DIOPHANTINE GEOMETRY OVER GROUPS III:

RIGID AND SOLID SOLUTIONS

Z. Sela^{1,2}

posted Nov. 26, 2000

This paper is the third 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 third paper we analyze exceptional families of solutions to parametric system of equations. The structure of the exceptional solutions, and the global bound on the number of families of exceptional solutions we obtain, play an essential role in our approach towards quantifier elimination in the elementary theory of a free group presented in the fifth paper of this sequence.

In the first paper in this sequence on Diophantine geometry over groups we studied sets of solutions to systems of equations defined over a free group and parametric families of such sets. With a given system of equations we associated a canonical Makanin-Razborov diagram. This Makanin-Razborov diagram 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 constructing the (ungraded) Makanin-Razborov diagram associated with a system of equations, we are guaranteed that the abelian JSJ decomposition associated with a limit group is non-trivial, and that a shortening quotient of a limit group is always a proper quotient of that limit group. In studying graded limit groups and their associated (canonical) graded Makanin-Razborov diagrams, we were forced to study rigid and solid graded limit groups. A rigid graded limit group is a graded limit group for which its associated graded abelian JSJ decomposition is trivial, and a solid graded limit group is a graded limit group with a non-trivial graded abelian JSJ decomposition, but for which there exists a unique maximal graded shortening quotient, and this shortening quotient is isomorphic to the original solid graded limit group (i.e., it is not a proper quotient of the graded solid limit group).

To study rigid and solid limit groups we have associated a (canonical) finite collection of (maximal) flexible quotients with either a rigid or a solid limit group, and showed that for each specialization of the defining parameters there are at most finitely many families of solutions that do not factor through these flexible quotients. Flexible quotients were shown to be proper quotients of the original rigid or solid limit group they are associated with, so they enabled us to use the ascending chain condition for limit groups to construct the graded Makanin-Razborov diagram. The families of solutions that do not factor through the flexible quotients were termed rigid solutions for a rigid limit group, and solid solutions for a solid one.

1Hebrew University, Jerusalem 91904, Israel.

2Partially supported by an Israel academy of sciences fellowship.

Understanding the exceptional rigid and solid solutions seems to be a necessity in any attempt to analyze elementary sets defined over a free group. In defining these families (section 11 of [Se1]), we have only shown that for each value of the defining parameters there exists at most finitely many families of either rigid or solid solutions. However, in analyzing elementary sets this finiteness is not sufficient. In fact a global bound (independent of the parameters value) on the number of families of rigid and solid solutions is required. Achieving such a global bound is the main goal of this paper.

We start this paper by studying the combinatorial structure of rigid solutions of a rigid limit group, and (strictly) solid solutions of a solid limit group that are the shortest in their solid family. In the first section we get a ”bound” on the combinatorial structure of such solutions. In the second section we use the combinatorial boundedness achieved in the first section to get the desired global bound on the number of rigid and families of (strictly) solid families of solutions. In the third section we use our basic understanding of rigid and solid solutions, to analyze the singular locus of the bundle of solutions of a parametric system of equations, and show that the various stratas in the basis of the bundle that correspond to the different possible degenerations that occur in the fibers of the bundle, are in the Boolean algebra of AE sets. The global bound on the number of families of rigid and solid solutions and the basic analysis of the singular locus play an essential role in our approach towards quantifier elimination in the elementary theory of free groups presented in the fifth paper in this sequence. In the fourth section we present solid configuration limit groups, which are limit groups that are used to study a finite collection of solid solutions simultaneously. Solid configuration limit groups can be used to show that in some cases the results of section 3 can be strengthened, and the stratas associated with the singular locus are in fact in the Boolean algebra of universal sets. This in combination with the procedure presented in the fifth paper of the sequence, can be used to show that certain elementary sets are in the Boolean algebra of universal sets. However, the construction of solid configuration limit groups seems to be restricted to the case of solid limit groups with no flexible quotients, which prevents us from using them in our general procedures for analyzing elementary sets. Still, they are an important part of the structure theory, and they are likely to be applicable elsewhere.

§1. Rigid and Solid Solutions and their Combinatorial Types

In the the first paper in this sequence we introduced graded limit groups, their graded Makanin-Razborov diagram, and the singular graded limit groups associated with each of its resolutions and rigid and solid graded limit groups. The structure of the singular locus gives us a basic understanding of the connection between the graded Makanin-Razborov diagram and the restricted diagrams for each possible specialization of the set of defining parameters. To get a better understanding of this connection, which will later be used to analyze the structure of elementary sets of a free group, we need to look more closely into the structure of rigid and solid graded limit groups.

Let X be the Cayley graph of a rigid graded limit group Rgd(x, p, a) with respect to the generating system Rgd(x, p, a) =< x, p, a >, and let T be the Cayley graph (tree) of the free group F_{k} with respect to the generating system F_{k} =< a >. Clearly, a homomorphism h : Rgd(x, p, a) → F_{k} corresponds to a natural equivariant map τ : X → T, where each edge in X is mapped to a (possibly degenerate) segment in T.

Definition 1.1. Let B_{R} be the ball of radius R in the Cayley graph X of Rgd(x, p, a). We say that a homomorphism h : Rgd(x, p, a) → F_{k} is R-AP -covered, if the image in T of the union of edges labeled by either an element of {a} or an element of {p} in B_{R} covers the entire image in T of the ball B_{1}.

To define and control the combinatorial types of rigid solutions we need the following basic theorem.

Theorem 1.2. Let Rgd(x, p, a) be a rigid limit group. There exists a constant R_{0} so that every rigid homomorphism h : Rgd(x, p, a) → F_{k} is R_{0}-AP -covered.

Proof: Suppose there is no such constant R_{0}. Let {h_{n} | h_{n} : Rgd(x, p, a) → F_{k}} be a sequence of rigid homomorphisms, so that for every integer n, the n-th homomorphism in the sequence, h_{n}, is not n-AP -covered. By ([Pa],2.3) from the sequence of rigid homomorphisms {h_{n}} one can extract a subsequence converging in the Gromov-Haussdorff topology into an action of Rgd(x, p, a) on a (pointed) real tree (Y, y_{0}). The action of Rgd(x, p, a) on the real tree Y satisfies the conclusions of lemmas 1.3 and 1.4 of [Se1], hence, it is in particular a stable action. If K_{∞} denotes the kernel of the action, then L_{∞} = Rgd(x, p, a)/K_{∞} is a limit group, so by ([Se1],4.4) it is finitely presented.

Let Y_{AP} ⊂ Y be the convex hull of the images of the base point y_{0} under the action of the subgroup AP =< a, p > of our rigid graded limit group Rgd(x, p, a). Clearly, either there exists a non-degenerate segment J ⊂ Y which is not in the orbit of Y_{AP} in Y, or the orbit of every non-degenerate segment in Y contains a non-degenerate subsegment which is in the subtree Y_{AP}.

Suppose that there exists a non-degenerate segment J ⊂ Y which does not contain a non-degenerate subsegment that is in the orbit of Y_{AP}. Suppose further that the action of L_{∞} on the real tree Y is geometric, i.e., the action of L_{∞} on Y is equivariantly isometric to the action of L_{∞} on some resolution of Y (see [Be-Fe2]). Note that in this (geometric) case, there exists a canonical graph of groups with fundamental group L_{∞} associated with the action of L_{∞} on the real tree Y (see [Be-Fe2]). Let Λ_{L_{∞}} be this graph of groups. If the orbit of Y_{AP} under the action of L_{∞} intersects a mixing component of Y in a non-degenerate segment, then the whole mixing component is in the orbit of Y_{AP}. If the orbit of Y_{AP} under the action of L_{∞} intersects an edge in the discrete part of Y in a non-degenerate segment, then the entire edge is in the orbit of Y_{AP}. Therefore, since the orbit of Y_{AP} misses a non-degenerate segment J ⊂ Y, the subgroup AP < L_{∞} is contained in a proper connected subgraph of groups Λ′ ⊂ Λ_{L_{∞}}.

If the action of L_{∞} on the real tree Y is not geometric, and the orbit of Y_{AP} misses a non-degenerate segment in the real tree Y, then there exists a resolution (see [Be-Fe2]) (Y^{r}, y^{r}) of (Y, y_{0}) equipped with an isometric action of L_{∞}, so that the orbit of the convex hull of the images of y^{r} under the action of L_{∞} in Y^{r}, denoted Y_{AP}^{r}, does not contain a non-degenerate segment J^{r} ⊂ Y^{r}, and no nondegenerate subsegment contained in J. Since the action of L_{∞} on the resolution Y^{r} of Y is geometric by construction, there is a canonical graph of groups Λ_{L_{∞}}^{r} associated with the action of L_{∞} on Y^{r}, and the subgroup AP < L_{∞} is contained in a proper connected subgraph of groups Λ′ ⊂ Λ_{L_{∞}}^{r}.

