which are mapped to the identity by the map τ_{1}. Let Q_{t} be the ambient QH subgroup in the principal cyclic JSJ decomposition of Rlim_{1}(y, a) that contains Q_{i}^{1}. Since by our assumptions the collection of curves ζ(S_{Q_{t}}) is a maximal collection, no s.c.c. on the surface Ŝ_{i}^{1} is mapped to the identity by the homomorphism τ_{1}, and by the construction of the decomposition Γ(S_{i}^{1}) all the boundary elements of the surface Ŝ_{i}^{1} are mapped to non-trivial elliptic elements in the decomposition Λ_{j}. With these properties of the image of the surface Ŝ_{i}^{1} under the map τ_{1}, the proof of the lemma is identical with the proof of lemma 2.7.

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If Q′ is a QH subgroup that appears along the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) so that no subsurface of a QH subgroup in the cyclic JSJ decomposition of Rlim_{1}(y, a) is mapped non-trivially into Q′, then we can bound the genus and Euler characteristic of the QH subgroup Q′ in terms of the genus and Euler characteristic of some QH subgroup in the cyclic decomposition of Rlim(y, a). Clearly, there exists a canonical map τ_{0} : Rlim(y, a) → T^{1}(s, z, y, r^{1}, p^{1}, a).

Lemma 2.14. With the notation above let Q′ be a quadratically hanging subgroup in the multi-graded principal JSJ decomposition of a factor D_{j}^{1} of T^{1}(s, z, y, r^{1}, p^{1}, a) with corresponding surface S′, and suppose that no subsurface of a QH subgroup Q in the cyclic JSJ decomposition of Rlim_{1}(y, a) is mapped non-trivially into a finite index subgroup of Q′ by the map τ_{1}. Then there exists some QH subgroup Q in the cyclic JSJ decomposition of Rlim(y, a) with a corresponding surface S_{Q}, and a subsurface S_{i}^{1} of S_{Q}, so that τ_{0}(π_{1}(S_{i}^{1})) is a finite index subgroup of Q′. In particular: genus(S′) ≤ genus(S_{i}^{1}) and |χ(S′)| ≤ |χ(S_{i}^{1})|. In addition either: genus(S′) < genus(S_{Q}) or: |χ(S′)| < |χ(S_{Q})|.

Proof: Let Λ_{j} be the multi-graded principal decomposition of the factor D_{j}^{1}. By the assumptions of the lemma, Q′ is a QH subgroup in the graph of groups Λ_{j}. By assumption, no subsurface of any QH subgroup Q in the principal cyclic JSJ decomposition of Rlim_{1}(y, a) is mapped into a finite index subgroup of Q′, so by lemma 2.13 each of the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} that is mapped by the homomorphism τ_{1} into the factor D_{j}^{1} by the homomorphism τ_{1}, is mapped into the fundamental group of a connected subgraph of Λ_{j} that does not contain the vertex stabilized by Q′. In addition, by part (v) of theorem 2.12, if there is a subgroup P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} that is mapped by τ_{1} into the factor D_{j}^{1}, then all the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} that are mapped into the factor D_{j}^{1} are mapped into the fundamental group of a connected subgraph of Λ_{j} that does not contain the vertex stabilized by the QH subgroup Q′. If none of the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} is mapped into the factor D_{j}^{1}, then by part (v) of theorem 2.12, there exists some subgroup Q_{c}^{1}, which is the fundamental group of a connected subsurface in the decomposition ζ(S_{Q_{i}}) corresponding to a QH subgroup Q_{i} in the cyclic JSJ decomposition of Rlim_{1}(y, a), so that the surface corresponding to Q_{c}^{1} in ζ(S_{Q_{i}}), S_{Q_{c}^{1}}, contains no boundary component of S_{Q_{i}}, and τ_{1}(Q_{c}^{1}) < D_{j}^{1}. By lemma 2.13 the subgroup Q_{c}^{1} is mapped into the fundamental group of a connected subgraph of Λ_{j} that does not contain the vertex stabilized by Q′.

In both cases the multi-graded limit group T^{1}(s, z, y, r^{1}, p^{1}, a) admits a graph of groups Δ, obtained from the free decomposition given in part (v) of theorem 2.12 and the graph of groups Λ_{j}, by collapsing a connected subgraph of Λ_{j}, so that the QH subgroup Q′ is a vertex group in Δ, and the homomorphism τ_{1} maps the entire limit group Rlim_{1}(y, a) into a vertex group in Δ, which is not the vertex stabilized by the QH subgroup Q′. Therefore, by lemma 2.7 there must exist a subgroup of a non-surviving QH subgroup Q in the cyclic JSJ decomposition of Rlim(y, a) that is mapped by the homomorphism: τ_{0} : Rlim(y, a) → T^{1}(s, z, y, r^{1}, p^{1}, a) into a finite index subgroup of Q′. In particular, if S_{Q} is the surface corresponding to the QH subgroup Q then: genus(S′) ≤ genus(S_{Q}) and |χ(S′)| ≤ |χ(S_{Q})|.

The homomorphisms h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗ F_{rk(Res(y,a))} used to construct the limit group T^{1}(s, z, y, r^{1}, p^{1}, a) are assumed rigid (or solid) with respect to the subgroups P, R_{2}, …, R_{μ}. Since Q is not a surviving QH subgroup, if genus(S′) = genus(S_{Q}) and |χ(S′)| = |χ(S_{Q})| then the homomorphisms h_{1} : T_{1}(s, z, y, r, p, a) → F_{k} ∗F_{rk(Res(y,a))} used to construct the limit group T^{1}(s, z, y, r^{1}, p^{1}, a) are all flexible multi-graded homomorphisms of the rigid or solid limit group T_{1}(s, z, y, r, p, a), a contradiction. Hence, genus(S′) < genus(S_{Q}) and |χ(S′)| < |χ(S_{Q})|, and the lemma follows.

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We continue the construction and the analysis of the next levels of the multigraded resolution MGRes(s, z, y, r^{1}, p^{1}, a) in the same way we have constructed and analyzed the first level. At each step we first construct a ”maximal” multigraded free decomposition of the multi-graded limit group MGlim_{j}(s, z, y, r^{1}, p^{1}, a) in question by repeated applications of theorem 2.12 and then apply the (multigraded) shortening procedure for each of the factors. Clearly, theorem 2.12 and lemmas 2.13 and 2.14 remain valid for the next levels of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a). After finitely many steps we are left with either a rigid or solid multi-graded limit group with respect to the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1}. Like in the minimal rank case, to analyze the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) we need to look at surviving surfaces.

Definition 2.15. Let Q be a quadratically hanging subgroup in the JSJ decomposition of Rlim_{1}(y, a), and let S be its corresponding (punctured) surface. The QH subgroup Q (and the corresponding surface S) is called surviving if along the multigraded resolution MGRes(s, z, y, r^{1}, p^{1}, a) there exists some quadratically hanging subgroup Q′ in the multi-graded principal JSJ decomposition of a factor of one of the multi-graded limit groups MGlim_{j}(s, z, y, r^{1}, p^{1}, a), with corresponding surface S′, so that τ_{j} : Rlim_{1}(y, a) → MGlim_{j}(s, z, y, r^{1}, p^{1}, a) 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 JSJ decomposition of Rlim_{1}(y, a), then every QH subgroup Q′ in any level of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, 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 multigraded resolution MGRes(s, z, y, r^{1}, p^{1}, a) to be smaller than that of the resolution Res(y, a) if one is able to ”isolate” the surviving surfaces.

Theorem 2.16. Let Q_{1}, …, Q_{r} be the surviving QH subgroups in the JSJ decomposition of Rlim_{1}(y, a), i.e., those QH vertex groups that are mapped non-trivially into QH subgroups of the same genus and Euler characteristic in some level of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a). Then in the taut Makanin-Razborov diagram of QRlim(s, z, y, a) associated with those specializations that are taut with respect to the closure Cl(Res)(s, z, y, a), the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) can be replaced by finitely many multi-graded resolutions, each composed from two consecutive parts. The first part is a multi-graded resolution of QRlim(s, z, y, a) with respect to the subgroups < P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1}, Q_{1}, …, Q_{r} >, which we denote MGRes(s, z, y, (r^{1}, p^{1}, Q_{1}, …, Q_{r}), a). The second part is a one step resolution that maps the rigid (solid) terminal multi-graded limit group of MGRes(s, z, y, (r^{1}, p^{1}, Q_{1}, …, Q_{r}), a) to the rigid (solid) terminal multi-graded limit group of the resolution MGRes(s, z, y, r^{1}, p^{1}, a). The two consecutive parts of the multi-graded resolution have the following properties:

(1) the multi-graded decomposition corresponding to the second part of the resolution contains vertices stabilized by the terminal rigid (solid) multi-graded limit group of the resolution MGRes(s, z, y, r^{1}, p^{1}, a) connected to r′ surviving QH subgroups Q_{i_{1}}, …, Q_{i_{r′}} for some r′ ≤ r and 1 ≤ i_{1} < … < i_{r′} ≤ r.

(2) if the terminal multi-graded limit group of the multi-graded resolution MGRes(s, z, y, (r^{1}, p^{1}, Q_{1}, …, Q_{r}), a) is rigid (solid) so is the terminal multi-graded limit group of the multigraded resolution MGRes(s, z, y, r^{1}, p^{1}, a). Furthermore, if they are both solid, their principal multi-graded decompositions are in correspondence. i.e., the decomposition differ only in the stabilizer of the vertices stabilized by the subgroups P and R_{d}^{1}’s. These vertices are stabilized by the subgroups P^{1} and R_{d}^{1}’s and the subgroups Q_{i_{1}}, …, Q_{i_{r′}} in the first part, and by the subgroups P^{1} and R_{d}^{1}’s in the second.

Proof: Identical with the proof of theorem 1.8.

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Since the resolution Res(y, a) is well-separated, the fundamental groups of the various connected components of the graph of groups Θ_{1}^{1}, W_{1}^{1}, …, W_{μ_{1}}^{1}, are mapped to different factors of the third restricted limit group along the resolution Res(y, a), Rlim_{2}(y, a). In accordance with this free decomposition, the completed limit group Comp(Rlim)_{2}(z, y, a) admits a free decomposition: Comp(Rlim)_{2} = P^{1} ∗ R_{2}^{1} ∗ … ∗ R_{μ_{1}}^{1}. Furthermore, for every taut maximal rank homomorphism: h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))}, which is taut with respect to the resolution Res(y, a), and its associated homomorphism: ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))}, we have:

h(Rlim(y, a)) = ĥ(Comp(Rlim)(z, y, a)) = ĥ(Comp(Rlim)_{1}(z, y, a)) ∗ ĥ(H^{1}) = ĥ(Comp(Rlim)_{2}(z, y, a)) ∗ ĥ(H^{1}) ∗ ĥ(H^{2})

where H^{1} is the free group freely generated by free groups ”dropped” in the top level and H^{1} is the free group freely generated by free groups ”dropped” in the second level of the resolution Res(y, a) from the circumference of the various QH vertex groups that appear in these levels. Let T_{2}(s, z, y, r, p, a) be the terminal rigid or solid multi-graded limit group of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a). To continue the analysis of the resolutions associated with the quotient limit group QRlim(s, z, y, a), we need a ”linkage” between the free decomposition of the completed limit group Comp(Rlim)_{2}(z, y, a) and the free decomposition of T_{2}(s, z, y, r, p, a), a correspondence between (the ranks of) the free groups ”dropped” along the second level of the resolution Res(y, a) and along the various levels of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a), and a correspondence between the image ĥ(Comp(Rlim)_{2}(z, y, a) and the image h_{2}(T_{2}(s, z, y, r^{1}, p^{1}, a)) where h_{2} : T_{2}(s, z, y, r^{1}, p^{1}, a) → F_{k} ∗ F_{rk(Res(y,a))} is a rigid or solid shortening of a maximal rank taut homomorphism h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} which is taut with respect to both the closure Cl(Res)(s, z, y, a) and the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a).

Theorem 2.17. Let T_{2}(s, z, y, r^{1}, p^{1}, a) be the solid or rigid terminal limit group of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a), let h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} be a homomorphism which is taut with respect to the resolution Res(y, a), let ĥ : Comp(Rlim)(z, y, a) → F_{k}∗F_{rk(Res(y,a))} be the corresponding homomorphism of the completed limit group, and suppose the homomorphism ĥ factors through the closure Cl(Res)(s, z, y, a) and is taut with respect to the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a). Let h_{2} : T_{2}(s, z, y, r^{1}, p^{1}, a) → F_{k} ∗ F_{rk(Res(y,a))} be a rigid or solid homomorphism obtained from the homomorphism ĥ by the shortening procedure applied along the levels of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a). Then the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) can be extended in finitely many ways, to give finitely many multi-graded resolutions with respect to the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1}, which we still denote MGRes(s, z, y, r^{1}, p^{1}, a), so that the terminal rigid or solid multi-graded limit group T_{2}(s, z, y, r^{1}, p^{1}, a) of each of these multi-graded resolutions, which we still denote T_{2}(s, z, y, r^{1}, p^{1}, a), has the following properties:

(i) the summation of the ranks of the free groups dropped along the various levels of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) is identical to the summations of the ranks of the free groups dropped along the top two levels of the resolution Res(y, a).

(ii) by our assumptions the completed limit group, Comp(Rlim)_{2}(z, y, a) admits the free decomposition: Comp(Rlim)_{2}(z, y, a) = P^{1}∗R_{2}^{1}∗…∗R_{μ_{1}}^{1}. Then the terminal solid or rigid multi-graded limit group T_{2}(s, z, y, r^{1}, p^{1}, a) admits a free decomposition: T_{2}(s, z, y, r^{1}, p^{1}, a) = N_{1} ∗ … ∗ N_{μ_{1}} ∗ F where F is a free group having a rank which is equal to the summation of the ranks of the free groups drooped along the levels of the multi-graded resolution MGRes(s, z, y, r, p, a), P < N_{1} and for every d ≥ 2, R_{d} can be conjugated into N_{d}.

(iii) h_{2}(T_{2}(s, z, y, r^{1}, p^{1}, a)) = ĥ(Comp(Rlim)_{2}(z, y, a) and with the notation of

part (ii): h_{2}(N_{1}) = ĥ(P^{1}), and for every d ≥ 2: h_{2}(N_{d}) = ĥ(R_{d}^{1}). (iv) the limit group Rlim_{1}(y, a) admits a natural map into each multi-graded

limit group along the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a). Given the map τ_{1} : Rlim_{1}(y, a) → T_{2}(s, z, y, r^{1}, p^{1}, a), the image of every QH subgroup Q in the second level of the resolution Res(y, a) under the map τ_{1} admits a (possibly trivial) free decomposition from the free decomposition T_{2}(s, z, y, r^{1}, p^{1}, a) = N_{1} ∗ … ∗ N_{μ_{1}} ∗ F. Let S be the (punctured) surface corresponding to Q, and let Γ(S) be the decomposition of the surface S corresponding to the free decomposition of τ_{1}(Q). Let S_{1}, …, S_{m} be the punctured subsurfaces of S in the decomposition Γ(S) that contain a boundary component of S, let Ŝ = S \ (S_{1} ∩ … ∩ S_{m}), and let S′ be the closed surface obtained from Ŝ by adding disks to its boundary components. Then the maximal rank of a free quotient of π_{1}(S′) plus the number of Bass-Serre generators connecting Ŝ to other parts of the surface S is precisely the rank of the free group dropped from the QH subgroup Q in the second level of the resolution Res(y, a).

Proof: Identical to the proof of theorem 2.10.

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We continue the analysis of the limit group QRlim(s, z, y, a) by reduce the subgroups that are used for the construction of the multi-graded resolutions sequentially. Recall that in order to obtain the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) we first used the subgroups P, R_{2}, …, R_{μ} for constructing a multi-graded resolution, where Comp(Rlim)_{1}(z, y, a) = P ∗ R_{2} ∗ … ∗ R_{μ} and for the connected component W_{d} of the graph of groups Θ_{1}^{0}, η_{0}(W_{1}) = P and for d ≥ 2: η_{0}(W_{d}) = R_{d}. Then we used the subgroups P^{1}, R_{2}^{1}, …, R_{μ_{1}}^{1} for constructing a multi-graded resolution, where Comp(Rlim)_{2}(z, y, a) = P^{1} ∗ R_{2}^{1} ∗ … ∗ R_{μ_{1}}^{1} and for the connected component W_{d}^{1} of the graph of groups Θ_{1}^{1}, η_{1}(W_{1}^{1}) = P^{1} and for d ≥ 2: η_{1}(W_{d}^{1}) = R_{d}^{1}. To continue the analysis of the resolutions limit group QRlim(s, z, y, a), we set the subgroups P^{ℓ−1}, R_{1}^{ℓ−1}, …, R_{μ_{ℓ−1}}^{ℓ−1} to be the factors in the free decomposition: Comp(Rlim)_{ℓ}(z, y, a) = P^{ℓ−1} ∗ R_{1}^{ℓ−1} ∗ … ∗ R_{μ_{ℓ−1}}^{ℓ−1} where for the connected components W_{d}^{ℓ−1} of the graphs of groups Θ_{j}^{ℓ−1} associated with Rlim_{ℓ−1}(y, a), η_{ℓ−1}(W_{1}^{ℓ−1}) = P^{1} and for d ≥ 2: η_{ℓ−1}(W_{d}^{ℓ−1}) = R_{d}^{1}. The subgroups P^{ℓ−1}, R_{1}^{ℓ−1}, …, R_{μ_{ℓ−1}}^{ℓ−1} are used in the ℓ-th part of our sequential construction of multi-graded resolutions. Let T_{2}(s, z, y, r^{1}, p^{1}, a) be the terminal rigid or solid graded limit group in the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) which is multi-graded with respect to the subgroups: P^{1}, R_{1}^{1}, …, R_{μ_{1}}^{1}. We continue the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) by viewing T_{2}(s, z, y, r^{1}, p^{1}, a) as a multi-graded limit group with respect to the subgroups P^{2}, R_{1}^{2}, …, R_{μ_{2}}^{2} and continue with the procedure used to construct the second part of the multi-graded resolution MGRes(s, z, y, r^{1}, p^{1}, a) to obtain a multi-graded resolution MGRes(s, z, y, r^{2}, p^{2}, a). We further continue with the rigid or solid terminal limit group T_{3}(s, z, y, r^{2}, p^{2}, a) by viewing it as a multi-graded limit group with respect to the subgroups P^{3}, R_{1}^{3}, …, R_{μ_{3}}^{3} and so on, until we exclude all the z variables appearing in the different levels of the completed resolution Comp(Res)(z, y, a). Note that the final resolution we obtain is an ungraded resolution of the limit group QRlim(s, z, y, a), or of some quotient of it. Also, note that the procedure described produces (canonically) finitely many such resolutions of the quotient limit group QRlim(s, z, y, a). We will denote an (ungraded) resolution obtained by our procedure QRes(s, z, y, a), and call a quotient resolution of the quotient limit group QRlim(s, z, y, a).