Now, since AP is contained in a proper connected subgraph of groups Λ′ of a graph of groups Λ_{L_{∞}} with fundamental group L_{∞}, and since the real tree Y was constructed from the homomorphisms {h_{n} : Rgd(x, p, a) → F_{k}}, for large enough n, the homomorphisms {h_{n}} are flexible homomorphisms, a contradiction to our assumption of the homomorphisms {h_{n} : Rgd(x, p, a) → F_{k}} being rigid.

By the argument given above, for the rest of the argument we may assume that the orbit of Y_{AP} in Y intersects any non-degenerate segment in the real tree (Y, y_{0}) in a non-degenerate segment. Furthermore, given any non-degenerate segment J in the real tree Y, the segment is covered in a finite time (depending only on the segment J) by the images of the edges (y_{0}, a_{i}(y_{0})) and (y_{0}, p_{j}(y_{0}), where the elements a_{i} are the generators of the coefficient group F_{k}, and the elements p_{j} are the given generators of the parameter subgroup P.

In case every non-degenerate segment in the real tree Y intersects the orbit of the tree Y_{AP} in a non-degenerate subsegment, we continue inductively by analyzing the actions of each of the point and edge stabilizers on corresponding real trees. To demonstrate our approach, we will first assume that the stabilizer of every nondegenerate segment in the real tree Y is trivial and the action of L_{∞} on Y contains no IET components, and then generalize our approach to a general stable action.

Let V_{1}, …, V_{q} be the (conjugacy classes of) point stabilizers in the real tree Y. Starting with the sequence of homomorphisms {h_{n} : Rgd(x, p, a) → F_{k}} we pass to a subsequence (still denoted {h_{n}}) for which each of the sequences of actions of the subgroups V_{i} on the Cayley graph T of F_{k} via the sequence of homomorphisms {h_{n}} converges into a (non-trivial) action of the subgroup V_{i} on some real tree Y_{i}. If at least one of the trees Y_{i} contains a non-degenerate segment that does not intersect the orbit of Y_{AP} in a non-degenerate segment, then the graph of groups Λ_{L_{∞}} of L_{∞} can be extended to a graph of groups Λ_{L_{∞}}^{1} of L_{∞} so that the subgroup AP is contained in a the fundamental group of a proper subgraph of the graph of groups Λ_{L_{∞}}^{1}. Therefore, for large enough n, each homomorphism h_{n} is a flexible homomorphism, a contradiction. In case every non-degenerate segment in the real trees Y_{i} intersects the orbit of the tree Y_{AP} in a non-degenerate subsegment, we continue inductively by analyzing the actions of each of the point and edge stabilizers in the real trees Y_{i} on corresponding real trees. In this case every non-degenerate segment J in one of the real trees Y_{i} is covered by non-degenerate segments that are contained in the orbit of Y_{AP} in a finite time (depending only on the segment J). Since we assumed that the stabilizer of each non-degenerate segment in the real tree Y is trivial, and the action of L_{∞} on Y contains no IET components, the rank of each of the point stabilizers V_{i} is strictly smaller than the rank of the ambient group L_{∞}. Hence, if we assume that throughout our inductive process, the action of each of the point stabilizers on the corresponding real trees satisfies the same assumptions, the inductive procedure terminates after finitely many steps. Since for each index n, the homomorphism h_{n} is not n-AP covered, in one of the steps of the inductive process at least in one of the trees corresponding to a point stabilizer, there must exist a non-degenerate segment containing no non-degenerate segments that are not included in the orbit of the subtree Y_{AP}. Hence, the graph of groups Λ_{L_{∞}} of L_{∞} can be extended to a graph of groups Λ_{L_{∞}}^{j} of L_{∞} so that the subgroup AP is contained in a the fundamental group of a proper subgraph of the graph of groups Λ_{L_{∞}}^{j}. Therefore, for large enough n, each homomorphism h_{n} is a flexible homomorphism, a contradiction, and the proof is completed in case the action of each of the point stabilizers on its corresponding tree contains no non-trivial stabilizers of non-degenerate segments and no IET components.

Concluding the proof of the theorem in case the action of the obtained limit groups on their associated real trees contain no non-trivial stabilizers of nondegenerate segments, and no IET components, we proceed to the general case. As we argued in the above special case, if there exists a non-degenerate segment J ⊂ Y which does not contain a non-degenerate subsegment that is in the orbit of Y_{AP}, then there exists a graph of groups Λ_{L_{∞}} associated with the action of L_{∞} on the real tree Y in case the action is geometric, or with a resolution of that action in case the action is not geometric, so that the subgroup AP < L_{∞} is contained in a proper connected subgraph of groups Λ′ ⊂ Λ_{L_{∞}}. Since AP is contained in a proper connected subgraph of groups Λ′ of the graph of groups Λ_{L_{∞}} with fundamental group L_{∞}, and since the real tree Y was constructed from the homomorphisms {h_{n} : Rgd(x, p, a) → F_{k}}, for large enough n, the homomorphisms {h_{n}} are flexible homomorphisms, a contradiction to our assumption of the homomorphisms {h_{n} : Rgd(x, p, a) → F_{k}} being rigid. Hence, for the rest of the argument we may assume that no non-degenerate subsegment J ⊂ Y contains no non-degenerate subsegment which is not in the orbit of the subtree Y_{AP}. Since every segment J ⊂ Y contains a non-degenerate subsegment which is in the orbit of the subtree Y_{AP}, every non-degenerate segment J ⊂ Y is covered by finitely many segments which are in the orbit of Y_{AP}, hence, every non-degenerate segment J ⊂ Y is covered in a finite time (depending only on the segment J) by the images of the segments corresponding to the generators {a_{i}} of F_{k} and the generators {p_{j}} of the parameter subgroup P in the real tree Y.

Our approach in the general case is similar to the approach used in case the action of the group L_{∞} on the real tree Y contains no non-degenerate segments with non-trivial stabilizers and no IET components. In the general case it is not possible to obtain a non-trivial free decomposition of L_{∞} from its action on the real tree Y, still by analyzing the action of vertex and edge stabilizers on corresponding real trees, we will be able to either reduce the first betti numbers of the limit groups in question, or replace a limit group by its proper quotient, or successively refine the graphs of groups associated with a given limit group, until we finally obtain a graph of groups of L_{∞} in which the subgroup AP is contained in a proper subgraph of groups of L_{∞}, which contradicts the rigidity of the homomorphisms {h_{n}}.

Let Λ_{L_{∞}} be the graph of groups associated with the action of L_{∞} on the real tree Y, or with a resolution of this action. If the graph of groups Λ_{L_{∞}} gives rise to a free decomposition of L_{∞}, or if Λ_{L_{∞}} collapses to a graph of groups with cyclic edge stabilizers and more than a single vertex group, then each factor in a free decomposition and each vertex group in a cyclic splitting with more than a single vertex has a strictly smaller first betti number than the first betti number of L_{∞}. If the action of L_{∞} on the real tree Y contains an axial component, or if there exists a non-cyclic free abelian subgroup of L_{∞} which does not fix a point in the real tree Y, then L_{∞} admits a graph of groups with a non-cyclic free abelian vertex group, so that the first betti number of the fundamental group of the subgraph that does not contain the abelian vertex group is smaller than the first betti number of L_{∞}. Hence, if each of these cases applies, we continue (inductively) with either a factor in a free decomposition, a vertex group in a cyclic splitting with more than a single vertex, or the fundamental group of the proper subgraph that does not contain the corresponding abelian vertex group. If non of these cases applies to the action of L_{∞} on the real tree Y, then the action of L_{∞} on the real tree Y is discrete, each edge group is f.g. free abelian subgroup of L_{∞}, and there exists a (possibly degenerate) spanning subtree in the graph of groups Λ_{L_{∞}} in which each edge group is a non-cyclic free abelian group.

With the graph of groups Λ_{L_{∞}} associated with the discrete action of L_{∞} on the real tree Y, we naturally associate the modular group Mod(Λ_{L_{∞}}), generated by Dehn twists along the edges of Λ_{L_{∞}}. For some index n_{0}, and for every index n > n_{0} each of the homomorphisms h_{n} : Rgd(x, p, a) → F_{k} splits through the (quotient) limit group L_{∞}, i.e., it is in fact a homomorphism h_{n} : L_{∞} → F_{k}. Hence, we may act on the homomorphisms {h_{n} : L_{∞} → F_{k}} by pre-composing each of them with modular automorphisms from the modular group Mod(Λ_{L_{∞}}), which allows us to apply the shortening procedure for the homomorphisms {h_{n}}. For each homomorphism h_{n} : L_{∞} → F_{k}, we choose a modular automorphism φ_{n} ∈ Mod(Λ_{L_{∞}}) so that the homomorphism h_{n}^{1} = h_{n} ∘φ_{n} is the shortest among all possible choices of an automorphism φ ∈ Mod(Λ_{L_{∞}}). Note that the modular automorphisms φ_{n} ∈ Mod(Λ_{L_{∞}}) preserve the conjugacy classes of the specializations of the various vertex groups in the graph of groups Λ_{∞}.