Proposition 2.18. With the notation above the resolution QRes(s, z, y, a) is a well-separated resolution. Furthermore, if h : Rlim(y, a) → F_{k} ∗ F_{rk(Res(y,a))} is a taut maximal rank homomorphism with respect to the resolution Res(y, a), and its corresponding (completed) homomorphism ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗ F_{rk(Res(y,a))} factors through the closure Cl(Res)(s, z, y, a), then ĥ is a taut maximal rank homomorphism with respect to at least one of the well-separated (quotient) resolutions QRes(s, z, y, a) constructed by our procedure.

Proof: By construction, a quotient resolution QRes(s, z, y, a) is a strict resolution. Theorems 2.6, 2.12, and 2.17, guarantee that it is also a well-separated resolution. The quotient resolutions QRes(s, z, y, a) were constructed from the entire collection of (completed) taut maximal rank homomorphisms ĥ : Comp(Rlim)(z, y, a) → F_{k} ∗F_{rk(Res(y,a))} that factor through the closure Cl(Res)(s, z, y, a) and the quotient limit group QRlim(s, z, y, a), hence, every such (completed) taut maximal rank homomorphism ĥ is a taut maximal rank homomorphism with respect to at least one of the quotient resolutions QRes(s, z, y, a).

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To analyze taut maximal rank homomorphisms, we define the complexity of a resolution in the same way we have defined it in the minimal rank case (definition 1.15). In order to ensure the termination of the iterative procedure presented in the next sections we need the complexities of the resolutions obtained in its different steps to decrease. Indeed, the complexity of the (well-separated) resolutions QRes(s, z, y, a) obtained through our procedure for analyzing taut maximal rank homomorphisms is strictly smaller than the complexity of the resolution Res(y, a) we started with.

Theorem 2.19. The complexity of the resolution QRes(s, z, y, a) is strictly smaller than the complexity of the resolution Res(y, a), i.e.,

Cmplx(QRes(s, z, y, a)) < Cmplx(Res(y, a)).

Proof: Let Q′ be a QH subgroup that appears in a principal graded abelian decomposition associated with one of the levels of the quotient resolution QRes(s, z, y, a), and let S′ be its associated punctured surface. By lemmas 2.7, 2.13 and 2.14 there exists some QH subgroup Q with associated punctured surface S that appears in a cyclic JSJ decomposition associated with one of the levels of the original resolution Res(y, a), so that the fundamental group of a subsurface of S is mapped into a finite index subgroup of S′, and in particular: genus(S′) ≤ genus(S) and |χ(S′)| ≤ |χ(S)|. Furthermore, if genus(S′) = genus(S) and |χ(S′)| = |χ(S)| then Q is by definition a surviving surface, so by theorems 2.9 and 2.16 the QH subgroup Q is mapped isomorphically onto Q′, Q is mapped to the (distinguished) vertex group containing F_{k} in all levels above the one in which Q′ appears, and in all the levels below the one containing Q′, Q is identified with its image in the completed resolution Comp(Res)(z, y, a).

Hence, for any surviving QH subgroup Q with associated punctured surface S, corresponds a unique QH subgroup Q′ in the resolution QRes(s, z, y, a) with associated punctured surface S′, so that S is homeomorphic to S′. If Q′ is a QH subgroup in the quotient resolution QRes(s, z, y, a) with an associated punctured surface S′, and no surviving surface Q is mapped onto a conjugate of Q′, then for any QH subgroup Q with an associated surface S in Res(y, a), for which the fundamental group of some subsurface of S is mapped into a finite index subgroup of a conjugate of Q′, genus(S′) ≤ genus(S), |χ(S′)| ≤ |χ(S)|a and either genus(S′) < genus(S) or |χ(S′)| < |χ(S)|. Therefore, if Q_{1}, …, Q_{m} are the QH subgroups that appear in the original resolution Res(y, a) and S_{1}, …, S_{m} are their associated punctured surfaces, and Q′_{1}, …, Q′_{m′} are the QH subgroups that appear in the quotient resolution QRes(s, z, y, a) and S′_{1}, …, S′_{m′} are their associated punctured surfaces, then the tuple ((genus(S_{1}), |chi(S_{1})|), …, (genus(S_{m}), |chi(S_{m}|)) is less than or equal in the lexicographical order than the tuple ((genus(S′_{1}), |chi(S′_{1})|), …, (genus(S′_{m′}), |chi(S′_{m′}|)), and the two tuples are equal if and only if all the QH subgroups Q_{1}, …, Q_{m} are surviving QH subgroups. Since the quotient limit group QRlim(s, z, y, a) is a proper quotient of the closure Cl(Res)(s, z, y, a), if all the QH subgroups Q_{1}, …, Q_{m} are surviving, then necessarily: Abrk(QRes(s, z, y, a)) < Abrk(Res(y, a)), and we finally get that: Cmplx(QRes(s, z, y, a)) < Cmplx(Res(y, a)).

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§3. Induced Resolutions

In the first section we used an iterative procedure for the validation of a sentence, which assumes all the limit groups containing the set of the ”remaining” y’s are of minimal rank. In order to construct an iterative procedure for validation of a sentence in the general case, i.e., when the limit groups containing the set of the ”remaining” y’s are not necessarily of minimal rank, 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 a procedure for validation of a sentence in the general case. We start with the definitions of a geometric subresolution of a resolution Res(s, z, y, a) and its complexity.

Definition 3.1. Let Res(t, y, a) be a a well-separated resolution, and let Comp(Res)(u, t, y, a) be its completion. Let GSRes(g, y, a) be a resolution with the following properties:

(i) the resolution GSRes(g, y, a) is a well-separated resolution.

(ii) the resolution GSRes(g, y, a) is a completed resolution, i.e., the completion

of GSRes(g, y, a) is the resolution GSRes(g, y, a) itself. (iii) there exists a (geometric) embedding ν : GSRes(g, y, a) → Comp(Res)(u, t, y, a)

that maps the subgroup < y, a > of the resolution GSRes(g, y, a) into the subgroup < y, a > of the completed resolution Comp(Res)(u, t, y, a) elementwise. In addition, the embedding ν has the following properties:

1) every QH subgroup in an abelian decomposition associated with one of the various levels of the resolution GSRes(g, y, a) is embedded geometrically into (a finite index subgroup of) a QH subgroup of Comp(Res)(u, t, y, a).

2) every abelian vertex group in an abelian decomposition associated with one of the levels of the resolution GSRes(g, y, a) is embedded into an abelian vertex group in one of the abelian decompositions associated with Comp(Res)(u, t, y, a).

3) except for the terminal free groups dropped along the various levels of the geometric subresolution, the free and abelian decompositions associated with each level of the resolution GSRes(g, y, a) are the decompositions induced (using Bass-Serre theory) from the embedding ν and the free and abelian decompositions of Comp(QRes)(u, t, y, a). Furthermore, the canonical maps between consecutive levels in the resolution GSRes(g, y, a) are the ones induced from the canonical maps

between consecutive levels in the completion Comp(Res)(u, t, y, a).

We call the resolution GSRes(g, y, a) together with the embedding ν : GSRes(g, y, a) → Comp(Res)(u, t, y, a) a geometric subresolution of the (completion of the) resolution Res(t, y, a). The modular groups associated with a geometric subresolution are set to be the modular groups induced from those of the completed resolution Comp(Res)(u, t, y, a), hence, the modular groups associated with each of the QH vertex groups in the abelian decompositions of GSRes(g, y, a) are set to be the modular groups of the corresponding QH vertex groups of Comp(Res)(u, t, y, a) into which they are embedded. Note that the completion Comp(Res)(u, t, y, a) itself is a geometric subresolution of the (completion of the) resolution Res(t, y, a).

To analyze resolutions in the general case, we need to measure the complexity of geometric subresolutions of their completion. To get such a complexity measure, we modify definition 1.15 to take into account the reduced modular groups associated with the various QH subgroups.

Definition 3.2. Let Res(t, a) be a well-separated completed resolution with (possibly) reduced modular groups associated with each of its various QH subgroups. Let Q_{1}, …, Q_{m} be the QH subgroups that appear in the completion Comp(Res)(t, y, a) and let S_{1}, …, S_{m} be the (punctured) surfaces associated with the reduced modular group associated with each of the QH vertex group. To 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(Res(t, a)) be the rank of the resolution Res(t, a), and let Abrk(Res(t, a)) be the abelian rank of the resolution (see definition 1.15). We set the complexity of the resolution Res(t, a), denoted Cmplx(Res(t, a)), to be:

Cmplx(Res(t, a)) = (rk(Res(t, a)), (genus(S_{1}), |χ(S_{1})|), …, (genus(S_{m}), |χ(S_{m})|), Abrk(Res(t, a))).

On the set of completed resolutions with (possibly) reduced modular groups we can define a partial order. Let Res_{1}(t_{1}, a) and Res_{2}(t_{2}, a) be two completed resolutions with (possibly) reduced modular groups. We say that Cmplx(Res_{1}(t_{1}, a)) = Cmplx(Res_{2}(t_{2}, a)) if the tuples defining the two complexities are identical. We say that Cmplx(Res_{1})(t_{1}, a)) < Cmplx(Res_{2}(t_{2}, a)) if:

(1) the rank rk(Res_{1}(t_{1}, a)) is smaller than the rank rk(Res_{2}(t_{2}, a)).

(2) the above ranks 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 ranks and tuples are equal and Abrk(Res_{1}(t_{1}, a)) < Abrk(Res_{2}(t_{2}, a)).

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 in the next level by the canonical epimorphisms between consecutive levels.

From the completed resolution Comp(Res)(t, v, a) we construct the induced resolution, Ind(Comp(Res))(u, v, a), of the limit group < v, a > < Rlim(t, v, a) iteratively. We start by describing the first step in the iterative construction.

(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 lemma 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 contained 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 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, nonabelian vertex group in the graph of groups Λ_{1}. Repeating this operation for all the edges connecting two non-QH, non-abelian 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 non-abelian vertex groups in Δ_{1}^{i}, we get a subgroup V̂_{i} < Comp(Res)(t, v, a), so that V_{i} < 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 in definition 2.2, 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, non-abelian 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).

(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 in Θ̃_{1}^{i}, let Γ_{1}^{(i,s)} be the subgraph of the corresponding connected component of Θ̃_{1}^{i} obtained by deleting all the abelian vertex groups, i.e., the subgraph containing the nonabelian vertex group in that connected component and all the QH vertex groups in it. We set J_{s}^{i} to be the subgroup of the (unique) non-abelian vertex group in the corresponding connected component of Θ̃_{1}^{i} generated by the image under the map η_{1} of the fundamental group of the graph of groups Γ_{1}^{(i,s)}.

(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 V_{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 our iterative procedure for constructing the induced 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(QRes)(t, v, a) and applying steps (i)-(v) above. To start the second step in our iterative procedure for constructing the induced 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) a free group F < Ṽ_{i} which is an infinite index subgroup in some QH subgroup Q in the decomposition Λ_{1} may be increased to become a subgroup Q′, so that F < Q′ and Q′ is a subgroup of finite index in the QH subgroup Q in Λ_{1}. This clearly reduces the rank of the free factor of in the decomposition IΔ_{1} generated by free groups which are infinite index subgroups of QH vertex groups in Λ_{1}, and perhaps reduces the no. of factors in the decomposition IΔ_{1} in comparison with the decomposition Δ_{1}. Also in this case every factor V_{i} is contained in a factor in the decomposition IΔ_{1}.

(3) The number of free 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 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 for the construction of the induced 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).

Proposition 3.3. The iterative procedure for the construction of the induced resolution of the subgroup < v, a > < Comp(Res)(t, v, a) terminates.

Proof: Suppose that for some index n, the subgroup IRes_{n}(u, v, a) properly contains the subgroup IRes_{n−1}(u, v, a). The subgroup IRes_{n−1}(u, v, a) is obtained by starting with the subgroup IRes_{n−2}(u, v, a) and applying steps (i)-(v), and the subgroup IRes_{n}(u, v, a) is obtained by starting with the subgroup IRes_{n−1}(u, v, a) and applying steps (i)-(v). Since the subgroup IRes_{n−1}(u, v, a) is properly contained in the subgroup IRes_{n}(u, v, a), at least one of the decompositions constructed from IRes_{n−1}(u, v, a) through steps (i)-(v) is not similar to the corresponding decomposition constructed from IRes_{n−2}(u, v, a) through steps (i)-(v). By properties (1)-(3) above, if we restrict ourselves to the highest level decompositions constructed from IRes_{n−1}(u, v, a) which are not similar to the corresponding decompositions constructed from IRes_{n−2}(u, v, a), then either the decompositions constructed from IRes_{n−1}(u, v, a) have fewer factors, or the ranks of the free groups generated by Bass-Serre generators of edges with trivial stabilizers and free groups which are infinite index subgroups in QH vertex groups in the corresponding (level) abelian decomposition of the completed resolution Comp(Res)(t, v, a) are smaller in the decompositions constructed from IRes_{n−1}(u, v, a) than they are in the decompositions constructed from IRes_{n−2}(u, v, a), or in case there are equalities in the number of factors and the ranks of the corresponding free groups, then the combinatorics of the graphs of groups constructed from IRes_{n−1}(u, v, a) is smaller than the combinatorics of the graphs of groups constructed from IRes_{n−2}(u, v, a). This decrease in either the number of factors, the ranks of the free factors, or the combinatorics in the highest level decompositions which are not similar forces the iterative procedure for the construction of the induced resolution to terminate.

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Lemma 3.4. Let IRes(u, v, a) be the terminal resolution obtained by the above procedure. Then IRes(u, v, a) is a geometric subresolution of the completed resolution Comp(Res)(t, v, a).

Proof: The resolutions IRes_{n}(u, v, a) constructed in the various steps of the iterative procedure are all well-separated by construction. The terminal resolution IRes(u, v, a) is a completed resolution, since if we start with the subgroup IRes(u, v, a) and apply steps (i)-(v) we get the same subgroup (and resolution) IRes(u, v, a), which implies that IRes(u, v, a) is equivalent to its completion. Since starting with the subgroup IRes(u, v, a) and applying steps (i)-(v) gives us the subgroup IRes(u, v, a) and the resolution used to get it in the first place, the natural map of the resolution IRes(u, v, a) is a ”geometric” embedding, i.e., it satisfies all the properties listed in part (iii) of definition 3.1.

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Definition 3.5. Let IRes(u, v, a) be the terminal resolution obtained by the above procedure. We say that the resolution obtained from the resolution IRes(u, v, a) by replacing the modular groups associated with each of its QH vertex groups Q with its subgroup corresponding to the modular group of the QH vertex group Q′ of the completed resolution Comp(Res)(t, v, a) it is associated with, is the induced resolution of the subgroup < v, a > extracted from the completed resolution Comp(Res)(t, v, a). We denote the induced resolution Ind(Comp(Res))(u, v, a).

The terminating iterative procedure presented above associates with a given completed well-separated resolution Comp(Res)(t, v, a), and a subgroup < v, a > of the limit group Rlim(t, v, a), the induced resolution Ind(Comp(Res))(u, v, a). Let GRes(t, v, p, a) be a completed graded well-separated resolution, and let < v, p_{1}, a > be a subgroup of the graded limit group GRlim(t, v, p, a), where < p_{1} > is a subgroup of the parameter subgroup < p > < GRlim(t, v, p, a). Applying the procedure used to construct an induced resolution with an (ungraded) completed well-separated resolution and a subgroup of its limit group, we associate a graded induced resolution, GInd(GRes)(u, v, p_{1}, a), with the graded resolution GRes(t, v, p, a) and the subgroup < v, p_{1}, a >. As in the ungraded case, the procedure for the construction of the graded induced resolution guarantees that the graded induced resolution, GInd(GRes)(u, v, p_{1}, a), is canonically embedded into the graded resolution GRes(t, v, p, a). In the sequel, we will need the following basic property of the canonical embedding of the graded induced resolution.