From the sequence of homomorphisms {h_{n}^{1}} we extract an infinite subsequence (still denoted {h_{n}^{1}}) that converges in the Gromov-Haussdorff topology into a faithful action of some limit group L_{∞}^{1} on some pointed real tree (Y_{1}, y_{1}). The limit group L_{∞}^{1} is naturally a quotient of the limit group L_{∞} we have started with. Let Λ_{L_{∞}^{1}} be the graph of groups associated with either the action of L_{∞}^{1} on the real tree Y_{1} in case this action is geometric, or with some resolution of the action in case it is not geometric. Each of the vertex groups V_{i} in the graph of groups Λ_{L_{∞}} is naturally embedded in the limit group L_{∞}^{1}, so it inherits an action on the tree Y_{1}, and a decomposition Λ_{i} associated with this action, which the same decomposition associated by the vertex group V_{i} from the graph of groups Λ_{L_{∞}^{1}}. If in one of the subtrees of Y_{1} associated with the various vertex groups of Λ_{L_{∞}}, {V_{i}}, there exists a non-degenerate segment J that does not intersect the orbit of the subtree Y_{AP} in a non-degenerate segment, then using the graphs of groups Λ_{1}, …, Λ_{q} of the vertex groups V_{1}, …, V_{q}, the graph of groups Λ_{L_{∞}} can be extended to a graph of groups Λ̂_{L_{∞}} in which the subgroup AP < L_{∞} is contained in a proper subgraph of groups, a contradiction to the rigidity of the homomorphisms {h_{n}}. Hence, we may assume that every subsegment J ⊂ Y intersects the orbit of the tree Y_{AP} in a non-degenerate subsegment in all the subtrees associated with the vertex groups V_{i} of Λ_{L_{∞}}. If L_{∞}^{1} is a proper quotient of L_{∞} we replace L_{∞} by L_{∞}^{1} and proceed inductively. If L_{∞}^{1} is isomorphic to L_{∞} and the graph of groups Λ_{L_{∞}^{1}} gives rise to a free decomposition of L_{∞} = L_{∞}^{1}, or if Λ_{L_{∞}^{1}} collapses to a graph of groups with cyclic edge stabilizers and more than a single vertex group, then each factor in a free decomposition and each vertex group in a cyclic splitting with more than a single vertex has a strictly smaller first betti number than the first betti number of L_{∞} = L_{∞}^{1}. If the action of L_{∞}^{1} on the real tree Y_{1} contains an axial component, or if there exists a non-cyclic free abelian subgroup of L_{∞}^{1} which does not fix a point in the real tree Y_{1}, then L_{∞} = L_{∞}^{1} admits a graph of groups with a non-cyclic free abelian vertex group, so that the first betti number of the fundamental group of the subgraph that does not contain the abelian vertex group is smaller than the first betti number of L_{∞}. Hence, if each of these cases applies, we continue (inductively) with either a factor in a free decomposition, a vertex group in a cyclic splitting with more than a single vertex, or the fundamental group of the proper subgraph that does not contain the corresponding abelian vertex group. If non of these cases applies to the action of L_{∞}^{1} on the real tree Y_{1}, then the action of L_{∞}^{1} on the real tree Y_{1} is discrete, each edge group is f.g. free abelian subgroup of L_{∞}, and there exists a (possibly degenerate) spanning subtree in the graph of groups Λ_{L_{∞}^{1}} in which each edge group is a non-cyclic free abelian group.

Let D_{∞} be the subgroup of the limit group L_{∞} which is the fundamental group of the subgraph of Λ_{∞} containing all the edges with non-cyclic stabilizer. By our assumptions, the graph of groups Λ_{∞} collapses to a graph of groups Λ_{D_{∞}} with one vertex stabilized by D_{∞} and (possibly) few loops with cyclic stabilizer based on this vertex. Clearly, b_{1}(D_{∞}) ≤ b_{1}(L_{∞}). Recall that with each vertex group V_{i} in the graph of groups Λ_{∞}, we have associated a graph of groups Λ_{i}, which is the graph of groups inherited by V_{i} from the graph of groups Λ_{L_{∞}^{1}}. By construction all the non-cyclic edge groups in the graph of groups Λ_{L_{∞}} are elliptic in each of the graphs of groups Λ_{i} of the vertex group V_{i}. If there exists an edge with trivial stabilizer in one of the graphs of groups Λ_{i}, then the graphs of groups Λ_{i} can be used to obtain a free decomposition of D_{∞}, clearly the first betti number of each of the factors in this free decomposition is strictly smaller than the first betti number of the limit group L_{∞}, so we proceed our analysis with each of the factors of this free decomposition. Hence, for the rest of our analysis we may assume the non of the edge groups in the graphs of groups Λ_{i} is trivial. If all the cyclic edge groups in the graph of groups Λ_{∞} are elliptic in all the graphs of groups Λ_{i} as well, then we can replace each of the vertex groups V_{i} by its associated graph of groups Λ_{i} and obtain a graph of groups Δ_{L_{∞}} which is a proper refinement of the graph of groups Λ_{L_{∞}}, i.e., the number of edges in Δ_{L_{∞}} is strictly bigger than the number of edges in Λ_{L_{∞}}. Hence, if all the edge groups in Λ_{L_{∞}} are elliptic in the graphs of groups Λ_{i}, we proceed our analysis by replacing the graph of groups Λ_{L_{∞}} with its proper refinement Δ_{L_{∞}}, and repeat the shortening procedure with the corresponding enlarged modular group. Since by the generalized accessibility of M. Bestvina and M. Feighn [Be-Fe1] there is a global bound on the combinatorics of small graphs of groups associated with L_{∞}, such a refinement procedure has to terminate in a finite time. Hence, for the rest of our analysis we may assume that non of the edge groups in each of the graphs of groups Λ_{i} is trivial, and that at least one of the cyclic edge groups in Λ_{L_{∞}} is not elliptic in at least one of the graphs of groups Λ_{i}.

Suppose that there exists a cyclic edge group Z_{e} in Λ_{L_{∞}} which is not elliptic in one of the graphs of groups Λ_{i}. Then Z_{e} is not elliptic in the graph of groups Λ_{L_{∞}^{1}} of L_{∞}^{1}. By our assumptions, Both graphs of groups Λ_{L_{∞}} and Λ_{L_{∞}^{1}} contain spanning trees where all edge groups are non-cyclic abelian. Let D_{∞}^{1} < L_{∞}^{1} be the fundamental group of the maximal subgraph of Λ_{L_{∞}^{1}} in which all edge groups are non-cyclic abelian, and let U_{∞} < L_{∞} be the fundamental group of the subgraph obtained from Λ_{L_{∞}} by taking out the edge stabilized by Z_{e}. By construction, the graph of groups Λ_{L_{∞}} collapses to a graph of groups Θ_{L_{∞}} with a single vertex stabilized by U_{∞} and single edge stabilized by the cyclic group Z_{e}. Each of the vertex groups in the graph of groups Λ_{L_{∞}^{1}}, being a subgroup of L_{∞} inherits a graph of groups from Λ_{L_{∞}}. We claim that at least one of these graph of groups gives rise to a free decomposition of the corresponding vertex group of Λ_{L_{∞}^{1}}. Suppose this is not the case, i.e., the stabilizers of all edges in the graphs of groups inherited by the vertex groups of Λ_{L_{∞}^{1}} from the graph of groups Λ_{L_{∞}} are non-trivial. Since the cyclic group Z_{e} can not be conjugated to any of these vertex groups by assumption, each of these vertex groups can be conjugated into the vertex groups U_{∞} in the graph of groups Θ_{L_{∞}}, which implies that the subgroup D_{∞}^{1} < L_{∞}^{1} can be conjugated into U_{∞} as well. By assumption, Λ_{L_{∞}^{1}} collapses to a graph of groups with one vertex group stabilized by D_{∞}^{1} and a bouquet of circles with cyclic stabilizers based on this vertex. Since non of the cyclic stabilizers of these circles can be conjugated into the cyclic group Z_{e}, if D_{∞}^{1} can be conjugated into U_{∞} then the ambient group L_{∞} = L_{∞}^{1} can be conjugated into U_{∞} as well, a contradiction. Hence, at least one of the vertex groups in Λ_{L_{∞}^{1}} inherits a graph of groups containing at least one edge with a trivial stabilizer, so the graphs of groups of inherited by all the vertices groups in Λ_{L_{∞}^{1}} can be combined to a graph of groups of D_{∞}^{1}, which gives rise to a free decomposition of D_{∞}^{1}. Clearly, b_{1}(D_{∞}^{1}) ≤ b_{1}(L_{∞}) so the first betti number of each of the factors in the free decomposition of D_{∞}^{1} obtained from the graph of groups Λ_{L_{∞}} is strictly smaller than the first betti number of L_{∞}. Therefore, in case there exists a cyclic edge group Z_{e} in Λ_{L_{∞}} which is not elliptic in Λ_{L_{∞}^{1}}, we proceed with each of the factors of D_{∞}^{1} separately.

Since at each step of our inductive procedure we either properly refine a given graph of groups of a limit group, replace a limit group by its proper quotient, or continue with limit groups which have strictly smaller betti number than the betti number of the limit group in the previous step of the procedure, the inductive procedure has to terminate. Since for each index n the homomorphism h_{n} : L_{∞} → F_{k} was assumed not n-AP -covered, at the terminating stage of the procedure we have to obtain an action of a limit group L_{∞}^{j} on some real tree Y^{j} that contains a non-degenerate segment that does not intersect the orbit of the subtree Y_{AP} in a non-degenerate segment. Now, using the graphs Λ_{L_{∞}^{i}} associated with the actions of the corresponding limit groups along our inductive procedure, the graph of groups Λ_{L_{∞}} of L_{∞} can be extended to a graph of groups Λ_{L_{∞}}^{j} of L_{∞} so that the subgroup AP is contained in a the fundamental group of a proper subgraph of the graph of groups Λ_{L_{∞}}^{j}. Therefore, for large enough n, each homomorphism h_{n} is a flexible homomorphism, a contradiction, and the proof of theorem 1.2 is completed.

□

In a similar way to rigid homomorphisms of rigid graded limit groups we analyze solid homomorphisms of solid graded limit groups. We start by defining a solid family of specializations of a solid graded limit group.

Definition 1.3. Let Sld(x, p, a) be a solid graded limit group. Let h : Sld(x, p, a) → F_{k} be a solid homomorphism of the solid graded limit group Sld(x, p, a). The family of homomorphisms containing the given homomorphism h, consists of the following specializations:

(i) all homomorphisms τ : Sld(x, p, a) → F_{k} for which τ = h ∘ φ, where φ ∈

GMod(Sld(x, p, a)). ii Let E_{1}, …, E_{q} be the edge groups in the graded abelian JSJ decomposition

of Sld(x, p, a). Let τ be a homomorphism obtained from the given solid homomorphism h by pre-composing with an element of the graded modular group GMod(Sld(x, p, a)) (as in part (i)). In the solid family associated with h we further include all homomorphisms ν : Sld(x, p, a) → F_{k} obtained from τ by pre-composing with Dehn twists along the maximal cyclic subgroups of F_{k} that contain the subgroups τ(E_{1}), …, τ(E_{q}).

(iii) Let L_{1}, …, L_{n} be the abelian vertex groups in the graded abelian splitting of Sld(x, p, a), so that the vertex stabilized by L_{i} is not the distinguished vertex stabilized by the subgroup AP. Suppose that L_{i} = L′_{i}+B_{i}, where the subgroup generated by the edge groups connected to the vertex stabilized by L_{i} is of finite index in L′_{i}. Let ν : Sld(x, p, a) → F_{k} be obtained from h by parts (i) and (ii). In the solid family associated with the given solid homomorphism h, we include all homomorphisms η : Sld(x, p, a) → F_{k} associated with the homomorphism ν : Sld(x, p, a) → F_{k}, for which for every element g ∈ B_{i}, η(g) is an element from the maximal cyclic subgroup in F_{k} that contains the subgroup ν(L′_{i}), and η(g) = ν(g) for every element g ∈ Sld(x, p, a) for which:

1) g is an element of a non-abelian vertex group or the distinguished vertex group stabilized by AP.

2) g is an element of L′_{i} for some i.

We say that the family of homomorphisms containing the homomorphism h is a solid family if the family of homomorphisms contains no flexible homomorphism.

An immediate, but important property of solid families is the following claim. Since it follows immediately from the definition we omit its proof and leave it for the interested reader.

Claim 1.4. Let Sld(x, p, a) be a solid graded limit group. Let h_{i} : Sld(x, p, a) → F_{k} be solid homomorphisms for i = 1, 2, 3. Then:

(i) if h_{1} belongs to the solid family associated with h_{2}, then h_{2} belongs to the solid family associated with h_{1}.

(ii) if h_{1} belongs to the solid family associated with h_{2}, and h_{3} belongs to the solid family associated with h_{2}, then h_{3} belongs to the solid family associated with h_{1}.

i.e., membership in solid families is an equivalence relation for solid specializations.

To get a bound for solid family of solutions that is parallel to the bound on the number of rigid solutions given by theorem 1.2, we need to define strictly solid solutions and families of such solutions.

Definition 1.5. Let Sld(x, p, a) be a solid graded limit group, and let F lx_{1}(x, p, a), …, F lx_{s}(x, p, a) be the canonical collection of (maximal) flexible quotients of the solid graded limit group Sld(x, p, a). A specialization (x_{0}, p_{0}, a) of the solid graded limit group Sld(x, p, a) is called a strictly solid specialization if it does not factor through a completion of all the graded resolutions that factor through the flexible quotients of Sld(x, p, a). Note that a strictly solid solution is, in particular, a solid solution, and if (x_{0}, p_{0}, a) is a strictly solid solution then all the solutions that belong to the same solid family are strictly solid.

Definition 1.6. Let Sld(x, p, a) be a solid graded limit group, let X be the Cayley graph of the solid graded limit group Sld(x, p, a) with respect to the generating set Sld(x, p, a) =< x, p, a >, and let T be the Cayley graph (tree) of the free group F_{k} with respect to the generating set F_{k} =< a >. With each abelian vertex group A_{i} in the graded abelian JSJ decomposition a of the solid limit group Sld(x, p, a), we associate a fixed retract r_{i} : A_{i} → A_{i}^{1}, where A_{i}^{1} is the subgroup generated by the edge groups connected to the vertex stabilized by A_{i}. Clearly, the collection of retracts {r_{i}} extends to a retract r : Sld(x, p, a) → Sld(x, p, a) which restricts to the identity on all the non-abelian vertex groups, and to the retract r_{i} on the abelian vertex group A_{i}.

We say that a solid homomorphism h : Sld(x, p, a) → F_{k} is among the shortest in its solid family, if h = ĥ ∘ r for some homomorphism ĥ : Sld(x, p, a) → Sld(x, p, a), and the image of the generating set of Sld(x, p, a) in the free F_{k}, is the shortest among all homomorphisms h_{1} : Sld(x, p, a) → F_{k} that are in the same solid family of the homomorphism h, and for which there exists some homomorphism ĥ_{1} for which h_{1} = ĥ_{1} ∘ r.

In the case of a solid limit group we are able to bound the combinatorial types of strictly solid solutions which are among the shortest in their corresponding solid families.

Theorem 1.7. Let Sld(x, p, a) be a solid graded limit group. There exists a constant R_{0} so that every strictly solid homomorphism h : Sld(x, p, a) → F_{k} which is among the shortest in its solid family, is R_{0}-AP -covered.

Proof: Our argument is a modification of the one used to study rigid solutions of rigid limit groups (theorem 1.2). Suppose there is no such constant R_{0}. Let {h_{n} | h_{n} : Sld(x, p, a) → F_{k}} be a sequence of strictly solid homomorphisms, so that for every index n, h_{n} is one of the shortest homomorphisms in its corresponding solid family, and so that for every index n, the n-th homomorphism in the sequence, h_{n}, is not n-AP -covered. By ([Pa],2.3) from the sequence of homomorphisms {h_{n}} one can extract a subsequence converging in the Gromov-Hausdorff topology into an action of Sld(x, p, a) on a (pointed) real tree (Y, y_{0}).