Lemma 3.6. Let GRes(t, v, p, a) be a completed graded well-separated resolution, and let < v, p_{1}, a > be a subgroup of the graded limit group GRlim(t, v, p, a), where < p_{1} > is a subgroup of the parameter subgroup < p > < GRlim(t, v, p, a). Let GInd(GRes)(u, v, p_{1}, a) be the graded induced resolution associated with GRes(t, v, p, a) and the subgroup < v, p_{1}, a >, let T erm(t, v, p, a) be the terminal (rigid or solid) graded limit group of the graded resolution GRes(t, v, p, a), and let GIndT erm(u, v, p_{1}, a) be the terminal (graded) limit group of the graded induced resolution GInd(GRes)(u, v, p_{1}, a). By construction, the terminal limit group of the graded induced resolution, GIndT erm(u, v, p_{1}, a), is canonically embedded into the terminal graded limit group of GRes(t, v, p, a), T erm(t, v, p, a). Let ν : GIndT erm(u, v, p_{1}, a) → T erm(t, v, p, a) be that canonical embedding. Then the image of the canonical embedding, ν(GIndT erm(u, v, p_{1}, a)), is contained in the subgroup generated by < v, p_{1}, a > in the terminal limit group T erm(t, v, p, a).

Proof: Follows immediately from the properties of the procedure for the construction of the graded induced resolution.

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In our iterative procedure for validation of a sentence, as well as in our iterative procedure for quantifier elimination, we will need to be able to ”compare” resolutions constructed at various steps of the iterative procedures. One of the basic tools for making such a ”comparison” possible is the existence of a decorated closure.

Theorem 3.7. Let Res(t, y, a) be a well-structured resolution, let Comp(Res)(u, t, y, a) be its completion, and let Rlim(y, a) be the subgroup generated by < y, a > in the limit group Rlim(t, y, a) associated with the resolution Res(t, y, a). Let Res_{1}(y, a), …, Res_{r}(y, a) be the resolutions in the taut Makanin-Razborov diagram of the limit group Rlim(y, a). There exists a (canonical) covering closure Cl_{1}(Res)(s, u, t, y, a), …, Cl_{d}(s, u, t, y, a) of the resolution Res(t, y, a), and for each index i, 1 ≤ i ≤ d, there exists an associated resolution Res_{δ(i)}(y, a), which is one of the resolutions in the taut Makanin-Razborov diagram of the limit group Rlim(y, a), and an associated canonical map:

λ_{i} : Comp(Res_{δ(i)})(z, y, a) → Cl_{i}(Res)(s, u, t, y, a)

that satisfy the following.

(1) For every index i, 1 ≤ i ≤ d, and every index j, 1 ≤ j ≤ k, λ_{i}(a_{j}) = a_{j}.

(2) For every index i, 1 ≤ i ≤ d, and every index j, 1 ≤ j ≤ ℓ, λ_{i}(y_{j}) = y_{j}.

(3) Let {(u_{j}, t_{j}, y_{j}, a)}_{j=1}^{∞} be an arbitrary test sequence that factors through the completed resolution Comp(Res)(u, t, y, a). Then there exists some index i, 1 ≤ i ≤ d, and a subsequence of the test sequence (still denoted {(u_{j}, t_{j}, y_{j}, a)}_{j=1}^{∞}) for which there exists a corresponding sequence of specializations {(z_{j}, y_{j}, a)}_{j=1}^{∞} that factor through and are taut with respect to the completed resolution Comp(Res_{δ(i)})(z, y, a), and so that the combined sequence {(λ_{i}(z_{j}), u_{j}, t_{j}, y_{j}, a)}_{j=1}^{∞} factors through the closure Cl_{i}(QRes)(s, u, t, y, a).

We call each of the closures Cl_{i}(QRes)(w, s, z, y, a) a decorated closure, denoted DecorCl(s, u, t, y, a), and its associated resolution Res_{δ(i)}(y, a), its reference resolution, denoted RefRes(y, a).

Proof: Given a test sequence {(u_{j}, t_{j}, y_{j}, a)}_{j=1}^{∞} that factors through the completion Comp(Res)(u, t, y, a), the universality of the taut Makanin-Razborov diagram implies that it is possible to find a subsequence, (still denoted) {(u_{j}, t_{j}, y_{j}, a)}_{j=1}^{∞}, for which there exists a corresponding shortest form sequence of specializations {(z_{j}, y_{j}, a)}_{j=1}^{∞} that factor through and are taut with respect to the completed resolution Comp(Res_{δ(i)})(z, y, a). Hence, the same proof that was used to prove the existence of a formal solution ([Se2],1.17) gives a proof of theorem 3.7.

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Induced resolutions, graded induced resolutions, and decorated closures, are the additional tools needed for presenting the iterative procedure for validation of a sentence in the general case. During the attempts to find such a procedure, we have constructed several additional tools that eventually were not useful in constructing a (terminating) procedure for validation of a sentence, but these tools seem rather basic and natural to us, and they are likely to be helpful in approaching other problems, even closely related ones. It is our aim to present some of these tools in forthcoming papers, still in the rest of this section we decided to include some notions that seem basic to us, although they are not going to be used in the sequel.

Let Comp(Res)(t, v, a) be a well-separated completed resolution, and let < v, a > < Rlim(t, v, a). The induced resolution Ind(Comp(Res))(u, v, a) is the ”basic” geometric subresolution of Comp(Res)(t, v, a) that contains the subgroup < v, a >. However, for certain applications, which include the possibility to bound the complexity of resolutions by the complexity of their reference resolutions, the induced resolution needs to be modified. Besides the geometric requirement we will need our (completed) resolutions to be firm. Intuitively, we would like the rank of the geometric subresolution we work with to be equal to the rank of the free group corresponding to a generic specialization of the group < v, a > that factors through the ambient resolution Comp(Res)(t, v, a).

Definition 3.8. Let Comp(Res)(t, v, a) be a completed resolution, and let < v, a > < Rlim(t, v, a). Let r be the maximal rank of all free groups which are specializations of the subgroup < v, a > that can be extended to a specialization of the limit group Rlim(t, v, a) that factors through the completed resolution Comp(Res)(t, v, a). We call r the rank of the subgroup < v, a > with respect to the completed resolution Comp(Res)(t, v, a). A test sequence {t_{n}, v_{n}, a} of the completed resolution Comp(Res)(t, v, a) is called a firm test sequence of the subgroup < v, a > if for every index n the rank of the free group < v_{n}, a > is equal to the rank of < v, a > with respect to the completed resolution Comp(Res)(t, v, a).

Using firm test sequences we are able to define firm resolutions which are the type of resolutions one needs in order to control the complexity of resolutions by the complexity of their reference resolutions.

Definition 3.9. Let Comp(Res)(t, v, a) be a completed resolution, let < v, a > be a subgroup of the limit group Rlim(t, v, a), and let F rmGSRes(g, v, a) be a geometric subresolution of Comp(Res)(t, v, a). We say that the resolution F rmGSRes(g, v, a) is a firm subresolution of the completed resolution Comp(Res)(t, v, a) if it has the following properties:

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

(ii) there exists a firm test sequence of the subgroup < 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}.

Given a completed resolution Comp(Res)(t, v, a) and a subgroup < v, a >, the induced resolution, Ind(Comp(Res))(u, v, a), is the basic geometric subresolution of Comp(Res)(t, v, a), which is a resolution of the subgroup < v, a >. However, in general the induced resolution is certainly not a firm subresolution of Comp(Res)(t, v, a). Indeed, it may be that rk(Comp(Res)(t, v, a)) = 0 whereas the rank of the induced resolution, Ind(Comp(Res))(u, v, a) is arbitrarily large. Hence, for the purpose of controlling the complexity of the resolutions we need a different type of resolution. Instead of actually construct a firm subresolution we present the notion of an extremal subresolution.

Definition 3.10. Let Comp(Res)(t, v, a) be a completed resolution, and let < v, a > be a subgroup of the limit group Rlim(t, v, a). A geometric subresolution of Comp(Res)(t, v, a) that contains the subgroup < v, a > is called extremal, and denoted ExtrmRes(u, v, a), if for every geometric subresolution of Comp(t, v, a) that contains the subgroup < v, a >, GSRes(g, y, a):

Cmplx(ExtrmRes(u, v, a)) ≤ Cmplx(GSRes(g, y, a)).

Before studying some of the basic properties of extremal resolutions we need to show their existence.

Proposition 3.11. Let Comp(Res)(t, v, a) be a completed resolution, and let < v, a > be a subgroup of the limit group Rlim(t, v, a). There exists an extremal subresolution ExtrmRes(u, v, a) of the completed resolution Comp(Res)(t, v, a) that contains the subgroup < v, a >.

Proof: The completed resolution Comp(Res)(t, v, a) itself, and (by lemma 3.4) the induced resolution Ind(Comp(Res))(u, v, a) are geometric subresolutions of Comp(Res)(t, v, a) that contain the subgroup < v, a >, hence, the set of geometric subresolutions of Comp(Res)(t, v, a) that contain the subgroup < v, a > is nonempty. By the way the complexity of a resolution is defined (definition 3.2), any decreasing sequence of complexities of resolutions terminates in a finite time. Hence, any (strictly) minimizing sequence of geometric subresolutions of Comp(t, v, a) that contain the subgroup < v, a > terminates, so the set of geometric subresolutions of Comp(Res)(t, v, a) that contain the subgroup < v, a > contains extremal subresolutions.

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As we have already pointed out, induced resolutions need not be firm. Extremal resolutions must be. Although we do not use extremal resolutions in our iterative procedures for validation of a sentence and for quantifier elimination, we list few basic properties of these resolutions. The proof of these properties will be presented in a forthcoming paper.

Theorem 3.12. Let Comp(Res)(t, v, a) be a completed resolution, and let < v, a > be a subgroup of the limit group Rlim(t, v, a). Then:

(1) An extremal subresolution ExtrmRes(u, v, a) of the completed resolution Comp(Res)(t, v, a) that contains the subgroup < v, a > is a firm subresolution of Comp(Res)(t, v, a).

(2) Suppose < v, a >=< a > ∗ < v > where < v > is isomorphic to a free group F_{s} of rank s. Then either an extremal subresolution ExtrmRes(u, v, a) of the completed resolution Comp(Res)(t, v, a) that contains the subgroup < v, a > is the free group < a, v > itself, or the rank of ExtrmRes(u, v, a) is strictly smaller than s.

(3) Let ExtrmRes(u, v, a) be an extremal subresolution of the completed resolution Comp(Res)(t, v, a) that contains the subgroup < v, a >. Let Q be a conjugate of a QH subgroup in one of the abelian decompositions associated with the various levels of the completed resolution Comp(Res)(t, v, a). Then:

(i) If one of the free groups dropped along the various levels of the extremal resolution ExtrmRes(u, v, a) intersects Q in a subgroup of finite index, then the extremal resolution ExtrmRes(u, v, a) contains the entire QH subgroup Q.

(ii) If Q is a QH subgroup in Comp(Res)(u, v, a) from which a free factor is dropped in the next level (a floating QH vertex group), and ExtrmRes(u, v, a) intersects Q in a subgroup of finite index, then the extremal resolution ExtrmRes(u, v, a) contains the entire subgroup Q.

(4) Let Res(t, y, a) be a well-separated resolution, Comp(Res)(u, t, y, a) its completion, Rlim(y, a) the subgroup generated by < y, a > in Rlim(t, y, a), and Res_{1}(y, a), …, Res_{r}(y, a) the resolutions in the taut Makanin-Razborov diagram of the limit group Rlim(y, a). Let DecorCl_{1}(s, u, t, y, a), …, DecorCl_{d}(s, u, t, y, a) be its canonical set of decorated closures, and let Res_{δ(1)}(y, a), …, Res_{δ(d)}(y, a) be their associated reference resolutions. For each index i, 1 ≤ i ≤ d, let λ_{i} : Comp(Res_{δ(i)})(z, y, a) → DecorCl(s, u, t, y, a) be the canonical embedding of the completion of the reference resolution into its associated decorated closure. For each level m of the completion of the reference resolution, Comp(Res_{δ(i)})(z, y, a), let < z_{m}, a > be the subgroup associated with the m-th level of Comp(Res_{δ(i)})(z, y, a). Let < λ_{i}(z_{m}), a > be the image of the m-th level subgroup < z_{m}, a > in the decorated closure DecorCl(s, u, t, y, a). Let ExtrmRes(u_{m}, λ_{i}(z_{m}), a) be an extremal resolution of the subgroup < λ_{i}(z_{m}), a > in the decorated closure DecorCl(s, u, t, y, a).

For each index i, 1 ≤ i ≤ d, and each level m of the completion of the reference resolution Comp(Res_{δ(i)})(z, y, a), let ExtrmRes(u_{m}, λ_{i}(z_{m}), a) be an extremal resolution of the subgroup < λ_{i}(z_{m}), a > in the decorated closure DecorCl_{i}(s, u, t, y, a). There exists a test sequence {(s_{j}, u_{j}, t_{j}, y_{j}, a)}_{j=1}^{∞} that satisfies the following:

(i) the test sequence factors through all the decorated closures: DecorCl_{1}(s, u, t, y, a), …, DecorCl_{d}(s, u, t, y, a). (ii) the test sequence is a firm test sequence with respect to all the ex-

tremal resolutions ExtrmRes(u_{m}, λ_{i}(z_{m}), a) of the various subgroups < λ_{i}(z_{m}), a >.

(iii) there exists some index i, 1 ≤ i ≤ d, so that for every index j the specializations {(λ_{i}(z)_{j}, y_{j}, a)} are shortest form and taut with respect to the completion of the i-th reference resolution Comp(Res_{δ(i)})(z, y, a).

(5) With the notation of (4), there exists some index i, 1 ≤ i ≤ r, for which:

Cmplx(ExtrmRes(u, y, a)) ≤ Cmplx(Comp(Res_{i})(t, y, a))

and in case of equality, there exist extremal subresolutions so that for every level m of the completion Comp(Res_{i})(t, y, a), the structure of the extremal resolution ExtrmRes(u_{m}, λ_{i}(z_{m}), a)) is identical to the structure of the part of the completion Comp(Res_{i})(z, y, a) corresponding to the m-th level subgroup < z_{m}, a >. In particular, the complexity of ExtrmRes(u_{m}, λ_{i}(z_{m}), a) is identical to the complexity of the corresponding part of Comp(Res_{i})(z, y, a).

§4. An Iterative Procedure for Validation of a Sentence

In section 1 we used an iterative procedure for validation of a sentence, assuming the restricted limit groups in question are of minimal rank. In section 2 we modified this procedure to analyze collections of maximal rank homomorphisms which are taut with respect to a given (well-separated) resolution. In section 3 we studied induced resolutions and decorated closures of a resolution. To obtain termination of an iterative procedure for validation of a sentence, which is crucial for our quantifier elimination ”trial and error” procedure, we must show that either the Zariski closures or the complexities of the resolutions associated with the set of specializations for which the validity of the sentence was not yet demonstrated, strictly decreases after finitely many steps of the procedure. To get the desired strict decrease in either the complexity of the associated resolutions or the associated Zariski closures, we combine the tools, ideas and concepts of the (terminating) procedure in the minimal rank case presented in section 1, with some additional properties of the (canonical) JSJ decomposition and the construction of the induced resolution presented in the previous section. To get this combination we are forced to construct certain ”data structure” that keeps track of all the previous steps of the procedure, and not only the last one.

Let:

∀y ∃x Σ(x, y, a) = 1 ∧ Ψ(x, y, a) ≠ 1

be a truth sentence over F_{k}. Let F_{y} =< y_{1}, …, y_{ℓ} > be the free group with a free basis y_{1}, …, y_{ℓ}. By proposition 1.1 of [Se2] there exists a formal solution x = x(y, a), and a finite set of restricted limit groups: Rlim_{1}(y, a), …, Rlim_{m}(y, a) for which:

(i) the words corresponding to the equations in the system Σ(x(y, a), y, a) represent the trivial word in the free group F_{k} ∗ F_{y}.

(ii) for every index i, Rlim_{i}(y, a) is a proper quotient of the free group F_{k} ∗ F_{y}. (iii) let B_{1}(y), …, B_{m}(y) be the basic sets corresponding to the restricted limit

groups Rlim_{1}(y, a), …, Rlim_{m}(y, a). If y ∉ B_{1}(y) ∪ … ∪ B_{m}(y), and ψ_{j}(x, y, a) is a word corresponding to one of the equations in the system Ψ(x, y, a), then: ψ_{j}(x(y, a), y, a) ≠ 1.