The action of Sld(x, p, a) on the real tree Y satisfies the conclusions of lemmas 1.3 and 1.4 of [Se1], hence, it is in particular a stable action. Let K_{∞} denote the kernel of the action, and L_{∞} = Sld(x, p, a)/K_{∞}. Let AP =< a, p >. By the proof of theorem 1.2, from the faithful action of the limit group L_{∞} on the real tree Y, the limit group L_{∞} inherits an abelian decomposition Λ_{L_{∞}}, a decomposition in which the subgroup AP is contained in a proper subgraph of groups Λ′. Since the homomorphisms {h_{n}} from which the action of L_{∞} on the real tree Y is constructed are assumed to be strictly solid homomorphisms, each of the homomorphisms h_{n} does not factor through a free group of the form F_{k}∗ < v > in which the subgroup AP is mapped into F_{k} ([Se1], 11.7), and the limit group L_{∞} is mapped onto F_{k}∗ < v >. Hence, every vertex group which is in Λ_{L_{∞}} and not in Λ′ corresponds to either a point stabilizer in the real tree Y, an IET component or an axial component that do not intersect the orbit of the subtree Y_{AP} in a nondegenerate segment, and every edge in Λ_{L_{∞}} that is not in Λ′ and is not connected to an IET or an axial vertex group in Λ_{L_{∞}} is an edge in the discrete part of the real tree Y, and its stabilizer is necessarily a non-trivial abelian subgroup of L_{∞}.

Let Δ_{L_{∞}} be the graph of groups obtained from Λ_{L_{∞}} by collapsing Λ′ to a vertex. In Δ_{L_{∞}} the subgroup AP is contained in a vertex stabilizer, each edge group is a non-trivial abelian subgroup of L_{∞} and each vertex group that does not contain a conjugate of AP corresponds to either a point stabilizer, an IET component, or an axial component in Y.

Let η : Sld(x, p, a) → L_{∞} be the canonical quotient map. Suppose that the quotient map η maps every edge group and every vertex group that is not abelian nor quadratically hanging in the graded JSJ decomposition of Sld(x, p, a), into an edge group or into a vertex group which does not correspond to an IET nor axial component in the graph of group Δ_{∞} of L_{∞}. Then for large enough n, either the homomorphism h_{n} factors through a free group of the form F_{k}∗ < v >, where the subgroup AP is mapped into the subgroup F_{k}, a possibility that contradicts our assumption that the homomorphisms h_{n} are strictly solid, or there exists a homomorphism ĥ_{n} : Sld(x, p, a) → F_{k} which is in the same solid family as the homomorphism h_{n} and is strictly shorter than h_{n}, a contradiction to the assumption that the homomorphisms h_{n} are among the shortest homomorphisms in their solid families.

Therefore, we may assume that there exists either an edge group or a non-abelian, non-QH vertex group in the graded JSJ decomposition of Sld(x, p, a) which is not mapped by the quotient map η into either a conjugate of an edge group nor into a conjugate of a vertex group which is not an abelian nor a QH vertex group in Δ_{L_{∞}}. In this case we do the following.

With the limit group L_{∞} and its abelian decomposition Δ_{L_{∞}} we associate a finite collection of well-structured resolutions of groups that contain L_{∞}. We view L_{∞} as an ungraded limit group, and start with the taut Makanin-Razborov (ungraded) resolutions of it. We continue only with those taut Makanin-Razborov resolutions for which the limit group corresponding to them is L_{∞} and not a proper quotient of it. With each such taut Makanin-Razborov resolution, Res(x, p, a), we associate its completion, Comp(Res)(u, x, p, a), which is by construction a well-structured resolution. We continue with each of the completions separately. Given a completion, Comp(Res)(u, x, p, a), we add to it an additional level as follows. For each QH vertex group in the decomposition Δ_{L_{∞}} we add (an additional) copy of that QH vertex group and connect it to the subgroups of L_{∞} to which the corresponding QH vertex group in Δ_{∞} is connected. For each abelian vertex group and each edge that is not connected to a QH vertex group in Δ_{∞}, we add an additional abelian vertex group (as in the construction of a completion), and connect it to its associated abelian subgroup of L_{∞} as in the decomposition Δ_{L_{∞}}. Altogether, we get a completed well-structured resolution, which we denote, ERes(v, x, p, a).

The limit group L_{∞}(x, p, a) is naturally a subgroup of the limit group associated with the resolution ERes(v, x, p, a), since it is the limit group associated with the top level of the completion, Comp(Res)(u, x, p, a), hence, it is associated with the second level of the resolution ERes(v, x, p, a). Let ψ denote this natural embedding of L_{∞} into the limit group associated with ERes(u, x, p, a). This natural embedding of L_{∞}(x, p, a) into the limit group associated with ERes(v, x, p, a) can be naturally lifted to another embedding of L(x, p, a), by replacing every abelian vertex group and every QH vertex group in Δ_{L_{∞}} with its corresponding additional copy in the top level of the resolution ERes(v, x, p, a). Let ν(L_{∞}) be this last embedding. Note that in this embedding of the limit group L_{∞}(x, p, a), the subgroup AP =< p, a > is embedded into levels that are below the top level of the resolution ERes(v, x, p, a), and there exists at least one edge or a non-abelian, non-QH vertex group in the abelian decomposition associated with the solid limit group Sld(x, p, a) which is not mapped into a conjugate of the subgroup generated by the limit groups associated with the levels below the top level of the resolution ERes(v, x, p, a).

At this point, we look at the collection of all the test sequences associated with the well-structured resolution ERes(v, x, p, a). From each test sequence of ERes(v, x, p, a), the subgroup L_{∞} inherits a sequence of specializations via the embedding ν. Hence, from each such test sequence the subgroup Sld(x, p, a) inherits a sequence of specializations using the composition ν ∘ η. Since the subgroup AP is mapped by ν into the levels below the top one in ERes(v, x, p, a), and since there exists an edge or a non-abelian, non-QH vertex group in the abelian decomposition associated with Sld(x, p, a) that is not mapped by the composition ν ∘ η into a conjugate of the subgroup generated by the limit group associated with the levels below the top level of ERes(v, x, p, a), the tail of the sequence of specializations inherited by Sld(x, p, a) from a test sequence of ERes(v, x, p, a) has to be a sequence of flexible specializations. Each flexible specialization can be appropriately conjugated into a flexible quotient of Sld(x, p, a). Hence, applying the construction of formal solutions, presented in section 1 of [Se2] (cf. theorem 1.17), from the collection of test sequences of ERes(v, x, p, a), we may associate a covering closure of ERes(v, x, p, a), where each covering closure contains variables that correspond to the appropriate flexible quotient of Sld(x, p, a), and additional variables that conjugate the image (under ν ∘ η) of every non-abelian, non-QH vertex group in the abelian decomposition associated with Sld(x, p, a) into the flexible quotient.

L_{∞} was constructed as a limit group from a sequence of strictly solid homomorphisms {h_{n} | h_{n} : Sld(x, p, a) → F_{k}}. Hence, there must exist a resolution in the (ungraded) taut Makanin-Razborov diagram of L_{∞}, through which a subsequence of the sequence of homomorphisms {h_{n}} factors, and so that this sequence converges into the limit group L_{∞} and not into its proper quotient. We still denote this subsequence {h_{n}}. Since the resolutions ERes(v, x, p, a) were constructed from the collection of (ungraded) taut Makanin-Razborov resolutions of L_{∞} for which their associated limit group is L_{∞} and not a proper quotient of it, the subsequence of homomorphisms {h_{n}} has to factor through one of the resolutions ERes(v, x, p, a) and its associated composition ψ∘η. Now, from the existence of the covering closure of ERes(v, x, p, a) constructed above, all the homomorphisms of Sld(x, p, a) into F_{k} that are inherited from specializations of the resolution ERes(v, x, p, a) through the map ψ ∘ η, are not strictly solid homomorphisms of Sld(x, p, a). This clearly contradicts our assumption on the homomorphisms {h_{n}} being solid.

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§2. A Bound on the Number of Rigid and Solid Solutions

In the previous section we have shown that for a rigid (solid) graded limit group Rgd(x, p, a) (Sld(x, p, a)) there exists a constant R_{0}, so that all the rigid homomorphisms h : Rgd(x, p, a) → F_{k} (strictly solid homomorphisms h : Sld(x, p, a) → F_{k} which are the shortest in their solid family) are R_{0}-AP -covered (theorems 1.2,1.7). In this section we use this bound on the possible combinatorial types of states of rigid and shortest strictly solid homomorphisms to get a global bound on the number of possible distinct rigid homomorphisms h : Rgd(x, p, a) → F_{k} (strictly solid families of homomorphisms: h : Sld(x, p, a) → F_{k}) for any specialization of the defining parameters h(p) = p_{0}. The existence of a global bound on the number of rigid (strictly solid families of) homomorphisms plays an essential role in our approach to the quantifier elimination of predicates defined over a free group. We first consider Rigid limit groups and their rigid specializations. Mainly for presentation purposes, we first prove the existence of a bound on the number of rigid solutions assuming a bound on the periodicity of the specialization p_{0} of the defining parameters p. Later on, we modify the argument to get a global bound on the number of rigid solutions without the periodicity assumption.

Theorem 2.1. Let Rgd(x, p, a) be a rigid graded limit group. For every integer s there exists a bound b_{s}(Rgd(x, p, a)) for which for any particular value of the defining parameters p_{0} with periodicity bounded by s, there are at most b_{s} distinct rigid homomorphisms h : Rgd(x, p, a) → F_{k} satisfying h(p) = p_{0}.

Proof: Suppose that for some given integer s, there is no bound on the number of distinct rigid solutions x for all possible specializations p_{0} of the defining parameters p with periodicity bounded by s. Then there must exist some sequence {p_{n}} of specializations of the defining parameters p with periodicity bounded by s, so that for every index n, the rigid graded limit group Rgd(x, p, a) admits n distinct rigid specializations (x_{1}^{n}, p_{n}, a), …, (x_{n}^{n}, p_{n}, a).

Let B be a ball in the Cayley graph of Rgd(x, p, a), so that the ball B contains all the defining relations of the rigid graded limit group Rgd(x, p, a). By theorem 1.2 there exists some constant R′, so that for any rigid specialization (x_{0}, p_{0}, a) the ball B is R′-AP covered. We set the constant R, to be R = 2 · R′, and for each rigid homomorphism h : Rgd(x, p, a) → F_{k}, we look at the image, h(B_{R}), of the Cayley graph of radius R, B_{R}, in the free group F_{k}. We call the combinatorial type of such an image, i.e., the finite tree it spans together with notation for the image of the corresponding vertices and edges of the ball B_{R}, an R-state. Clearly, there exist only finitely many combinatorial types of R-states. Hence, we may further assume that for every index n, the combinatorial types of the R-states associated with the specializations (x_{1}^{n}, p_{n}, a), …, (x_{n}^{n}, p_{n}, a) are identical.

Furthermore, we may also assume that after rescaling the metric on the R-states associated with the specializations {(x_{i}^{n}, p_{n}, a)} by dilatation constants depending only on the specializations of the defining parameters p_{n}, for every sequence of indices {i_{n} | 1 ≤ i_{n} ≤ n}, the (rescaled) R-states associated with the sequence of specializations {(x_{i_{n}}^{n}, p_{n}, a)} converge in the Gromov-Hausdorff topology on metric spaces, and the limit (finite) tree Y is independent of the particular choice of indices {i_{n}}.

We have chosen the sequence of specializations {(x_{i_{n}}^{n}, p_{n}, a)} to have the same combinatorial type of R-state. We denote by Mp_{i_{n}}^{n} the maximal length of a segment between branching points in the R-state corresponding to the specialization (x_{i_{n}}^{n}, p_{n}, a) which is mapped to a point in the (finite) limit tree Y (note that this length may be 0). For each two specializations (x_{i_{n_{1}}}^{n} , p_{n}, a), (x_{i_{n_{2}}}^{n} , p_{n}, a) we have associated R-states State_{1} and State_{2} in correspondence. Given a branching point in these states and a copy of one of the defining parameters that passes through this branching point, we can measure the difference in the placement of the branching point on the copy of the defining parameter in the two different R-states State_{1} and State_{2}. We denote by Mfluct_{i_{n_{1}} ,i_{n_{2}}}^{n} the maximum among all these (finite set of) differences. Note that by construction both Mp_{i_{n}}^{n} and Mfluct_{i_{n_{1}} ,i_{n_{2}}}^{n} are asymptotically o(1).

Definition 2.2. With the notation above, we say that the sequence of specializations {(x_{i_{n}}^{n}, p_{n}, a)} is tame with respect to segments in the R-state that correspond to edges in the real tree Y, if there exists a constant c and a subsequence of indices {n_{m}}_{m=1}^{∞}, so that for every index m there are m distinct indices 1 ≤ i_{1} < … < i_{m} ≤ n_{m} for which:

c · Mp_{i_{j_{1}}}^{n_{m}} > Mfluct_{i_{j_{2}} ,i_{j_{3}}}^{n_{m}}

for every possible indices j_{1}, j_{2}, j_{3}, 1 ≤ j_{1}, j_{2}, j_{3} ≤ m.

If the sequence of specializations {x_{i_{n}}^{n}, p_{n}, a)} is tame, we continue with the tame subsequence of specializations. By possibly passing to a further subsequence, we may assume that the maximum Mp_{i_{nm}}^{n} is always obtained for the same segment in the corresponding R-state, that the maximal fluctuation Mfluct_{i_{1},i_{2}}^{n_{m}} is asymptotically o(Mp_{i_{j}}^{n_{m}}), and that the ratio between the segments in the R-state that are asymptotically Ω(Mp_{i_{j}}^{n_{m}}) converge.

If the original sequence of specializations is tame with respect to the segments in R-state corresponding to edges in the tree Y, we check if the obtained subsequence is tame with respect to segments in the R-state corresponding to both edges in Y and segments for which their length grow as Ω(Mp_{i_{j}}^{n_{m}}). Since there are only finitely many segments in the R-state, and since there are m distinct solutions for each step m in our sequence, after finitely many ”uncoverings” we obtain a sequence which is not tame with respect to the previously ”uncovered” segments in the R-state.

Once we obtain a sequence of rigid specializations {(x_{i_{n}}^{n}, p_{n}, a)} with no tame subsequences with respect to the previously ”uncovered” segments in the corresponding R-states, we can choose a subsequence of indices {n_{m}}_{m=1}^{∞}, so that for every index m there are m distinct indices 1 ≤ i_{1} < … < i_{m} ≤ n_{m} for which:

m · Mp_{i_{j_{1}}}^{n_{m}} < Mfluct_{i_{j_{2}} ,i_{j_{3}}}^{n_{m}}

for every possible indices j_{2}, j_{3}, and an index j_{1} for which the minimal Mp_{i_{j}}^{n_{m}} is obtained. We call such a sequence of specializations a perturbed sequence.

Given a perturbed sequence, for each index m we construct the R-state corresponding to the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a) for which Mp_{i_{j(m)}}^{n_{m}} is minimal among all possible indices j. Clearly, given an R-state we can present each of the variables x in terms of fractions of few of the defining parameters. Hence, for every integer m we can represent the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a) in terms of fractions of the parameters p_{n_{m}} that appear in the R-state corresponding to this specialization, and for any other possible index j, we can represent the specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) in terms of the same fractions of the parameters p_{n_{m}} that appear in the R-state corresponding to the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a), together with elements u_{r}^{m}(j) that encode the fluctuations in the corresponding branching points along the variables x between the R-state associated with the specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) and the ”reference” R-state associated with the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a).

Each relation from the defining relations of the rigid graded limit group Rgd(x, p, a) corresponds to a close loop on the R-state, and since the combinatorial type of the R-states associated with all specializations from the perturbed sequence {(x_{i_{j}}^{n_{m}}, p_{n_{m}}, a)} are assumed to be identical, the ”combinatorial type” of the loop corresponding to a given relation is identical for all the specializations from the perturbed sequence. Furthermore, since we assume the periodicity of the defining parameters is bounded, every part of a loop that starts at some point on any of the ”uncovered” segments of the defining parameters and get back to the same point, represent the identity element in all R-states corresponding to specializations (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) from our perturbed sequence. Since for every index m such part of a loop of a defining relation represents the identity element in the R-state associated with the (distinguished) specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a), and the same holds for every other R-state associated with a specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a), every such part of a loop of one of the defining relations corresponds to a (parametric) equation involving (possibly) coefficients from the free group F_{k} =< a >, parameters corresponding to segments in the R-state associated with the (distinguished) specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a) which have not been uncovered, and variables corresponding to the ”fluctuations” u_{r}(m). Since the combinatorics of the R-states associated with all the specializations from our perturbed sequence is identical, the set of parametric equations corresponding to different parts of (closed) partial loops is identical for all specializations from the perturbed sequence. We denote this system of equations by Σ_{1}(u, p̂, a) = 1. To the system of equations Σ_{1}(u, p̂, a) that correspond to the loops around each of the branching points, we add equations that guarantee partial equivariance. i.e., the additional equations guarantee that the specializations of each of the variables x_{i} in the image of the ball of radius R′ = \frac{R}{2}, B_{R′}, are identical. These equations can also be written in terms of the same fractions of the defining parameters p̂ and the variables corresponding to the various fluctuations u that were used in formulating the equations corresponding to the loops around the branching points. We denote the system of equations that combine the equations corresponding to the loops around branching points and the equations coming from the partial equivariance condition by Σ(u, p̂, a) = 1.