Proposition 1.1 of [Se2] gives a formal solution that proves the validity of the given sentence on a co-basic set (F_{k})^{ℓ} \ (B_{1}(y) ∪ … ∪ B_{m}(y)), hence, the rest of the procedure needs to construct formal solutions that prove the validity of the sentence on the remaining basic sets B_{1}(y), …, B_{m}(y). Since our treatment of the restricted limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a) which define the basic sets B_{1}(y), …, B_{m}(y) is independent, we will continue with one of these limit groups, which for brevity we denote Rlim(y, a). Note that Rlim(y, a) is a proper quotient of the free group F_{k} ∗ F_{y}, hence, the rank of every resolution in the (taut) Makanin-Razborov diagram of Rlim(y, a) is strictly less than ℓ, the rank of the free group F_{y}. Let Rlim(y, a) be one of the restricted limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a). By proposition 2.5 with Rlim(y, a) one associates the (canonical) taut Makanin-Razborov diagram. Let Res_{1}(y, a), …, Res_{r}(y, a) be the resolutions in this taut diagram. Each resolution in the diagram is a well-separated Makanin-Razborov resolution of either the restricted limit group Rlim(y, a) or of a proper quotient of it, and every specialization of the restricted limit group Rlim(y, a) is taut with respect to at least one of the resolutions Res_{1}, (y, a), …, Res_{r}(y, a). Since every resolution in the taut Makanin-Razborov diagram is a well-separated resolution, with each resolution Res_{i}(y, a) we can associate its completed resolution Comp(Res_{i})(z, y, a). By lemma 1.13 of [Se2] every specialization y_{0} that factors through the resolution Res_{i}(y, a), there exists some (shortest form) specialization (z_{0}, y_{0}, a) that factors through the completed resolution Comp(Res_{i})(z, y, a).

I: The first step

As in the minimal rank case, the continuation of the iterative procedure in the general case is conducted independently for the different limit groups and each of their resolutions. Hence, for the rest of the procedure, we will denote the restricted limit group in question Rlim(y, a), and its (well-separated) resolution Res(y, a) omitting their indices. By theorem 1.17 of [Se2] from the validity of our given sentence, we get the existence of a covering closure Cl_{1}(Res)(s, z, y, a), …, Cl_{q}(Res)(s, z, y, a) of the resolution Res(y, a), and for each index 1 ≤ n ≤ q there exists a formal solution x_{n}(s, z, y, a) for which:

(i) the words corresponding to the equations in the system Σ(x_{n}(s, z, y, a), y, a) represent the trivial word in the closure Cl_{n}(Res)(s, z, y, a).

(ii) there exists some specialization (s_{0}^{n}, z_{0}^{n}, y_{0}^{n}) for which Ψ(x_{n}(s_{0}^{n}, z_{0}^{n}, y_{0}^{n}, a), y_{0}^{n}, a) ≠ 1.

Definition 4.1. Suppose the closure Cl_{n}(Res)(s, z, y, a) contains ℓ levels, and the limit group associated with the i-th level in the resolution Res(y, a) is Rlim(z_{i}, a). With the closure Cl_{n}(Res)(s, z, y, a), we associate a taut graded Makanin-Razborov diagram with respect to the parameter subgroup < z_{1}, a >=< y, a >. We call each of the resolutions in this graded diagram an auxiliary resolution, and its terminating solid or rigid limit group an auxiliary limit group, which we denote: AuxCl(Res)(s, z, y, a). Note that an auxiliary limit group is a quotient of the closure Cl_{n}(Res)(s, z, y, a) we have started with. With an auxiliary limit group we further associate it’s graded decompositions with respect to the various parameter subgroups < z_{1}, …, z_{j}, a >, which we call the auxiliary graded decompositions. To the list of the above auxiliary resolutions, limit groups, and graded decompositions, we add the list of (graded) resolutions corresponding to the singular locus of all possible graded auxiliary decompositions. We call such a resolution an auxiliary resolution as well. With each such resolution we associate its corresponding auxiliary limit group and auxiliary graded decomposition.

Let a closure Cl_{n}(s, z, y, a), a formal solution x_{n}(s, z, y, a) defined over it, an auxiliary resolution, and its associated auxiliary limit group AuxCl(Res)(s, z, y, a) be given. We can clearly use the modular groups associated with the various levels of the auxiliary resolution to construct a family of formal solution x_{α}^{n}(s, z, y, a) defined over the auxiliary limit group AuxCl(Res)(s, z, y, a). In the steps of the iterative procedure for validation of a sentence we present, we always start with this setup, which includes a closure, one of its auxiliary resolutions, its associated auxiliary limit group, and a family of formal solutions associated with the auxiliary resolution.

The first formal solution defined over the free group F_{y} =< y_{1}, …, y_{ℓ} > proves the validity of our given sentence on a co-basic set of y’s, and we are still required to prove the validity of the sentence on the basic set, consists of all the specializations of the collection of limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a). The formal solutions described above for each of the closures Cl_{n}(Res)(s, z, y, a), are defined using the sets of variables (s, z, y, a). Hence, they prove the validity of the given sentence on the intersection of modular blocks with certain co-Diophantine sets, defined by the various sets of variables (s, z, y, a).

To analyze the remaining set of y’s, we construct an iterative procedure that produces a sequence of well-separated resolutions and their completions. The iterative procedure for analyzing general resolutions is based on the procedures presented in the first two sections, but the analysis of the set of the remaining y’s in each step of the procedure needs to be modified to guarantee (eventual) strict reduction in complexity. Given a resolution Res(y, a), we go over all the indices n, 1 ≤ n ≤ q, and each of the auxiliary resolutions associated with the closure Cl_{n}(Res)(s, z, y, a) and their associated auxiliary limit groups AuxCl(Res)(s, z, y, a), and analyze all the specializations (s_{0}, z_{0}, y_{0}, a) that are taut with respect to the closure Cl_{n}(Res)(s, z, y, a) (hence, in particular, factor through Cl_{n}(Res)(s, z, y, a)), for which the tuple (s_{0}, z_{0}, y_{0}, a) factors through the auxiliary limit group AuxCl(Res)(s, z, y, a) and satisfies the system of equations:

Σ(x_{α}^{n}(s_{0}, z_{0}, y_{0}, a), y_{0}, a) = 1 ∧ ψ_{j}(x_{α}^{n}(s_{0}, z_{0}, y_{0}, a), y_{0}, a) = 1

for some index j and every index α. Note that since the equation: ψ_{j}(x_{n}(s_{0}, z_{0}, y_{0}, a), y_{0}, a) = 1, defined over the set of specializations that factor through the closure Cl_{n}(Res)(s, z, y, a) is a non-trivial equation by construction, the system of equations: ψ_{j}(x_{α}^{n}(s_{0}, z_{0}, y_{0}, a), y_{0}, a) = 1 (for all possible values of the parameter α), is a non-trivial system of equations defined on the set of specializations that factor through the auxiliary limit group AuxCl(Res)(s, z, y, a), in case the limit group associated with the corresponding auxiliary resolution and the associated modular groups is the closure itself (i.e., not a proper quotient of the closure). Clearly, if an element y_{0} ∈ F_{k} factors through the resolution Res(y, a), and y_{0} does not belong to the Diophantine sets defined by the above systems of equations, then the sentence is valid for y_{0}.

If we fix the index n and the auxiliary resolution, and the index j of the equation ψ_{j}(x, y, a) = 1, then the entire collection of (shortest form) specializations (s_{0}, z_{0}, y_{0}, a) that factor through the auxiliary limit group AuxCl(Res)(s, z, y, a) and satisfy the corresponding system of equations is contained in a finite set of maximal limit groups QRlim_{1}^{n}(s, z, y, a), …, QRlim_{u_{n}}^{n}(s, z, y, a). Since our analysis of these quotient limit groups is conducted in parallel for the different tuples of indices (n, t), we will omit the indices from the quotient limit groups and denote them QRlim(s, z, y, a).

The resolution Res(y, a) is well-separated, so by construction, the subgroup of the auxiliary limit group Aux(Cl)(s, z, y, a) generated by < s_{2}, z_{2}, …, s_{ℓ}, z_{ℓ} >, i.e., the subgroup generated by the variables associated with all the levels in the closure Cl(Res)(s, z, y, a) except the top level, decomposes into a free product, where each factor corresponds to a stabilizer of a non-abelian, non-QH vertex group in the abelian decomposition associated with the top level of the completion, Comp(Res)(z, y, a). Let Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} be these factors of the subgroup generated by: < s_{2}, z_{2}, …, s_{ℓ}, z_{ℓ} >. As we did in sections 1 and 2, with the quotient limit group QRlim(s, z, y, a) we associate canonically the multi-graded taut Makanin-Razborov diagram with respect to the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}. Let:

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)

be the quotient multi-graded resolutions in this taut Makanin-Razborov diagram.

We continue with a subset of the resolutions that appear in the multi-graded taut Makanin-Razborov diagram of the quotient limit group QRlim(s, z, y, a), a subset through which all specializations (y_{0}, a) that is taut with respect to Res(y, a) and for which there exists a specialization (s_{0}, z_{0}, y_{0}, a) that factor through QRlim(s, z, y, a) do factor, and not with all the quotient multi-graded resolutions in this taut multigraded diagram. If the subgroup of generated by < y, a > of the limit group < s, z, y, a > associated with a quotient multi-graded resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is a proper quotient of the limit group Rlim(y, a) associated with the resolution Res(y, a) we have started this branch of the first step of the procedure, we include the quotient multi-graded resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in the subset we continue with. Hence, to define the other quotient multi-graded resolutions in the subset we continue with, we may assume that the subgroup generated by < y, a > in the limit group associated with a multi-graded resolution is isomorphic to the limit group Rlim(y, a) associated with the resolution Res(y, a).

For each QH vertex group Q in the abelian decomposition associated with the top level of the closure Cl_{n}(Res)(s, z, y, a), the boundary elements of Q can be conjugated (in the limit group associated with the closure Cl_{n}(Res)(s, z, y, a)) into one of subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}. Since the closure Cl_{n}(Res)(s, z, y, a) is canonically mapped onto the quotient limit group QRlim(s, z, y, a), each of the QH subgroups Q in the abelian decomposition associated with the top level of Cl_{n}(Res)(s, z, y, a) are naturally mapped into the quotient limit group QRlim(s, z, y, a), hence, the QH subgroup Q and its corresponding quotients naturally inherits a sequence of abelian decompositions from the multi-graded abelian decompositions associated with the various levels of a multi-graded resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}), abelian decompositions in which the boundary elements of Q are all elliptic. The resolution Res(y, a) was assumed to be well-separated, hence, on each of the QH vertex groups Q associated with its various levels, there exists an additional indication for a collection of s.c.c. on its associated surface S that is mapped to a trivial element in the limit group associated with the next level of the resolution Res(y, a). Given a multi-graded resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}), if for some QH vertex group Q in the abelian decomposition associated with the top level of the closure Cl_{n}(Res)(s, z, y, a), the sequence of abelian decompositions Q and its corresponding quotients inherit from the multi-graded resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}), is not compatible with the specific indication of the collection of s.c.c. on the surface S associated with the QH vertex group Q that are mapped to the trivial element in the resolution Res(y, a), we omit the quotient multi-graded resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) from our list of quotient multi-graded resolutions. Otherwise, we include the quotient multi-graded resolution in the subset of multi-graded resolutions we continue with. By construction, every specialization (y_{0}, a) that factors and is taut with respect to the resolution Res(y, a) and for which there is a specialization (s_{0}, z_{0}, y_{0}, a) that factors through the closure Cl_{n}(Res)(s, z, y, a), there exists a quotient multi-graded resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) that is included in the subset of quotient multi-graded resolutions we continue with, and for which the specialization (s_{0}, z_{0}, y_{0}, a) factors through and is taut with respect to that quotient multi-graded resolution.

Let Q_{1}, …, Q_{r} be the QH vertex groups in the abelian decomposition associated with the top level of the resolution Res(y, a) we have started with, i.e., the QH vertex groups in the abelian JSJ decomposition of Rlim(y, a). By the same arguments used in proving theorems 1.8 and 2.9, we can assume that each multigraded quotient resolution MGQRes_{i}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) terminates in a multi-graded abelian limit group with a multi-graded abelian decomposition containing the entire collection of surviving QH vertex groups Q_{i_{1}}, …, Q_{i_{r′}}, and these surviving QH subgroups are either closed surface subgroups or (in case they correspond to punctured surfaces) they mapped to their image in the subgroup < z_{2}, a > in either a rigid or solid multi-graded limit group with respect to the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}. Furthermore, the surviving QH vertex groups Q_{i_{1}}, …, Q_{i_{r′}} are subgroups of the non-abelian, non-QH vertex groups, the ones stabilized by the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, in all levels above the two terminating ones (see theorem 2.9). A basic property of a multi-graded quotient resolution, that was proved for the entire resolution (and not just its top part) in section 1 under the minimal rank assumption, is the following property. It’s one of the main properties used in our approach to validation of a sentence.

Proposition 4.2. Suppose the subgroup generated by < y, a > in the limit group < s, z, y, a > associated with a multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is isomorphic to the limit group Rlim(y, a) associated with the resolution Res(y, a) we have started the first step with. Then the complexity of each of the multigraded abelian decompositions associated with the various levels of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is bounded by the complexity of the abelian decomposition associated with the top level of the completion Comp(Res)(z, y, a). In case of equality, the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) has only one level and its structure is identical to the structure of the abelian decomposition associated with the top level of the completion Comp(Res)(z, y, a).

Proof: Since the sequence of abelian decompositions inherited by each of the QH vertex groups Q in the abelian decomposition associated with the top level of Res(y, a), from the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), are compatible with the indicated collection of s.c.c. that are mapped to the trivial element in the resolution Res(y, a), the (multi-graded) rank of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is bounded by the rank of the free factor that is dropped at the top level of the completion Comp(Res)(z, y, a). If the rank of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is strictly smaller than the rank of the free factor dropped at the top level of the completion Comp(Res)(z, y, a), the proposition follows. If these ranks are equal, the proposition follows from our analysis of taut homomorphisms of maximal rank (theorem 2.10).

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Proposition 4.2 shows that, in particular, the complexity of each of the abelian decompositions associated with the various levels of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is strictly smaller than the complexity of the resolution Res(y, a) we have started with. A multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) has few other basic properties which are crucial in our approach.

Proposition 4.3. Let MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be one of the multigraded quotient resolutions, for which the subgroup generated by < y, a > in the limit group < s, z, y, a > associated with that multi-graded resolution is isomorphic to the limit group Rlim(y, a) associated with the resolution Res(y, a). By construction, the original limit group Rlim(y, 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_{j}(y, a) be the image of Rlim(y, a) in the limit group associated with the j-th level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Then:

(i) If one of the abelian decomposition associated with a higher level than the j-th level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is not of minimal rank, then either Q_{j}(y, a) is a proper quotient of Rlim(y, a), or Q_{j}(y, a) inherits a non-trivial free decomposition from the free decomposition 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), or for each specialization y_{0} for which there exists a specialization (s_{0}, z_{0}, y_{0}, a) that factors through the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), the specialization (s_{0}, z_{0}, y_{0}, a) factors through an auxiliary resolution of higher complexity than the auxiliary resolution we have started with.

(ii) Let Q_{term}(y, a) be the image of Rlim(y, a) in the terminal (rigid or solid) multi-graded limit group of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). If the limit group Rlim(y, a) is not the (coefficient) free group F_{k} =< 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 a collection of finitely many terminal multi-graded limit groups in which Q_{term}(y, a) is a proper quotient of Rlim(y, a).

(iii) Suppose that for some level j, Q_{j}(y, a) is isomorphic to Rlim(y, a). Let MGQlim_{j}(s, z, y, a) be the quotient 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). Then either the (ungraded) abelian JSJ decomposition of MGQlim_{j}(s, z, y, a) is obtained from the multi-graded abelian decomposition 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) together with the auxiliary decomposition associated with the auxiliary resolution we have started with, or each of the specializations (s_{0}, z_{0}, y_{0}, a) that factor through and are taut with respect to the closure Cl(Res)(s, z, y, a) and the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) factor through an auxiliary resolution of higher complexity.

Proof: Suppose that the abelian decomposition associated with level ℓ, that lies above level j of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), is not of minimal rank, and let QRlim_{ℓ}(s, z, y, a) be the limit group associated with that level. If the abelian decomposition associated with level ℓ collapses to a non-trivial free decomposition of QRlim_{ℓ}(s, z, y, a), then either Q_{ℓ}(y, a) inherits a non-trivial free decomposition from the free decomposition associated with the ambient limit group QRlim_{ℓ}(s, z, y, a), or it inherits a trivial free decomposition, which implies that for every specialization y_{0} for which there exists a specialization (s_{0}, z_{0}, y_{0}, a) that factors through the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), the specialization (s_{0}, z_{0}, y_{0}, a) factors through a non-trivial free product in which the subgroup < y > is mapped into one of the factors, hence, the specialization (s_{0}, z_{0}, y_{0}, a) is not rigid or solid with respect to the subgroup < y >, so it factors through an auxiliary resolution of higher complexity than the auxiliary resolution we have started with. If the abelian decomposition associated with QRlim_{ℓ}(s, z, y, a) do not collapse to a non-trivial free decomposition, since the abelian decomposition associated with the ℓ-th level is not of minimal rank, QRlim_{ℓ}(s, z, y, a) maps canonical into a limit group of the form QRlim_{ℓ+1}(s, z, y, a) ∗ H^{ℓ+1} (where H^{ℓ+1} is free, and perhaps trivial), with an associated abelian decomposition that collapses to a non-trivial free decomposition. Again, either Q_{ℓ}(y, a) inherits a non-trivial free decomposition from the free decomposition associated with the ambient limit group QRlim_{ℓ}(s, z, y, a), or it inherits a trivial free decomposition, which implies that for every specialization y_{0} for which there exists a specialization (s_{0}, z_{0}, y_{0}, a) that factors through the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), the specialization (s_{0}, z_{0}, y_{0}, a) factors through a non-trivial free product in which the subgroup < y > is mapped into one of the factors, hence, the specialization (s_{0}, z_{0}, y_{0}, a) is not rigid or solid with respect to the subgroup < y >, so it factors through an auxiliary resolution of higher complexity, and we get part (i) of the proposition.

To prove (ii), note that if Rlim(y, a) is a free product of free groups and (closed) surface groups, then Q(y, a) is necessarily a proper quotient of Rlim(y, a). If Rlim(y, a) is a free product of free groups, free abelian groups and (closed) surface groups, then Q_{term}(y, a) is necessarily the (coefficient) free group F_{k} =< a >, and since Rlim(y, a) not the coefficient free group F_{k}, Q_{term}(y, a) = F_{k} is a proper quotient of Rlim(y, a). Also, note that Q_{term}(y, a) is a quotient of Q(y, a), the limit group generated by < y, a > in the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Hence, to prove part (ii) we may assume that Q(y, a) is isomorphic to Rlim(y, a), and the groups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} are not trivial. For each subgroup Base_{2,j}^{1} let DBase_{2,v_{j}}^{1} be the subgroup of Base_{2,j}^{1} that stabilizes the corresponding distinguished vertex (i.e., the vertex group in the decomposition inherited by Base_{2,j}^{1} from the auxiliary decomposition that intersects Q(y, a) non-trivially).

Suppose that Q_{term}(y, a) is isomorphic to Rlim(y, a), and let T erm(s, z, y, 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 T erm(s, z, y, a) we associate the resolutions that appear in its (ungraded) taut Makanin-Razborov diagram. We treat these resolutions in parallel, so let T Res(s, z, y, a) be one of these resolutions. Let Λ_{top} be the abelian decomposition associated with the top level of the (ungraded) resolution T Res(s, z, y, a). Q_{term}(y, a) inherits an abelian decomposition from Λ_{top}. Since Q_{term}(y, a) is isomorphic to Rlim(y, a), the non-QH, non-abelian vertex groups in the abelian JSJ decomposition of Q_{term}(y, a) have to be elliptic in the abelian decomposition inherited from Q_{term}(y, a) from Λ_{top}. Hence, if any of the subgroups DBase_{2,j}^{1} is not elliptic in the abelian decomposition Λ_{top}, then for each specialization y_{0}, for which there exists a specialization (s_{0}, z_{0}, y_{0}, a) that factors and is taut with respect to the 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 T Res(s, z, y, a), the specialization (s_{0}, z_{0}, y_{0}, a) factors through an auxiliary resolution of higher complexity than the auxiliary resolution we have started with. In this case we can omit the resolution T Res(s, z, y, a) from our list of (ungraded) resolutions of the terminal limit group T erm(s, z, y, a), hence, for the continuation we may assume that each of the subgroups DBase_{2,j}^{1} is elliptic in the abelian decomposition Λ_{top}.

Since the subgroups DBase_{2,j}^{1} are elliptic in Λ_{top}, each of the other non-abelian, non-QH vertex groups and all the edge groups in the abelian decomposition inherited by the subgroups Base_{2,j}^{1} from the auxiliary decomposition have to be 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 the Λ_{top} then for each specialization y_{0}, for which there exists a specialization (s_{0}, z_{0}, y_{0}, a) that factors and is taut with respect to the 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 T Res(s, z, y, a), the specialization (s_{0}, z_{0}, y_{0}, a) factors through a multi-graded resolution MGQRes^{1}(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 nonabelian, non-QH vertex groups and all the edge groups in the abelian decomposition inherited by each of the subgroups Base_{2,j}^{1} from the graded auxiliary decomposition, 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 multigraded 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 T erm(s, z, y, a) into the limit group associated with the second level of the (ungraded) resolution T Res(s, z, y, a). If the image of Q_{term}(y, a) in that limit group is a proper quotient of Q_{term}(y, a), we replace the limit group T erm(s, z, y, a) with the limit group associated with the second level of T Res(s, z, y, a) and the claim of part (ii) follows. If the image of Q_{term}(y, a) in the limit group associated with the second level of T Res(s, z, y, a) is isomorphic to Q_{term}(s, z, y, a), we apply the same argument to show that the multigraded 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 T erm(s, z, y, a) into the limit group associated with the third level of the (ungraded) resolution T Res(s, z, y, a). Applying this argument inductively, finishes the proof of part (ii).

To prove part (iii), let Λ_{j} be the (ungraded) abelian JSJ decomposition of MGQlim_{j}(s, z, y, a), and let Θ_{j} be the multi-graded abelian decomposition 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).

Since, by the assumptions of part (iii), Q_{j}(s, z, y, a) is isomorphic to Rlim(y, a), every non-abelian, non-QH vertex group in the abelian JSJ decomposition of Rlim(y, a) has to be elliptic in both abelian decompositions Λ_{j} and Θ_{j}. With the notation used to prove part (ii), if any of the subgroups DBase_{2,1}^{1}, …, DBase_{2,v_{1}}^{1} is not elliptic in the abelian decomposition Λ_{j}, then for each specialization y_{0} for which there exists a specialization (s_{0}, z_{0}, y_{0}, a) that factors through the top j levels of the multi-grade resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), the specialization (s_{0}, z_{0}, y_{0}, a) factors through an auxiliary resolution of higher complexity than the auxiliary resolution we have started with. If any non-abelian, non-QH vertex group in a decomposition inherited by one of the subgroups Base_{2,i}^{1} from the auxiliary decomposition is not elliptic in Λ_{j}, then the same conclusion applies. Therefore, the abelian JSJ decomposition Λ_{j} is obtained from the multi-graded abelian decomposition Θ_{j} by refining it using the auxiliary decompositions associated with each of the subgroups MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), and part (iii) follows.

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In section 1 we have started the procedure for validation of a sentence with a (minimal rank) modular block, and showed that the complexities of the various (quotient) modular blocks associated with it 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 modular blocks in each step of our iterative procedure. To ”force” the ”size” of the set of the remaining specializations y_{0} to actually decrease, we need to associate with each modular block information about certain (multi-graded) resolutions and abelian decompositions associated with it together with Zariski closures of some subgroups associated with the modular block. To carry all the information attached to a modular block, we associate a staggered data-structure, and (canonical) resolutions which we call developing resolutions, with each of the quotient resolutions QRes(s, z, y, a), which are associated, or rather derived, from the various multi-graded quotient resolutions MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a).

We construct the staggered data-structure and developing resolutions iteratively. We start with the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), and continue with Zariski closures and (multi-graded) resolutions of (the image of) various subgroups of the completion Comp(Res)(z, y, a) inherited from the multigraded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) and (multigraded quotient) resolutions derived from it. We describe the iterative procedure for constructing the staggered data-structure and developing resolutions starting with the initial step, the second step, and then the general step.

(1) Let Rlim(y, a) be the restricted limit group we have started with. Let Q(y, a) be the limit group generated by < y, a > 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). If Q(y, a) is a proper quotient of the subgroup Rlim(y, a), we continue this branch of the iterative procedure, by replacing the limit group Rlim(y, a) by Q(y, a), and continue with the finite collection of resolutions that appear in the taut Makanin-Razborov diagram of the limit group Q(y, a) in the same way we handled the resolutions in the taut Makanin-Razborov diagram of Rlim(y, a). Note that this is always the case if the limit group we have started with, Rlim(y, a), is a free group.

(2) At this stage we may assume that Q(y, a) is isomorphic to Rlim(y, a). At this part we will also assume that the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is not of maximal complexity, i.e., that the complexities of the abelian decompositions associated with its various levels are strictly smaller than the complexity of the abelian decomposition associated with the top level of the resolution Res(y, a). To continue handling a multi-graded quotient resolutions MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) which are not of maximal complexity, we need the following observation.

Proposition 4.4. Let MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be a multi-graded quotient resolution which is not of maximal complexity. By construction, the original limit group Rlim(y, a) and the subgroup QRlim(s, z, y, a) are mapped into the limit group MGQlim_{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) associated with the second level of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Let Q_{2}(y, a) and Q_{2}(s, z, y, a) be the images of Rlim(y, a) and QRlim(s, z, y, a) in the limit group MGQlim_{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Then either Q_{2}(y, a) is a proper quotient of Rlim(y, a), or Q_{2}(s, z, y, a) is a proper quotient of the subgroup QRlim(s, z, y, a).

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

□

By proposition 4.4 either the image of Rlim(y, 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}(y, a), is a proper quotient of Rlim(y, a), or the image of QRlim(s, z, y, 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}(s, z, y, a), is a proper quotient of QRlim(s, z, y, a). Suppose that the subgroup Q_{2}(y, a) is a proper quotient of Rlim(y, a). Let QRes_{1}(y, a), …, QRes_{q}(y, a) be the resolutions in the taut Makanin-Razborov diagram of Q_{2}(y, a). We continue with each of the resolutions QRes_{i}(y, a) separately. With the resolution QRes(y, a) we associate a resolution CRes(y, a), where CRes(y, a) is composed from two parts, the top being the resolution induced by the subgroup < y, a > from the completion of the top level of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), and the tail being the resolution QRes(y, a). By theorem 1.17 of [Se2], with the resolution CRes(y, a) we can associate a set of closures Cl_{1}(CRes)(u, y, a), …, Cl_{d}(CRes)(u, y, a) and for each closure a corresponding formal solution defined over it. We continue with each of these closures in parallel. With a closure Cl(CRes)(u, y, a), we associate its auxiliary resolutions and with each auxiliary resolution we associate its corresponding auxiliary limit group and auxiliary multi-graded decomposition. We continue with each of the auxiliary resolutions in parallel. Fixing a closure Cl(u, y, a) and one of its associated auxiliary limit groups we set the developing resolution to be the auxiliary resolution. If the subgroup generated by < y, a > in the developing resolution is a proper quotient of the limit group Rlim(y, a) we have started with, we replace the particular developing resolution by starting the initial step of the procedure with the subgroup generated by < y, a > in it, hence, for the continuation we can assume that the subgroup generated by < y, a > in the developing resolution is isomorphic to the limit group Rlim(y, a) we have started with. With the closure Cl(u, y, a) there is an associated formal solution x(u, y, a). With the developing resolution we associate a family of formal solutions {x_{α}(u, y, a)}, where each formal solution x_{α}(u, y, a) is obtained from the formal solution x(u, y, a) by applying corresponding elements of the modular groups associated with the fixed auxiliary resolution.

We further set the anvil to be the limit group generated by the developing resolution and the completion of the top level of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), amalgamated along the top part of the developing resolution. We denote the anvil: Anv(MGQRes)(w, y, a). Note that the developing resolution is canonically embedded into the anvil, hence, the family of formal solutions defined over the developing resolution, can be naturally defined on the anvil.

Suppose that Q_{2}(y, a) is isomorphic to Rlim(y, a). In this case we continue to the next levels of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). By part (ii) of theorem 4.3, we are guaranteed that at some level j of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), the image of Rlim(y, a) in the limit group associated with the j-th level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), Q_{j}(y, a), is a proper quotient of Rlim(y, a). At this point we continue in a similar way to what we did in case Q_{2}(y, a) is a proper quotient of Rlim(y, a). We associate with Q_{j}(y, a) the resolutions that appear in its taut Makanin-Razborov diagram, and with each resolution we associate an auxiliary resolution, and a corresponding auxiliary limit group. We set the developing resolution to be the resolution obtained from the resolution induced by the subgroup < y, a > from the completion of the resolution obtained from the levels of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) that lie above the j-th level, continued with the auxiliary limit group associated with the specific auxiliary resolution of Q_{j}(y, a). We set the anvil to be the subgroup generated by the developing resolution and the completion of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) amalgamated over their common part, i.e., over the subgroup generated by images of the subgroup < y, a > in the all levels of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), that lie above the j-th level (including the j-th level itself). With the anvil, Anv(MGQRes)(w, y, a), we associate a family of formal solutions {x_{α}(w, y, a)} as we did in case Q_{2}(y, a)

is a proper quotient of Rlim(y, a).

(3) By part (1) we may assume that Q(y, a) is isomorphic to Rlim(y, a). Part (2) treat the case in which the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is not of maximal complexity. Hence, the only case left in presenting the initial step of our general procedure for validation of a sentence, is the case of a multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) of maximal possible complexity, i.e., a multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, , 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 completion Comp(Res)(z, y, a) we have started with. We treat this case as we treated it in the minimal rank case. In case the abelian decompositions associated with MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) and the top level of Res(y, 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 Rlim(y, a) into the limit group associated with the second level of Res(y, a), < z_{2}, a >, which is a subgroup of the subgroup generated by: Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} and the free factors dropped at the top level of the resolution Res(y, a). We now set the subgroups Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1} to be the subgroups corresponding to the non-abelian, non-QH vertex groups in the multi-graded abelian decomposition associated with the second level of Cl(Res)(s, z, y, a). By construction, these subgroups form a (canonical) free decomposition of the subgroup generated by: < s_{3}, z_{3}, …, s_{ℓ}, z_{ℓ}, a > in the closure Cl(Res)(s, z, y, a). 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}. Since the quotient limit group we have started with, QRlim(s, z, y, a) is a proper quotient of the closure, Cl(Res)(s, z, y, a), and since the abelian decomposition associated with the (only) level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is identical to the abelian decomposition associated with the top level of Cl(Res)(s, z, y, a), the subgroup QRlim_{2}(s, z, y, a), which is the subgroup generated by < s_{2}, z_{2}, …, s_{ℓ}, z_{ℓ}, a > in QRlim(s, z, y, a), is a proper quotient of its corresponding preimage in the closure Cl(Res)(s, z, y, a). Hence, at this point we can analyze the quotient limit group QRlim_{2}(s, z, y, 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, 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, a) all its multi-graded quotient resolutions with respect to the subgroups Base_{3,1}^{1}, …, Base_{3,t_{1}}^{1} which 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, 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, 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 resolution Res(y, a) we have started with, until one of the parts (1)-(2) applies to the corresponding multi-graded quotient resolution. At the level that one of the parts (1)-(2) applies, we set the developing resolution to be the resolution induced by the subgroup < y, a > from the parts of the resolution above that level followed by the developing resolution associated with that level according to one of the parts (1)-(2). We set the anvil, Anv(MGQRes)(w, y, a), to be the subgroup generated by the developing resolution and the completion of the multi-graded resolution associated with the relevant level, amalgamated along the image of the subgroup < y, a > in the limit group associated with the relevant level. With the developing resolution and its associated anvil, we naturally associate a family of formal solutions, x_{α}(w, y, a).

Definition 4.5. Starting with the limit group QRlim(s, z, y, 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, 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 auxiliary resolutions, and with each auxiliary resolution its corresponding auxiliary limit group and auxiliary multi-graded decomposition, and the associated anvil. We call the entire collection of the original limit group Rlim(y, a) and the resolution we have started with Res(y, a), the closure Cl(Res)(s, z, y, a) and the associated auxiliary resolution, auxiliary limit group and auxiliary multi-graded decomposition, together with the quotient limit group QRlim(s, z, y, a), its chosen quotient and multi-graded quotient resolution, the associated developing resolution, and the corresponding auxiliary resolution, auxiliary limit group and graded auxiliary decomposition, and the associated anvil, a staggered data-structure.

In order to force the iterative procedure to terminate, we need to use formal solutions defined over covering closures of the developing resolutions. By theorem 1.17 of [Se2], given a developing resolution, there exists a formal solution defined over the closure associated with the developing resolution. Therefore, starting with this formal solution, we may apply the modular groups associated with the auxiliary resolution corresponding to the developing resolution to obtain a family of formal solutions defined over the developing resolution, hence, a family of formal solutions defined over the anvil associated with the developing resolution. We denote this family of formal solutions x_{α}(w, y, a). By construction, for any choice of a modular element α, Σ(x_{α}(w, y, a), y, a) = 1. If the Zariski closure of the auxiliary resolution associated with the developing resolution is the closure associated with the developing resolution, and not a proper quotient of this closure, then there must exist a modular element α_{0}, and a specialization (w_{0}, y_{0}, a) of the anvil, so that the corresponding specialization factors through and is taut with respect to the developing resolution, for which for every index j, ψ_{j}(x_{α_{0}}(w_{0}, y_{0}, a), y_{0}, a) ≠ 1. We continue to the second step of the iterative procedure for validation of a sentence, with each of the staggered data-structure and its associated anvil, and the family of the formal solutions defined over the anvil.

II: The second step

In the initial step of the general iterative procedure for validation of a sentence, we used formal solutions defined over the entire set of y’s, and then formal solutions defined over closures of the various resolutions of the ”remaining” limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a). These formal solutions left us with the quotient limit groups QRlim(s, z, y, a) and their various multi-graded quotient resolutions. For presentation purposes, we present the second step of the iterative procedure. Later on we present the general step of the iterative procedure, and finally obtain its termination.