By our construction of maximal graded limit groups, there are finitely many maximal graded limit groups corresponding to all the specializations derived from the perturbed sequence {(x_{i_{jm}}^{n_{m}} , p_{n_{m}}, a)}. Since for every index m and any possible j_{1}, j_{2}:

m · Mp_{i_{j(m)}}^{n_{m}} < Mfluct_{i_{j_{1}} ,i_{j_{2}}}^{n_{m}}

at least one of these maximal graded limit groups is not rigid. We denote this nonrigid maximal graded limit group MLim(u, p̂, a). Since the limit group MLim(u, p̂, a) is not rigid, either it admits a non-trivial graded free decomposition, i.e., a nontrivial free decomposition in which the subgroup < p̂, a > can be conjugated into one of the factors, or it admits a non-trivial graded abelian JSJ decomposition, which implies that there is a non-trivial graded modular group associated with it.

The limit group MLim(u, p̂, a) was obtained as a limit group from a sequence of specializations {(u_{m}, p̂_{m}, a)}, that were derived from a sequence {(x_{i_{jm}}^{n_{m}} , p_{n_{m}}, a)} of rigid specializations of the rigid limit group Rgd(x, p, a) we have started with. If the limit group MLim(u, p̂, a) admits a non-trivial graded free decomposition, then all the specializations in the sequence {(u_{m}, p̂_{m}, a)}, split through epimorphisms: η_{m} : MLim(u, p̂, a) → F_{k} ∗ F_{s}, where F_{s} is a non-trivial free group, and so that the epimorphisms {η_{m}} map the subgroup < p̂, a > onto the factor F_{k}. Hence, all the specializations in the sequence {(x_{i_{jm}}^{n_{m}} , p_{n_{m}}, a)} split through epimorphisms: ν_{m} : Rgd(x, p, a) → F_{k} ∗ F_{t}, where F_{t} is a non-trivial free group, and so that the epimorphisms {ν_{m}} map the subgroup < p, a > onto the free group F_{k}. This clearly contradicts the specializations {(x_{i_{j}}^{n_{m}}, p_{n_{m}}, a)} being rigid. Hence, we may assume that the limit group MLim(u, p̂, a) admits no non-trivial graded free decomposition.

Suppose MLim(u, p̂, a) does not admit a non-trivial graded free decomposition and let m be an index for which specialization (u_{m}, p̂_{m}, a) associated with the specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) is not degenerate, i.e., the action of the graded modular group associated with the limit group MLim(u, p̂, a) on the specialization (u_{m}, p̂_{m}, a) is non-trivial. Since the action of the graded modular group associated with MLim(u, p̂, a) on the specialization (u_{m}, p̂_{m}, a) is non-trivial, we can act with various elements of this graded modular group on the specialization (u_{m}, p̂_{m}, a) to obtain a sequence of specializations (u′_{i}, p̂_{m}, a), so that if |u′_{i}| denotes the length (in the free group F_{k}) of the element u′_{i}, then: |u′_{1}| < |u′_{2}| < |u′_{3}| < . . ..

Now, the specializations {u′_{i}} denote the fluctuations in the intersection points between segments of the defining parameters on the R-state corresponding to a specialization of the tuple (x, p, a). Since the length of the specializations {u′_{i}} grows to infinity, the length of the corresponding specializations of the tuple (x, p, a), (x′_{i}, p_{m}, a), grows to infinity as well. Hence, by possibly taking a subsequence (still denoted {(x′_{i}, p_{m}, a)}), the sequence of specializations {(x′_{i}, p_{m}, a)} converge into a flexible quotient of the rigid limit group Rgd(x, p, a) we have started with. Since the specializations (x′_{i}, p_{m}, a) were obtained from the specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) using elements from the modular group associated with the limit group MLim(u, p̂, a), if all the elements {(x′_{i}, p_{m}, a)} factor through a flexible quotient of Rgd(x, p, a), so is the specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) we have started with, which clearly contradicts our assumption of the specializations {(x_{i_{j}}^{n_{m}}, p_{n_{m}}, a)} being rigid.

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Theorem 2.1 proves the existence of a bound on the number of rigid solutions assuming a bound on the periodicity of the specialization of the defining parameters. To get a global bound on the number of rigid solutions, regardless of the periodicity of the defining parameters, a slight modification of the argument is required. In fact, the only places a bound on the periodicity was used is the derivation of an equation from any partial closed loop of any of the loops corresponding to one of the defining relations on a R-state, and in deriving the equations that guarantee that the specializations of a fixed variable x_{i} in all places it appears along the defining relations are indeed identical (the partial equivalence condition). The way these equations are derived is no longer true in the presence of high periodicity of the defining parameters. However, a modification of the argument used to prove theorem 2.1 that takes into account the length of periods of the defining parameters is suitable for proving the existence of a global bound.

Theorem 2.3. Let Rgd(x, p, a) be a rigid graded limit group. There exists a global bound b_{R}(Rgd(x, p, a)) for which for any particular value of the defining parameters p_{0}, there are at most b_{R} distinct rigid homomorphisms h : Rgd(x, p, a) → F_{k} satisfying h(p) = p_{0}.

Proof: The argument we use is a modification of the argument used to prove theorem 2.1. Suppose there is no bound on the number of rigid solutions x for all specializations p_{0} of the defining parameters p. Then there must exist some sequence {p_{n}} of specializations of the defining parameters p, so that for every index n, the rigid graded limit group Rgd(x, p, a) admits n distinct specializations (x_{1}^{n}, p_{n}, a), …, (x_{n}^{n}, p_{n}, a).

As in the proof of theorem 2.1 let B be a ball in the Cayley graph of Rgd(x, p, a) that contains all the defining relations of Rgd(x, p, a). By theorem 1.2 there exists some constant R′, so that for any rigid specialization (x_{0}, p_{0}, a) the ball B is R′-AP covered. Let R = 2 · R′. Clearly, there exist only finitely many combinatorial types of R-states. Hence, we may further assume that for every index n, the combinatorial type of the R-states associated with the specializations (x_{1}^{n}, p_{n}, a), …, (x_{n}^{n}, p_{n}, a) are identical.

Furthermore, we may also assume that after rescaling the metric on the R-states associated with the specializations {(x_{i}^{n}, p_{n}, a)} by dilatation constants depending only on the specializations of the defining parameters p_{n}, for every sequence of indices {i_{n} | 1 ≤ i_{n} ≤ n}, the (rescaled) R-states associated with the sequence of specializations {(x_{i_{n}}^{n}, p_{n}, a)} converges in the Gromov-Hausdorff topology on metric spaces, and the limit (finite) tree Y is independent of the particular choice of indices {i_{n}}.

Definition 2.4. Let p_{r}(i_{n}, n) be a segment in the R-state corresponding to the specialization (x_{i_{n}}^{n}, p_{n}, a) that is mapped to the edge e_{r} in the finite tree Y (p_{r}(i_{n}, n) is a subword of a specialization of one of the defining parameters). We say that p_{r}(i_{n}, n) is pseudo-periodic if:

(i) p_{r}(i_{n}, n) = v_{1}α^{s}v_{2} and the equality is graphical (i.e., with no cancellations). (ii) s > 10 and the length of α^{s} is at least three quarters of the length of p_{r}(i_{n}, n).

It follows from the convergence of the R-states corresponding to the specializations {(x_{i_{n}}^{n}, p_{n}, a)} that for large enough n, if p_{r}(i_{n}, n) is pseudo-periodic for some index 1 ≤ i_{n} ≤ n, then p_{r}(i_{n}, n) is pseudo-periodic for any 1 ≤ i_{n} ≤ n (perhaps for slightly smaller pseudo-periodicity constants). If p_{r}(i_{n}, n) is pseudo-periodic, we set period(p_{r}(i_{n}, n)) to be the minimal length of an element α ∈ F_{k} that satisfies condition (i) above. We call period(p_{r}(i_{n}, n)) the pseudo-period of the pseudoperiodic segment p_{r}(i_{n}, n). Clearly, for large enough n, the pseudo-period does not depend on the specific index 1 ≤ i_{n} ≤ n. We call the maximal possible s appears in condition (i) above, the pseudo-periodicity of p_{r}(i_{n}, n).

We continue by going over the segments in the tree Y. If for such segment e_{r}, there exists a subsequence {p_{r}(i_{n}, n)}_{n=1}^{∞} which is either not pseudo-periodic or with bounded pseudo-periodicity, we restrict our attention to this subsequence and declare the particular segment as not truly periodic. After possibly restricting our attention to some subsequence, we may assume that the sequence of specializations corresponding to each segment is either not truly periodic, or it is pseudo-periodic with no bounded subsequence of pseudo-periods. We call these last segments truly periodic. We have chosen the sequence of specializations {(x_{i_{n}}^{n}, p_{n}, a)} to have the same combinatorial type of R-states. We denote by Mper^{n} the maximum among all pseudoperiods of truly periodic segments in the R-state corresponding to the specialization (x_{i_{n}}^{n}, p_{n}, a) (note that this maximum is independent of the index 1 ≤ i_{n} ≤ n). We denote by Mp_{i_{n}}^{n} the maximum between Mper^{n} and the maximum length of a segment between branching points in the R-state corresponding to the specialization (x_{i_{n}}^{n}, p_{n}, a) which is mapped to a point in the limit tree Y. For each two specializations (x_{i_{n_{1}}}^{n} , p_{n}, a), (x_{i_{n_{2}}}^{n} , p_{n}, a) we have associated R-states State_{1} and State_{2} in correspondence. Given a branching point in these states and a copy of one of the defining parameters that passes through this branching point, we can measure the difference in the placement of the branching point on the copy of the defining parameter in the two different R-states State_{1} and State_{2}. We denote by Mfluct_{i_{n_{1}} ,i_{n_{2}}}^{n} the maximum among all these (finite set of) differences. Note that by construction both Mp_{i_{n}}^{n} and Mfluct_{i_{n_{1}} ,i_{n_{2}}}^{n} are asymptotically o(1).