Since each of the remaining specializations of the variables y factors through at least one of the anvils and its associated developing resolution, we start the second step of the iterative procedure with the (canonical) finite set of anvils. Since we treat the anvils in parallel, we present the second step of the procedure with one of the anvils Anv(MGQRes)(w, y, a). With each anvil we have associated a family of formal solutions x_{α}(w, y, a), that (as a family) satisfy the properties listed in theorem 1.17 of [Se2]. Starting with the anvil Anv(MGQRes)(w, y, a) we impose the family of formal solutions x_{α}(w, y, a) associated with the corresponding developing resolution. We look at the set of specializations (w_{0}, y_{0}, a) that factor through and are taut with respect to the multi-graded resolution associated with the anvil Anv(GMQRes)(w, y, a) and factor through and are taut with respect to the developing resolution, for which for some index j: ψ_{j}(x_{α}(w_{0}, y_{0}, a), y_{0}, a) = 1 for all possible modular elements α. Clearly, if an element y_{0} ∈ F_{k} factors through one of the staggered resolutions, and y_{0} does not belong to any of the Diophantine sets defined by the above systems of equations and associated with the various staggered resolutions, then the sentence is valid for y_{0}. Hence, in the next steps of our iterative procedure it is enough to study these Diophantine sets.

If we fix the index j of the equation ψ_{j}(x_{α}, y, a) = 1 associated with the entire family of formal solutions {x_{α}(w, y, a)}, then the entire collection of specializations (w_{0}, y_{0}, a) that satisfy the corresponding system of equations is contained in a finite set of maximal (second quotient) limit groups Q^{2}Rlim_{1}(w, y, a), …, Q^{2}Rlim_{u_{j}}(w, y, a). Since our analysis of these (second) quotient limit groups is conducted in parallel, we will omit the indices from these (second) quotient limit groups and denote them Q^{2}Rlim(w, y, a).

We construct the staggered data-structure and developing resolutions associated with the anvil and the (second) quotient limit group Q^{2}Rlim(w, y, a) iteratively, similarly to the way we have analyzed quotient resolutions in the initial step of the procedure, though in somewhat modified way. The analysis we carry out in the second step depends on the structure of the staggered data-structure and the developing resolution constructed in the initial step of the procedure, in addition to various possibilities parallel to the ones handled in the initial step. As in the initial step 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(y, a) be the restricted limit group generated by < y, a > in the anvil Anv(MGQRes)(w, y, a). Let Q^{2}(y, a) be the limit group generated by < y, a > in the second quotient limit group Q^{2}Rlim(w, y, a). If Q^{2}(y, a) is a proper quotient of the subgroup Q(y, a), we continue this branch of the iterative procedure, by replacing the limit group Q(y, a) by Q^{2}(y, a), and continue with the finite collection of resolutions that appear in the taut Makanin-Razborov diagram of the limit group Q^{2}(y, a) in the same way we handled the resolutions in the taut Makanin-Razborov diagram of the original limit group Rlim(y, a). Note that this is always the case if the limit group Q(y, a) was a free group.

(2) At this stage we may assume that Q^{2}(y, a) is isomorphic to Q(y, a). At this part we assume that the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), associated with the anvil, Anv(MGQRes), is of maximal possible complexity. Let:

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

be the multi-graded resolutions in the multi-graded taut Makanin-Razborov diagram of Q^{2}Rlim(w, y, a) with respect to the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}. We will treat the multi-graded quotient resolutions MGQ^{2}Res_{j}(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in parallel, hence, we omit their index.

By proposition 4.2 the complexities of the abelian decompositions associated with the various levels of each of the (second) multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}a) is bounded by the complexity of the multi-graded quotient abelian decomposition associated with the top level of MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), and if the complexity of the abelian decomposition associated with some level of MGQ^{2}Res(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is equal to the complexity of the abelian decomposition associated with the top level of MGQRes(s, z, y, Base_{2,1}^{1} …, Base_{2,v_{1}}^{1}, a) then the second quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, 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 MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). In this part of the second step of the procedure we will also assume that the complexities of the abelian decompositions associated with the various levels of the second quotient multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) are strictly smaller than the complexity of the abelian decomposition associated with the top level of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) associated with the anvil. In this case we treat the (second) multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) as in part (2) of the initial step of the procedure, and associate with it finitely many anvils, which we denote Anv(MGQ^{2}Res), and with each anvil we associate a developing resolution, auxiliary resolution and auxiliary limit group as we did in part (2) of the initial step of the procedure.

(3) At this stage we may assume that Q^{2}(y, a) is isomorphic to Q(y, a). At this part we assume that the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is not of maximal complexity. Let Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2} be the non-abelian, non-QH vertex groups in the abelian decomposition associated with the top level of the anvil Anv(MGQRes)(w, y, a), i.e., the subgroups of the anvil generated by the variables w that belong to the limit groups associated with all the anvil except the top level. Note that the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} which are all subgroups of the closure Cl(Res)(s, z, y, a), are naturally mapped into the subgroups Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}. Let:

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

be the resolutions in the taut Makanin-Razborov diagram of the limit group Q^{2}Rlim(w, y, a) with respect to the subgroups Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}. As in the first step of the iterative procedure, we do not really need to consider all the multi-graded resolutions in this taut Makanin-Razborov diagram, but only those that are compatible with the indicated collections of s.c.c. that are being mapped to the trivial element on each of the QH vertex groups in the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) we have started this branch of the second step with. Since we treat these multi-graded quotient resolutions in parallel, we will omit their indices in the sequel.

Let MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) be one of these multi-graded quotient resolutions. Let Q(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be the subgroup generated by < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a > in the anvil Anv(MGQRes)(w, y, a), and let Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be the subgroup generated < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, , y, a > in the limit group corresponding to the multi-graded quotient MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a). By construction, Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is a quotient of Q(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). If Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is a proper quotient of Q(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), we replace the multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) by starting the initial step of the procedure with the subgroup generated by < s, z, y, a > in the completion of GQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), Q^{2}(s, z, y, a). Since the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) we have started with is not of maximal possible complexity, in analyzing the limit group Q^{2}(s, z, y, a) we treat only those multi-graded resolutions MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) which are not of maximal possible complexity, i.e., we do not (need to) treat those multi-graded quotient resolutions of Q^{2}(s, z, y, a) which have only one level above the terminating rigid or solid limit group, and the structure of the abelian decomposition associated with that level is identical to the structure of the abelian decomposition associated with the completion, Comp(Res)(z, y, a), we have started with.

(4) With the notation of part (3), at this part we 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, and that for our given second quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), the corresponding subgroup Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is isomorphic to Q(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). At this part we assume that the (second) graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, 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 MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, , a). The case of maximal complexity will be treated in the next part. To treat a (second) multigraded quotient resolution which is of maximal possible complexity we need the following observation, which is similar to part (ii) of proposition 4.3.

Lemma 4.6. Let MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) be one of the multigraded quotient resolutions in our list of second multi-graded quotient resolutions, and suppose MGQ^{2}Res is not of maximal complexity. Let MGQ^{2}lim_{term}(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) be the terminal rigid or solid limit group of the multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), and let Q_{term}^{2}(y, a) and Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be the images of Q^{2}(y, a) and Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in the limit group GQ^{2}lim_{term}(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a). Then the terminal limit group of the multi-graded resolution MGQ^{2}Res(s, z, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) can be replaced by a collection of finitely many terminal multi-graded limit groups in which either Q_{term}^{2}(y, a) is a proper quotient of Q(y, a), or Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is a proper quotient of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a).

Proof: The proof is similar to the proof of part (ii) of proposition 4.3. Suppose that Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is isomorphic to Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). For each subgroup Base_{2,j}^{2} let DBase_{2,j}^{2} be the subgroup of Base_{2,j}^{2} that stabilizes the corresponding distinguished vertex (i.e., the vertex group in the decomposition inherited by Base_{2,j}^{2} from the auxiliary decomposition that intersects Q(y, a) nontrivially). Let T erm(w, y, a) be the terminal rigid or solid limit group of the multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a). With the terminal limit group T erm(w, y, a) we associate the resolutions that appear in its (ungraded) taut Makanin-Razborov diagram. We treat these resolutions in parallel, so let T Res_{(}w, y, a) be one of these resolutions. Let Λ_{top} be the abelian decomposition associated with the top level of the (ungraded) resolution T Res(s, z, y, a). Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) inherits an abelian decomposition from Λ_{top}. Since Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is isomorphic to Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), and since by part (iii) of proposition 4.3 the abelian JSJ decomposition of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is obtained from the multi-graded abelian decomposition associated with the top level of the multi-graded resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) by further refining it using the auxiliary decompositions associated with each of the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, the non-QH, non-abelian vertex groups in the abelian JSJ decomposition of Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) have to be elliptic in the abelian decomposition inherited by Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) from Λ_{top}. Hence, if any of the subgroups DBase_{2,j}^{2} is not elliptic in the abelian decomposition Λ_{top}, then for each specialization y_{0}, for which there exists a specialization (w_{0}, y_{0}, a) that factors through and is taut with respect to the resolution obtained from the second multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) followed by the resolution T Res(w, y, a), the specialization (w_{0}, y_{0}, a) factors through an auxiliary resolution of higher complexity than the auxiliary resolution we have started the second step with. In this case we can omit the resolution T Res(w, y, a) from our list of (ungraded) resolutions of the terminal limit group T erm(w, y, a), hence, for the continuation we may assume that each of the subgroups DBase_{2,j}^{2} is elliptic in the abelian decomposition Λ_{top}. Since the subgroups DBase_{2,j}^{2} are elliptic in Λ_{top}, each of the other non-abelian, non-QH vertex groups and all the edge groups in the abelian decomposition inherited by the subgroups Base_{2,j}^{2} from the auxiliary decomposition have to be 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 second multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) is not elliptic in the abelian decomposition Λ_{top} then for each specialization y_{0}, for which there exists a specialization (w_{0}, y_{0}, a) that factors through and is taut with respect to the resolution obtained from the second multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) followed by the resolution T Res(w, y, a), the specialization (w_{0}, y_{0}, a) factors through a multi-graded resolution MGQ^{2}Res′(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, 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 MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), and all the non-abelian, non-QH vertex groups and all the edge groups in the abelian decomposition inherited by each of the subgroups Base_{2,j}^{1} from the graded auxiliary decomposition, 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 multi-graded modular group associated with the terminal level of the second multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a). Hence, the multi-graded modular group associated with the terminal level of MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) can be applied to map the terminal limit group T erm(w, y, a) into the limit group associated with the second level of the (ungraded) resolution T Res(w, y, a). If the image of Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in that limit group is a proper quotient of Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), we replace the limit group T erm(w, y, a) with the limit group associated with the second level of T Res(w, y, a) and the lemma follows. If the image of Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in the limit group associated with the second level of T Res(w, y, a) is isomorphic to Q_{term}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), we apply the same argument to show that the multi-graded modular groups associated with the terminal level of MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) can be used to map T erm(w, y, a) into the limit group associated with the third level of the (ungraded) resolution T Res(w, y, a). Applying this argument inductively, finishes the proof of the lemma.

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By lemma 4.6 we can either omit the (second) multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) from our list of (second) multi-graded quotient resolutions, since every specialization that factors through and is taut with respect to it factors through an auxiliary resolution with higher complexity, or the image of Q^{2}(y, a) or of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in the terminal rigid or solid limit group MGQ^{2}lim_{term}(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) of MGQ^{2}Res(w, y, Base_{2}^{2}2, 1, …, Base_{2,v_{2}}^{2}, a) are proper quotients of Q^{2}(y, a) or Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in correspondence. To continue our treatment of the (second) multi-graded resolution MGQ^{2}Res(w, y, Base_{2}^{2}2, 1, …, Base_{2,v_{2}}^{2}, a) we need the following observation, which is similar to proposition 4.4.

Lemma 4.7. Let MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) be a (second) multigraded quotient resolution from our list of (second) quotient multi-graded resolutions, and suppose MGQ^{2}Res is not of maximal complexity. Let Q^{2}(y, a), Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}), Q^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}) be the subgroups generated by < y, a >, < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} >, < y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2} > in correspondence, in the limit group associated with the second multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a). Let Q_{2}^{2}(y, a), Q_{2}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) and Q_{2}^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}) be the images in the limit group associated with the second level of MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), MGQ^{2}lim_{2}(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), of the subgroups: Q^{2}(y, a), Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}), and Q^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}) in correspondence. Then either Q_{2}^{2}(y, a) is a proper quotient of Q^{2}(y, a), or Q_{2}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) is proper quotient of the subgroup Q^{2}(y, Base_{2,1}^{1}, …, Base^{1}2, v_{1}), or Q_{2}^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}) is a proper quotient of the subgroup Q^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}).

Proof: The claim is simply one of the basic properties of a multi-graded resolution.

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By lemma 4.7 either the image of the subgroup Q^{2}(y, a) in the limit group associated with the second level of MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), Q_{2}^{2}(y, a), is a proper quotient of Q^{2}(y, a), or the image of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) in the limit group associated with the second level of MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), Q_{2}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{2}), is a proper quotient of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), or the image of Q^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) in the limit group associated with the second level of MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), Q_{2}^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}), is a proper quotient of Q^{2}(y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}). Suppose that the subgroup Q_{2}^{2}(y, a) is a proper quotient of Q^{2}(y, a). In this case we continue as we did in part (2) of the initial step. Let QRes_{1}(y, a), …, QRes_{q}(y, a) be the resolutions in the taut Makanin-Razborov diagram of Q_{2}^{2}(y, a). We continue with each of the resolutions QRes_{i}(y, a) separately, so we omit its index. With the resolution QRes(y, a) we associate a resolution CRes(y, a), where CRes(y, a) is composed from two parts, the top being the resolution induced by the subgroup < y, a > from the completion of the top level of the second multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), and the tail being the resolution QRes(y, a). By theorem 1.17 of [Se2], with the resolution CRes(y, a) we can associate a set of closures Cl_{1}(CRes)(u, y, a), …, Cl_{d}(CRes)(u, y, a) and for each closure a corresponding formal solution defined over it. We continue with each of these closures in parallel. With a closure Cl(CRes)(u, y, a), we associate its auxiliary resolutions and with each auxiliary resolution we associate its corresponding auxiliary limit group and auxiliary graded decomposition. We continue with each of the auxiliary resolutions in parallel. Fixing a closure Cl(u, y, a) and one of its associated auxiliary limit groups we set the developing resolution to be the auxiliary resolution. If the subgroup generated by < y, a > in the developing resolution is a proper quotient of the limit group Q^{2}(y, a) we have started with, we replace the particular developing resolution by starting the initial procedure with the subgroup generated by < y, a > in it, hence, for the continuation we can assume that the subgroup generated by < y, a > in the developing resolution is isomorphic to the limit group Q^{2}(y, a) we have started with. With the closure Cl(u, y, a) there is an associated formal solution x(u, y, a). With the developing resolution we associate a family of formal solutions {x_{α}(u, y, a)}, where each formal solution x_{α}(u, y, a) is obtained from the formal solution x(u, y, a) by applying corresponding elements of the modular groups associated with the fixed auxiliary resolution. We further set the anvil to be the limit group generated by the developing resolution and the completion of the top level of the (second) quotient multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), amalgamated along the top part of the developing resolution. We denote the anvil: Anv(MGQ^{2}Res)(w, y, a). Note that the developing resolution is canonically embedded into the anvil, hence, the family of formal solutions defined over the developing resolution, can be naturally defined over the anvil.

Suppose that Q_{2}^{2}(y, a) is isomorphic to Q^{2}(y, a), and Q_{2}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) is a proper quotient of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}). Let Q_{2}^{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) be the image of the subgroup Q^{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in the limit group associated with the second level of the second multigraded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a). Note that by our assumptions, Q_{2}^{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is a proper quotient of the limit group Q^{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). With the subgroup Q_{2}^{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) we associate the multi-graded resolutions that appear in its multi-graded taut Makanin-Razborov diagram with respect to the subgroups Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}:

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

We continue with each of the multi-graded resolutions MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in parallel.

Suppose that a multi-graded quotient resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is of maximal possible complexity, i.e., it consists of one level with an abelian decomposition which has the same structure as the abelian decomposition associated with the top level of the resolution Res(y, a). The group Q_{2}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is naturally mapped into the limit group Q^{2}Rlim(MGQRes_{j})(s, z, y, a) corresponding to the multi-graded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). Since Q_{2}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is a proper quotient of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) by our assumption, and since the multi-graded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is of maximal possible complexity, and since limit groups are residually finite, so they are necessarily Hopf, the image of the the subgroup generated by: < Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} > in Q^{2}Rlim(MGQRes_{j})(s, z, y, a) is necessarily a proper quotient of the image of the subgroup generated by: < Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1} > in the quotient limit group Q^{2}Rlim(w, y, a) we have started the second step with. Hence, we can replace the resolution MGQRes_{j}(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) by starting the initial step of the procedure with the subgroup generated by < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a > in the limit group corresponding to the resolution induced by the subgroup < s, z, y, a > from the top level of the multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) continued with the resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) which is necessarily a proper quotient of the limit group QRlim(s, z, y, a) we have started with. As we remarked in part (3), we need to treat only those multi-graded resolutions of the limit group < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a > which are not of maximal possible complexity. If the subgroup generated by < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a > in the limit group corresponding to the graded resolution CRes_{j}(u, y, a), where CRes_{j}(u, y, a) is the multi-graded resolution obtained from the resolution induced by the subgroup < s, z, y, a > from the top level of the completion of the multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) followed by the multi-graded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), is a proper quotient of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), we replace the multigraded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) by starting the initial step of the procedure with the subgroup generated by < s, z, y, a > in the limit group corresponding to the resolution CRes_{j}(u, y, a), and treat only those multi-graded resolutions of this limit group that are not of maximal possible complexity. Hence, we may assume that for the rest of this part the subgroup generated by < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a > in the limit group corresponding to CRes_{j}(u, y, a) is isomorphic to Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). In particular we may assume that each of the multi-graded resolutions MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in question is not of maximal possible complexity.

We now treat each of the multi-graded resolutions MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) and their associated resolution CRes_{j}(u, y, a) in a similar way to our treatment of multi-graded quotient resolutions in the initial step of the procedure. Every QH vertex group Q in the abelian decomposition associated with the top level of the resolution Res(y, a), we have started this branch of the procedure with, inherits a sequence of abelian decompositions from the various levels of the resolution CRes_{j}(u, y, a). Since the resolution Res(y, a) is well-separated, with each such QH vertex group Q, there is an associated collection of s.c.c. that are mapped into the trivial element in the next level of the multi-graded resolution. If for some such QH vertex group Q, its sequence of inherited abelian decompositions is not compatible with its associated collection of s.c.c. that are mapped into the trivial element in the next level of the resolution Res(y, a), we omit the multigraded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) from our list of multi-graded resolutions. Hence, for the rest of this part we assume the resolution CRes_{j}(u, y, a) is compatible with the collections of s.c.c. associated with each of the QH vertex groups in the abelian decomposition associated with the top level of the resolution Res(y, a).

At this point we apply the procedure presented in part (2) of the initial step of our iterative procedure, and associate with the multi-graded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) a finite collection of anvils and developing resolutions. With an anvil associated with the multigraded resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), we associate its graded Makanin-Razborov diagram with respect to the subgroup < s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a >. We call each resolution in this graded Makanin-Razborov diagram an auxiliary resolution. With each auxiliary resolution we associate its terminal (rigid or solid) limit group which we call auxiliary limit group, and its ”tower” of modular groups which we call auxiliary modular groups. Given the developing resolution, the anvil, and an auxiliary resolution associated with the resolution MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), we set the developing resolution of the multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) to be the resolution obtained from the resolution induced by the subgroup < y, a > from the completion of the multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) continued by the image of the developing resolution associated with MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in the auxiliary limit group which is the terminal limit group of the auxiliary resolution associated with MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) with respect to the parameter subgroup < s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a >. We set the anvil, Anv(MGQ^{2}Res)(w, y, a), to be the group obtained as the amalgamated product of the completion of the (second) multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), with the auxiliary limit group associated with MGQRes_{j}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a), along the corresponding copies of the image of the subgroup Q_{2}^{2}(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a). With the developing resolution we associate a family of formal solutions x_{α}(w, y, a), parameterized by the auxiliary modular groups, which are naturally defined over the anvil as well.

We still need to treat the case in which both Q_{2}^{2}(y, a) is isomorphic to Q^{2}(y, a) and Q_{2}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) is isomorphic to Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}). In this case, we just continue to the next level of the multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), and proceed with the same analysis we applied for the top level of this multi-graded resolution. By lemma 4.6, there must exist some level j of the second multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), for which either Q_{j}^{2}(y, a) is a proper quotient of Q^{2}(y, a) or Q_{j}^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) is a proper quotient of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}). If level j is the highest level of the multi-graded resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a), for which such proper quotients are obtained, we repeat the procedure described above and associate with the multi-graded resolution MGQ^{2}Res a finite collection of anvils, developing resolutions and auxiliary resolutions.

(5) By part (1) we may assume that Q^{2}(y, a) is isomorphic to the limit group Q(y, a) associated with the anvil, Anv(MGQRes)(w, y, a), and by part (3) we may assume that Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) is isomorphic to the subgroup Q(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}) associated with the anvil. Part (4) treat the case in which the second multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) is not of maximal complexity. Hence, the only case left in presenting the second step of our general procedure for validation of a sentence, is the case of a second multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) (or MGQ^{2}Res(w, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) in case MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) is of maximal possible complexity - see part (2)) of maximal possible complexity. i.e., a multi-graded quotient resolution MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{1}}^{2}, 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 completion of the multi-graded quotient resolution MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) we have started the second step of the procedure with. We treat this case as we treated it in the minimal rank case and in the initial step of the general procedure. In case the abelian decompositions associated with MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a) and the top level of (the completion of) MGQRes(s, z, y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}, a) are identical, we use the modular groups associated with the abelian decomposition associated with MGQ^{2}Res(w, y, Base_{2,1}^{2}, …, Base_{2,v_{1}}^{2}, a) to map the subgroup < w, y, a > into the limit group associated with the (appropriate) next level of the corresponding part of the anvil Anv(MGQRes)(w, y, a). By construction either the image of the subgroup generated by < y, a > in that limit group is a proper quotient of Q^{2}(y, a) or the subgroup generated by < y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{2} > in that limit group is a proper quotient of Q^{2}(y, Base_{2,1}^{1}, …, Base_{2,v_{1}}^{1}). We now analyze the remaining part of the anvil in the same way we have analyzed the second multi-graded quotient resolution, according to parts (1)-(5) above. 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 with respect to the parameter subgroup < w, y, Base_{2,1}^{2}, …, Base_{2,v_{2}}^{2}, a >. 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 second step).

Given an auxiliary resolution we set the developing resolution to be the resolution induced by the subgroup < y, 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^{2}Res)(w, y, a), to be the subgroup generated by the the auxiliary limit group associated with the auxiliary resolution and the subgroup Q^{2}(w, y, a), amalgamated along the copies of the limit group associated with the level to which one of the steps (1)-(4) was applied. With the developing resolution and its associated anvil, we naturally associate a family of formal solutions, x_{α}(w, y, 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, 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 4.8. With the notation above, 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, a > be the subgroup generated by the closure of the developing resolution in the anvil Anv(MGQRes)(w, y, 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, a >.

Then there exists some level j, so that the structure of the abelian decompositions associated with the resolutions induced by the subgroup < v, y, a > above level j are identical to the structure of the abelian decompositions associated with the corresponding resolution induced by the subgroup < y, 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, a > is strictly smaller than the number of factors in the corresponding free decomposition associated with the resolution induced by the subgroup < y, 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, a > is strictly smaller than the complexity of the abelian decomposition associated with the resolution induced by < y, a >.

Proof: The subgroup generated by < y, a > is naturally embedded in the subgroup generated by < v, y, a > in the anvil Anv(MGQRes)(w, y, a). If the subgroup generated by < y, a > is isomorphic to the subgroup generated by < v, y, a >, then the resolutions induced by the subgroup < v, y, a > are identical to those induced by the subgroup < y, a >, hence, the closure of the developing resolution associated with Anv(MGQRes)(w, y, a) is embedded in the quotient limit group QRlim(w, y, a) we have started the second step of the procedure with, a contradiction to the properties of the family of formal solutions associated with the developing resolution that is associated with the anvil Anv(MGQRes)(w, y, a). Hence, the resolutions induced by the subgroup < v, y, a > differs from the resolutions induced by the subgroup < y, a >.

As in the proof of the termination of the construction of the induced resolution (proposition 3.3), if we restrict ourselves to the highest level of the developing resolution, then the subgroup generated by < v, y, a > differs from the subgroup generated by < y, a > only in stabilizers of some of the distinguished vertices in each of the connected components of the graph of groups associated with the highest level. By properties (1)-(3) of the construction of the induced resolution, if we restrict ourselves to the highest level, if the decomposition associated with the subgroup < v, y, a > differs from the decomposition associated with the subgroup < y, a >, then either it contains fewer connected components, or in case of equality, the complexity of the decompositions associated with the subgroup < v, y, a > is strictly smaller than the complexity of the decompositions associated with the subgroup generated by < y, a >. In case of equality in the complexities of the decompositions associated with the subgroups < v, y, a > and < y, a > we continue to the next level, until we get to a level in which either a decrease in the number of connected components or a decrease in the complexity of the associated decompositions occurs. This completes the proof of the proposition.

□

Using proposition 4.8, in case all the multi-graded resolutions used in the construction of the developing resolution are of maximal possible complexity, we replace the resolutions induced by the subgroup < y, a > used to construct the developing resolution, by the resolutions induced by the subgroup < v, y, a >. With the obtained (modified) developing resolution we associate a covering closure according to theorem 1.17 of [Se2], with each closure we naturally associate an anvil, denoted Anv(MGQ^{2}Res(w, y, a), obtained from the anvil Anv(MGQRes)(w, y, a) by adding the appropriate roots corresponding to the specific closure, and with the closure and its associated anvil we associate a family of formal solutions x_{α}(w, y, a), parameterized by the auxiliary modular groups associated with the developing resolution associated with the anvil Anv(MGQRes)(w, y, a).

Given the developing resolution, the anvil, and the family of formal solutions defined over them, we construct the staggered data-structure according to definition 4.5. With the anvil constructed in the second step of the iterative procedure, we have associated a family of formal solutions {x_{α}(w, y, a)}. We start the third step of the iterative procedure, by imposing on the specializations that factor through and are taut with respect to the multi-graded quotient resolutions associated with the anvil, the (finitely many) systems of equations: ψ_{j}(x_{α}(w, y, a), y, a) = 1, a system for each fixed index j, and the entire family of formal solutions {x_{α}(w, y, a)}.

III: The general step

In the initial steps of the iterative procedure for validation of a sentence we used formal solutions defined over the entire set of y’s, and then families of formal solutions defined over closures of the various resolutions of the ”remaining” limit groups Rlim_{1}(y, a), …, Rlim_{m}(y, a) and closures of developing resolutions associated with quotient resolutions obtained from these resolutions. At each step, we have finally obtained finitely many staggered data-structures and their associated developing resolutions, auxiliary resolutions, and anvils. With each tuple of a developing resolution, an auxiliary resolution and anvil we have associated a family of formal solutions defined over the auxiliary resolution, which is embedded into the anvil. By construction, each specialization of the variables y remained after the initial steps factors through at least one of the modular blocks associated with the various anvils constructed in these steps. After presenting the initial steps, we finally present the general step of the procedure for validation of a sentence, 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(w, y, 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 developing resolution, auxiliary resolution, and an anvil, which we denote Anv(MGQ^{m}Res)(w, y, a). By construction, each of the specializations of the variables y factors through at least one of the modular blocks associated with the various multi-graded quotient resolutions and their associated anvils. We start the general step of our iterative procedure for validation of a sentence with the (finite) collection of multi-graded quotient resolutions constructed in the previous step, and their associated developing resolutions, 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)(w, y, a). With each anvil we have associated a family of formal solutions x_{α}(w, y, a), that (as a family) satisfy the properties listed in theorem 1.17 of [Se2]. Starting with the anvil Anv(MGQ^{n−1}Res)(w, y, a) we impose the family of formal solutions x_{α}(w, y, a) associated with the corresponding closure of the developing resolution. We look at the set of specializations (w_{0}, y_{0}, a) that factor through and are taut with respect to the resolutions associated with the anvil Anv(MGQ^{n−1}Res)(w, y, a) (which includes its associated developing resolution), for which for some index j: ψ_{j}(x_{α}(w_{0}, y_{0}, a), y_{0}, a) = 1 for all possible modular elements α. Clearly, if an element y_{0} ∈ F_{k} factors through one of the anvils, and y_{0} does not belong to any of the Diophantine sets defined by the above systems of equations and associated with the various staggered datastructures, then the sentence is valid for y_{0}. Hence, in the next steps of our iterative procedure it is enough to study these Diophantine sets.

If we fix the index j of the equation ψ_{j}(x_{α}(w, y, a), y, a) = 1 associated with the entire family of formal solutions {x_{α}(w, y, a)}, then the entire collection of specializations (w_{0}, y_{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}(w, y, a), …, Q^{n}Rlim_{u_{j}^{n}}(w, y, 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(w, y, 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 second step of the procedure. The analysis we carry out in the general step depends on the structure of the staggered data-structure, the developing resolutions, and the auxiliary resolutions constructed in the previous steps of the procedure. As in the 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, a) be the restricted limit group generated by < y, a > in the anvil Anv(MGQ^{n−1}Res)(w, y, a). Let Q^{n}(y, a) be the limit group generated by < y, a > in the n-th quotient limit group Q^{n}Rlim(w, y, a). If Q^{n}(y, a) is a proper quotient of the subgroup Q^{n−1}(y, a), we continue this branch of the iterative procedure, by replacing the limit group Q^{n−1}(y, a) by Q^{n}(y, a), and continue with the finite collection of resolutions that appear in the taut Makanin-Razborov diagram of the limit group Q^{n}(y, a) in the same way we handled the resolutions in the taut Makanin-Razborov diagram of the original limit group Rlim(y, a). Note that this is always the case if the limit group Q^{n−1}(y, a) was a free group.

(2) At this stage we may assume that Q^{n}(y, a) is isomorphic to Q^{n−1}(y, a). Along the process used to construct the anvil, Anv(MGQ^{n−1}Res)(w, y, 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 corre-

sponding multi-graded quotient resolution to be: MGQ^{m}Res(w_{m}, y, 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}}(w_{m_{1}}, y, a) be the subgroup generated by < w_{m_{1}}, y, a > in the limit group Q^{m_{2}}Rlim(w_{m_{2}}, y, a).

In this part of the general step we assume that the multi-graded quotient resolution MGQ^{n−1}Res(w, 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}(w_{ℓ(s)}, y, a) is a proper quotient of Q^{ℓ(s)}(w_{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(w, y, 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}(w_{ℓ(s)}, y, a) instead of the ℓ(s)-th limit group Q^{ℓ(s)}Rlim(w_{ℓ(s)}, y, a) used in the s-th step of the process that lead to the construction of the anvil, Anv(MGQ^{n−1}Res)(w, y, 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(w_{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(w_{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)(w, y, a).

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

MGQ^{n}Res_{1}(w, 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(w, y, 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 resolutions MGQ^{n}Res_{j}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) in parallel, hence, we omit their index.

If for some QH vertex group Q in the abelian decomposition associated with the top level of the multi-graded resolution MGQ^{n−1}Res(w, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a) the sequence of abelian decompositions Q inherits from the a multi-graded resolution MGQ^{n}Res_{j}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) is not compatible with the 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(w, 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_{j}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) from our list of multi-graded resolutions of the n-th quotient limit group Q^{n}(w, y, a). Otherwise, by proposition 4.2 the complexities of the abelian decompositions associated with the various levels of each of the n-th multi-graded quotient resolution MGQ^{n}Res(w, 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(w, 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(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) is equal to the com-

plexity of the abelian decomposition associated with the top level of MGQ^{n−1}Res(w, 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(w, 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(w, 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(w, 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(w, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a).

In this case we treat the multi-graded resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) according to part (4) of step n − 1 of the procedure, and associate with it a finite collection of anvils, developing resolutions, and auxiliary resolutions.

(3) At this stage we may assume that Q^{n}(y, a) is isomorphic to Q^{n−1}(y, a). At this part we assume that the multi-graded quotient resolution MGQ^{n−1}Res(w, 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)}(w_{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(w, y, 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}(w_{ℓ(s)}, y, a), the subgroup generated by < w_{ℓ(s)}, y, a > in the nth quotient limit group Q^{n}Rlim(w, y, a), instead of the ℓ(s)-th limit group Q^{ℓ(s)}Rlim(w_{ℓ(s)}, y, a) used in the ℓ(s)-th step of the process that lead to the construction of the anvil, Anv(MGQ^{n−1}Res)(w, y, 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(w, 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(w_{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(w_{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)(w, y, a).

(4) At this part we may assume that the multi-graded quotient resolution MGQ^{n−1}Res(w, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a) is not of minimal rank and is not of maximal complexity, and that Q^{n}(w_{n−1}, y, a) is isomorphic to Q^{n−1}(w_{n−1}, y, 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)(w, y, a). Let:

MGQ^{n}Res_{1}(w, 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(w, y, a) with respect to the parameter subgroups Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}.

If a multi-graded resolution MGQ^{n}Res_{j}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) is not compatible with the collections of s.c.c. associated with the various QH vertex groups in the abelian decomposition associated with the top level of the multi-graded resolution MGQ^{n−1}Res(w, y, Base_{2,1}^{s(n−1)}, …, Base_{2,v_{s(n−1)}}^{s(n−1)}, a), and are mapped to trivial element in the next level of that multi-graded resolution, we omit the multi-graded resolution MGQ^{n}Res_{j}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) from our list of n-th quotient multi-graded resolutions. We analyze the n-th multi-graded quotient resolutions MGQ^{n}Res_{j}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) in parallel, hence, we will omit their index.

At this part we will also assume that the n-th multi-graded quotient resolution MGQ^{n}Res(w, 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(w, 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. As in the second step of the procedure, to treat an n-th multigraded quotient resolution which is not of maximal possible complexity we need the following observation, which is similar to lemma 4.6 and part (ii) of theorem 4.3.

Lemma 4.9. Let MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) be one of the resolutions in our list of n-th multi-graded quotient resolutions and suppose MGQ^{n}Res is not of maximal complexity. By construction, the limit group Q^{n}(w_{n−1}, y, a) is mapped into the limit group associated with each of the levels of the multi-graded quotient resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a). Let Q_{term}^{n}(w_{n−1}, y, a)

be the image of Q^{n}(w_{n−1}, y, a) in the terminal (rigid or solid) limit group of MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a).

Then the terminal limit group of the n-th multi-graded resolution MGQ^{n}Res(w, 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}(w_{n−1}, y, a) is a proper quotient of Q^{n}(w_{n−1}, y, a).

Proof: Identical to the proof of lemma 4.6.

□

To handle an n-th multi-graded quotient resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a)

which is not of maximal complexity, we also need the following lemma which

is similar to lemma 4.7.

Lemma 4.10. Let MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) be one of the resolutions in our list of n-th multi-graded quotient resolutions, and suppose MGQ^{n}Res is not of maximal complexity. By construction, the limit group Q^{n}(w, y, a) is mapped onto the limit group associated with each of the levels of the multi-graded quotient resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a). Let Q_{2}^{n}(y, a), Q_{2}^{n}(w_{n−1}, y, a), Q_{2}^{n}(w_{n}, y, a) be the images of the subgroups generated by Q^{n}(y, a), Q^{n}(w_{n−1}, y, a), Q^{n}(w_{n}, y, a) in correspondence, in the limit group associated with the second level of the multi-graded quotient resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a). Then either Q_{2}^{n}(y, a) is a proper quotient of Q^{n}(y, a), or Q_{2}^{n}(w_{n−1}, y, a) is proper quotient of the subgroup Q^{n}(w_{n−1}, y, a), or Q_{2}^{n}(w_{n}, y, a) is a proper quotient of the subgroup Q^{n}(w_{n}, y, a).

Proof: The claim is simply one of the basic properties of a multi-graded resolution.

□

By lemma 4.10 either the image of the subgroup Q^{n}(y, a) in the limit group associated with the second level of MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), Q_{2}^{n}(y, a), is a proper quotient of Q^{n}(y, a), or the image of Q^{n}(w_{n−1}, y, a) in the limit group associated with the second level of MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), Q_{2}^{n}(w_{n−1}, y, a), is a proper quotient of Q^{n}(w_{n−1}, y, a), or the image of Q^{n}(w_{n}, y, a) in the limit group associated with the second level of MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), Q_{2}^{n}(w_{n}, y, a), is a proper quotient of Q^{n}(w_{n}, y, a). Suppose that the subgroup Q_{2}^{n}(y, a) is a proper quotient of Q^{n}(y, a). Let QRes_{1}(y, a), …, QRes_{q}(y, a) be the resolutions in the taut Makanin-Razborov diagram of Q_{2}^{n}(y, a). We continue with each of the resolutions QRes_{i}(y, a) separately, hence, we omit their index. With the resolution QRes(y, a) we associate a resolution CRes(y, a), where CRes(y, a) is composed from two parts, the top being the resolution induced by the subgroup < y, a > from the completion of the top level of the multi-graded quotient resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), and the tail being the resolution QRes(y, a). By theorem 1.17 of [Se2], with the resolution CRes(y, a) we can associate a set of closures Cl_{1}(CRes)(u, y, a), …, Cl_{d}(CRes)(u, y, a) and for each closure a corresponding formal solution defined over it. We continue with each of these closures in parallel. With a closure Cl(CRes)(u, y, a), we associate its auxiliary resolutions and with each auxiliary resolution we associate its corresponding auxiliary limit group and auxiliary multi-graded decomposition. We continue with each of the auxiliary resolutions in parallel. Fixing a closure Cl(u, y, a) and one of its associated auxiliary limit groups we set the developing resolution to be the auxiliary resolution. If the subgroup generated by < y, a > in the developing resolution is a proper quotient of the limit group Q^{2}(y, a) we have started with, we replace the particular developing resolution by starting the initial step of the procedure with the subgroup generated by < y, a > in it, hence, for the continuation we can assume that the subgroup generated by < y, a > in the developing resolution is isomorphic to the limit group Q^{n}(y, a) we have started with. With the closure Cl(u, y, a) there is an associated formal solution x(u, y, a). With the developing resolution we associate a family of formal solutions {x_{α}(u, y, a)}, where each formal solution x_{α}(u, y, a) is obtained from the formal solution x(u, y, a) by applying corresponding elements of the modular groups associated with the fixed auxiliary resolution. We further set the anvil to be the limit group generated by the developing resolution and the completion of the top level of the (second) multi-graded resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), amalgamated along the top part of the developing resolution. We denote the anvil: Anv(MGQ^{n}Res)(w, y, a). Note that the developing resolution is canonically embedded into the anvil, hence, the family of formal solutions defined over the developing resolution, can be naturally defined over the anvil.

Suppose that Q_{2}^{n}(y, a) is isomorphic to Q^{n}(y, a), and Q_{2}^{n}(w_{n−1}, y, a) is a proper quotient of Q^{n}(w_{n−1}, y, a). In this case we set s, 1 ≤ s ≤ s(n − 1), to be the minimal index for which: Q_{2}^{n}(w_{ℓ(s)}, y, a) is a proper quotient of Q^{n}(w_{ℓ(s)}, y, a) = Q^{ℓ(s)}(w_{ℓ(s)}, y, a). Let:

MGQRes_{1}(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), …, MGQRes_{d}(w_{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}(w_{ℓ(s)}, y, a) with respect to the parameter subgroups Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}. Suppose that a multi-graded quotient resolution MGQRes(w_{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(w_{ℓ(s)−1}, y, Base_{2,1}^{s(ℓ(s)−1)}, …, Base_{2,v_{s(ℓ(s)−1)}}^{s(ℓ(s)−1)}, a). Let Dw_{ℓ(s)} be the subgroup that stabilizes the distinguished vertex in the anvil Anv(MGQ^{ℓ(s)−1}Res)(w_{ℓ(s)}, y, a). Since the subgroup Q_{2}^{n}(w_{ℓ(s)}, y, a) is a proper quotient of Q^{n}(w_{ℓ(s)}, y, a), and since limit groups are residually finite, hence, satisfy the Hopf property, the image of the subgroup Dw_{ℓ(s)} in the (closure of the) multi-graded quotient resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is a proper quotient of the subgroup Dw_{ℓ(s)}. In this case of MGQRes(w_{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(w_{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 < w_{ℓ(s)}, y, a > from the top level of the completion of the multi-graded quotient resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) followed by the multi-graded resolution MGQRes(w_{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 < w_{ℓ(q)}, y, a > in (the closure of) CRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is a proper quotient of Q^{n}(w_{ℓ(q)}, y, a) = Q^{ℓ(q)}(w_{ℓ(q)}, y, a). We now replace the multi-graded quotient resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) of the limit group Q_{2}^{n}(w_{ℓ(s)}, y, a), by starting the ℓ(q)-th step of our process with the limit group generated by < w_{ℓ(q)}, y, a > in (the closure of) CRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) instead of the limit group Q^{ℓ(q)}Rlim(w_{ℓ(q)}, y, a) = Q^{n}Rlim(w_{ℓ(q)}, y, a) used in the ℓ(q)-th step of the procedure.

By the above argument, for the rest of part (24 of the general step of the procedure we may consider only those multi-graded quotient resolutions MGQRes(w_{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 multi-graded quotient resolutions MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) as we did in step ℓ(s) of our iterative procedure. First, we associate with the multi-graded resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) a multi-graded resolution CRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), obtained from the resolution induced by the subgroup < w_{ℓ(s)}, y, a > from the top level of the completion of the multi-graded resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) followed by the multi-graded resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a). If the multi-graded resolution CRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) is not compatible with the collections of s.c.c. associated the various QH vertex groups in the multi-graded abelian decomposition associated with the top level of the multi-graded resolution: MGQ^{ℓ(s−1)}Res(w_{ℓ(s−1)}, y, Base_{2,1}^{s−1}, …, Base_{2,v_{s−1}}^{s−1}, a), we omit the multi-graded resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) from our list of multi-graded resolutions of Q_{term}^{n}(w_{ℓ(s)}, y, a). Otherwise, we continue as in step ℓ(s) of the iterative procedure, and associate with the multi-graded resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) a canonical collection of developing resolutions and their associated anvils and families of formal solutions. As we did in part (4) of the second step of our iterative procedure, given an anvil associated with the graded resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a), we associate with it its graded taut Makanin-Razborov diagram with respect to the parameter subgroup < w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a >. We call each resolution in this graded taut Makanin-Razborov diagram, an auxiliary resolution, its terminal rigid or solid limit group, auxiliary limit group, and the modular groups associated with the auxiliary resolution, auxiliary modular groups. Given an auxiliary resolution associated with MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) ,we set a resolution CRes(u, y, a) obtained from the resolution induced by the subgroup < y, a > from the n-th multi-graded quotient resolution MGQ^{n}Res(w, 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 multigraded resolution: MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a). We set the developing resolution to be the resolution CRes(u, y, a). We set the anvil, Anv(MGQ^{n}Res)(w, y, a), to be the group generated by the corresponding closure of the multi-graded resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) and the auxiliary limit group associated with MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a) amalgamated along the corresponding images of the subgroup Q_{2}^{n}(w_{ℓ(s)}, y, a) in the second level of the completion of Q_{term}^{n}(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) and in the auxiliary limit group associated with the multi-graded resolution MGQRes(w_{ℓ(s)}, y, Base_{2,1}^{s}, …, Base_{2,v_{s}}^{s}, a). With the developing resolution we associate the family of formal solutions x_{α}(w, y, a) parameterized by the elements of the various modular groups associated with the auxiliary resolution from which the resolution CRes(u, y, a) was constructed. Since the developing resolution is embedded into the anvil, Anv(MGQ^{n}Res)(w, y, a), the family of formal solutions is naturally defined over the anvil as well.

We still need to treat the case in which both Q_{2}^{n}(y, a) is isomorphic to Q^{n}(y, a) and Q_{2}^{n}(w_{n−1}, y, a) is isomorphic to Q^{n}(w_{n−1}, y, a). In this case, we just continue to the next level of the multi-graded quotient resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), and proceed with the same analysis we applied for the top level of this multi-graded resolution. By lemma 4.6, there must exist some level j for which either Q_{j}^{n}(y, a) is a proper quotient of Q^{n}(y, a) or Q_{j}^{n}(w_{n−1}, y, a) is a proper quotient of Q^{n}(w_{n−1}, y, a). If j is the highest such level in the multi-graded resolution MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a), we associate with it a finite collection of developing resolutions and anvils according to the various cases described above.

(5) By part (1) we may assume that Q^{n}(y, a) is isomorphic to the limit group Q(y, a) associated with the anvil, Anv(MGQ^{n−1}Res)(w, y, a), and by parts (2)-(3) we may assume that Q^{n}(w_{s(n)−1}, y, a) is isomorphic to the subgroup Q^{n−1}(w_{s(n)−1}, y, a) associated with the anvil. Part (4) treat the case in which the n-th multi-graded quotient resolution MGQ^{n}Res(w, 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(w, 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(w, 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(w, 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 and in the initial steps of the general procedure. In case the abelian decompositions associated with MGQ^{n}Res(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) and the top

level of MGQ^{n−1}Res(w, 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(w, y, Base_{2,1}^{s(n)}, …, Base_{2,v_{s(n)}}^{s(n)}, a) to map the subgroup Q^{n}(w, y, a) into the limit group associated with the next level of the corresponding part of the anvil, Anv(MGQ^{n−1}Res)(w, y, a), i.e., into 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)(w, y, a), this image of the appropriate subgroup of Q^{n}(w, y, a) is a proper quotient of itself, and the relevant part of the developing resolution associated with the anvil Anv(MGQ^{n−1}Res)(w, y, 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 with respect to the parameter subgroup < w, y, Base_{ℓ,1}^{s}, …, Base_{ℓ,t_{s}}^{s}, a > 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 (as in part (4) of the general step).

Given an auxiliary resolution we set the developing resolution to be the resolution induced by the subgroup < y, 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)(w, y, a), to be the subgroup generated by the the auxiliary limit group associated with the auxiliary resolution and the subgroup Q^{n}(w, y, a), amalgamated along the copies of the limit group associated with the level to which one of the steps (1)-(4) was applied. With the developing resolution and its associated anvil, we naturally associate a family of formal solutions, x_{α}(w, y, 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, a >, and with each level of the induced resolution there is associated an (induced) abelian decomposition (see section 3 for the construction of the induced resolution).

Proposition 4.11. With the notation above, 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, a > be the subgroup generated by the closure of the developing resolution in the anvil Anv(MGQ^{n−1}Res)(w, y, 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, a >.

Then there exists some level j, so that the structure of the abelian decompositions associated with the resolutions induced by the subgroup < v, y, a > above level j are identical to the structure of the abelian decompositions associated with the corresponding resolution induced by the subgroup < y, 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, a > is strictly smaller than the number of factors in the corresponding free decomposition associated with the resolution induced by the subgroup < y, 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, a > is strictly smaller than the complexity of the abelian decomposition associated with the resolution induced by < y, a >.

Proof: Identical to the proof of proposition 4.8.

□

Using proposition 4.11, in case all the multi-graded resolutions used in the construction of the developing resolution are of maximal possible complexity, we replace the resolutions induced by the subgroup < y, a > used to construct the developing resolution, by the resolutions induced by the subgroup < v, y, a >. With the obtained (modified) developing resolution we associate a covering closure according to theorem 1.17 of [Se2], with each closure we naturally associate an anvil, denoted Anv(MGQ^{n}Res(w, y, a), obtained from the anvil Anv(MGQ^{n−1}Res)(w, y, a) by adding the appropriate roots corresponding to the specific closure, and with the closure and its associated anvil we associate a family of formal solutions x_{α}(w, y, a), parameterized by the auxiliary modular groups associated with the developing resolution associated with the anvil Anv(MGQ^{n−1}Res)(w, y, a).

Given the developing resolution, the anvil, and the family of formal solutions defined over them, we construct the staggered data-structure according to definition 4.5. With the anvil constructed in the general step of the iterative procedure, we have associated a family of formal solutions {x_{α}(w, y, 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}(w, y, a), y, a) = 1, a system for each fixed index j, and the entire family of formal solutions {x_{α}(w, y, a)}.

IV: Termination of the iterative procedure

Defining the initial, second and general steps of our iterative procedure for validation of a sentence, 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. 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 data-structures 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 group Rlim(y, a) but rather to specializations of a proper quotient of it. Clearly, once we show that, the ascending chain condition for limit groups ([Se1],5.1) guarantees the termination of our iterative procedure for validation of a sentence.

Theorem 4.12. The iterative procedure for validation of an AE sentence terminates after finitely many steps.

Proof: Suppose that the iterative procedure for validation of an AE sentence does not terminate after finitely many steps, which implies that the procedure must contain an infinite path. Each time part (1) of the general step of the procedure is applied along the given infinite path of the procedure, the limit group Q^{n}(y, 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 along our given path of the procedure. If part (3) is applied to an anvil along the infinite path, then the limit group Q^{ℓ(s)}(w_{ℓ(s)}, y, a) is replaced by its proper quotient, Q^{n}(w_{ℓ(s)}, y, 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}(w_{ℓ(s)}, y, a) 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 along the infinite path, the index of the based subgroups, s(n), is increased by 1, and the complexity of the associated (multi-graded) abelian decomposition strictly decreases. Since a strictly decreasing sequence of complexities of (multi-graded) abelian decompositions 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) along our given infinite path of the iterative procedure only finitely many times.

Hence, there exists some step n_{0} along the given path of the iterative procedure, so that for every step n > n_{0}:

(i) s(n) = s(n_{0}).

(ii) for every 1 ≤ s ≤ s(n_{0})−1 Q^{ℓ(s)}(w_{ℓ(s)}, y, a) is isomorphic to Q^{n}(w_{ℓ(s)}, y, a). (iii) the multi-graded resolution associated with the top level of the developing

resolution constructed in step n of the sieve procedure, MGQ^{n}Res, is of maximal complexity.

Since for every n > n_{0}, the multi-graded resolutions MGQ^{n}Res associated with the top level of the developing resolution are of maximal complexity, for every n > n_{0}, the procedure is effectively applied to the limit group associated with the terminal (second) level of the maximal complexity (one level) multi-graded 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}(w_{ℓ(s(n_{0})−1)}, y, a), Q_{term}^{n}(w_{ℓ(s(n_{0})−1)}, y, a), is a proper quotient of Q^{ℓ(s)}(w_{ℓ(s(n_{0})−1)}, y, a). If at some step n > n_{0}, the image of Q^{n}(y, a) in the terminal level of MGQ^{n}Res, Q_{term}^{n}(y, a), is a proper quotient of Q^{1}(y, 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}(y, a) we have started with. Otherwise, let s_{1} ≤ s(n_{0}) − 1 be the minimal index s, for which Q_{term}^{n}(w_{ℓ(s)}, y, a), is a proper quotient of Q^{ℓ(s)}(w_{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) along our given infinite path of the iterative 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) along the given infinite path of the iterative procedure are of maximal complexity.

By the ascending chain condition for limit groups ([Se1],5.1), if we repeat this argument inductively, we obtain either:

(i) a level d, and an index n_{d}, for which the multi-graded resolutions associated with the top d levels of the associated developing resolutions are of maximal complexity at all steps n > n_{d}, and the image of Q^{n}(y, a) in the limit group associated with the d + 1 level of the developing resolution, is a proper quotient of Q^{1}(y, a) for all n > n_{d}.

(ii) an infinite sequence of limit groups Q(w_{ℓ(s)}, y, a), a corresponding sequence of (multi-graded) abelian decompositions (the multi-graded abelian JSJ decompositions of the (multi-graded) limit groups Q(w_{ℓ(s)}, y, a)), so that the sequence of complexities of abelian decompositions of these abelian decompositions is strictly decreasing.

Since a sequence of strictly decreasing complexities of abelian decompositions 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 Rlim(y, a) we have started the procedure with. Therefore, the ascending chain condition for limit groups, contradicts the existence of an infinite path, which implies that the iterative procedure for validation of a sentence terminates after finitely many steps.

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