We define a sequence of specializations {(x_{i_{n}}^{n}, p_{n}, a)} to be tame exactly as we did in definition 2.2. If the sequence of specializations {(x_{i_{n}}^{n}, p_{n}, a)} is tame, we continue with the tame subsequence of specializations. By possibly passing to a further subsequence, we may assume that the maximum Mp_{i_{nm}}^{n} is always obtained for the same segment in the corresponding R-state or for a pseudo-period of the same truly periodic segment, that the maximal fluctuation Mfluct_{i_{1},i_{2}}^{n_{m}} is asymptotically o(Mp_{i_{j}}^{n_{m}}), and that the ratio between the length of segments that are between branching points and pseudo-periods of truly periodic segments in the R-state that are asymptotically Ω(Mp_{i_{j}}^{n_{m}}) converge. If the original sequence of specializations is tame with respect to the segments in the R-state corresponding to edges in the tree Y, we declare each of the original segments in Y that are truly periodic, and for which their pseudo-period is Ω(Mp_{i_{j}}^{n_{m}}) as not truly periodic for the continuation. We also check which of the ”newly uncovered” segments are truly periodic according to definition 2.4, and pass to an appropriate subsequence. We check if the obtained subsequence is tame with respect to segments in the R-state corresponding to both edges in Y and segments and pseudo-periods of truly periodic segments for which their length grow as Ω(Mp_{i_{j}}^{n_{m}}). Since there are only finitely many segments in the R-state, and since there are m distinct solutions for each step m in our sequence, after finitely many ”uncoverings” we obtain a sequence which is not tame with respect to the previously ”uncovered” segments, and pseudo-periods of truly periodic segments in the R-state.

Once we obtain a sequence of rigid specializations {(x_{i_{n}}^{n}, p_{n}, a)} with no tame subsequences with respect to the previously ”uncovered” segments and pseudoperiods of truly periodic segments in the corresponding R-states, we can choose a subsequence of indices {n_{m}}_{m=1}^{∞}, so that for every index m there are m distinct indices 1 ≤ i_{1} < … < i_{m} ≤ n_{m} for which:

m · Mp_{i_{j_{1}}}^{n_{m}} < Mfluct_{i_{j_{2}} ,i_{j_{3}}}^{n_{m}}

for every possible indices j_{2}, j_{3}, and an index j_{1} for which the minimal Mp_{i_{j}} is obtained. We call such a sequence of specializations a perturbed sequence.

Given a perturbed sequence, for each index m we construct the R-state corresponding to the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a) for which Mp_{i_{j(m)}}^{n_{m}} is minimal among all possible indices j. Clearly, given an R-state we can present each of the variables x in terms of fractions of few of the defining parameters. Hence, for every integer m we can represent the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a) in terms of fractions of the parameters p_{n_{m}} that appear in the R-state corresponding to this specialization, and for any other possible index j, we can present the specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) in terms of the same fractions of the parameters p_{n_{m}} that appear in the R-state corresponding to the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a), together with elements u_{r}^{m}(j) that encode the fluctuations in the corresponding branching points along the variables x between the R-state associated with the specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) and the ”reference” R-state associated with the specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a).

Each relation from the defining relations of the rigid graded limit group Rgd(x, p, a) corresponds to a close loop on the R-state, and since the combinatorial type of the R-states associated with all specializations from the perturbed sequence {(x_{i_{j}}^{n_{m}}, p_{n_{m}}, a)} are assumed to be identical, the ”combinatorial type” of the loop corresponding to a given relation is identical for all the specializations from the perturbed sequence. Furthermore, by our ”uncovering” procedure, every part of a loop that starts at some point on any of the ”uncovered” segments of the defining parameters, which are not truly periodic, and get back to the same point, represent the identity element in all R-states corresponding to specializations (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) from our perturbed sequence. Also, every part of a loop that starts at some point on any of the ”uncovered” segments of the defining parameters, which are truly periodic, and get back to the same point, represent an element that commutes with a pseudoperiod of that truly periodic segment in all R-states corresponding to specializations (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a) from our perturbed sequence. Since for every index m such part of a loop of a defining relation represent either the identity element or an element that commutes with a pseudo-period of a truly periodic ”uncovered” segment in the R-state associated with the (distinguished) specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a), and the same holds for every other R-state associated with a specialization (x_{i_{j}}^{n_{m}}, p_{n_{m}}, a), every such part of a loop of one of the defining relations corresponds to a (parametric) equation involving (possibly) coefficients from the free group F_{k} =< a >, fractions of parameters corresponding to segments in the R-state associated with the (distinguished) specialization (x_{i_{j(m)}}^{n_{m}}, p_{n_{m}}, a) which have not been uncovered and pseudo-periods of truly periodic ”uncovered” segments, and variables corresponding to the ”fluctuations” u_{r}(m). Since the combinatorics of the R-states associated with all the specializations from our perturbed sequence is identical, the set of parametric equations corresponding to different parts of (closed) partial loops corresponding to the defining relations is identical for all specializations from the perturbed sequence. We denote this system of equations by Σ_{1}(u, p̂, a) = 1.

To the system of equations Σ_{1}(u, p̂, a) that involve the loops around each of the branching points, we add equations that guarantee partial equivariance. i.e., the additional equations guarantee that the specializations of each of the variables x_{i} in all places it appears in the image of the ball of radius R′ = \frac{R}{2}, B_{R′}, are identical. These equations can also be written in terms of the same fractions of the defining parameters p̂ and the variables corresponding to the various fluctuations u that were used in formulating the equations that involve the loops around the branching points. We denote the system of equations that combine the equations corresponding to the loops around branching points and the equations coming from the partial equivariance condition by Σ(u, p̂, a) = 1. Given this (modified) system of equations, the rest of the argument is identical with the one described in the proof of theorem 2.1.

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Theorem 2.3 bounds the number of rigid homomorphisms of a rigid graded limit group for any possible value of the defining parameters. We continue by proving the existence of a similar bound for families of strictly solid homomorphisms of solid graded limit groups. For presentation purposes we first prove the existence of such bound for solid limit groups for which their associated graded abelian decomposition contains no QH vertex group, and afterwards gradually generalize the argument to all solid limit groups.

Theorem 2.5. Let Sld(x, p, a) be a solid graded limit group, and suppose the graded abelian decomposition associated with Sld(x, p, a) contains no QH vertex groups. There exists a global bound b_{S}(Sld(x, p, a)) for which for any particular value of the defining parameters p_{0}, there are at most b_{S} distinct strictly solid families of homomorphisms h : Sld(x, p, a) → F_{k} satisfying h(p) = p_{0}.

Proof: The argument we use is a modification of the argument used to prove theorem 2.3. Before presenting the argument, it is convenient to work with homomorphisms which are not necessarily the shortest in their corresponding solid families of homomorphisms, but rather with almost shortest homomorphisms.

Definition 2.6. Let Sld(x, p, a) be a solid graded limit group, and let Λ be its associated abelian decomposition. Let T be a rooted maximal tree in Λ, where the root of T is the distinguished vertex (the one containing the subgroup AP =< a, p > in Λ). We fix a generating set for each of the vertex groups in Λ, and Bass-Serre generators that correspond to the graph of groups Λ and its maximal subtree T. We assume that a generating set of an edge group is contained in each vertex group adjacent to the corresponding edge. With each abelian vertex group in Λ we associate a fix retract from the abelian vertex group onto the subgroup generated by the edges connected to that abelian vertex group. We say that a specialization (x_{0}, p_{0}, , a) of the solid graded limit group Sld(x, p, a) is almost shortest if the following hold:

(i) The specialization of each abelian vertex group in Λ factors through the retract associated with this abelian vertex group.

(ii) Let v_{0} be the distinguished vertex in the tree T (the one stabilized by AP =< a, p >). With each edge e_{i} in the tree of groups T ⊂ Λ that connects the distinguished vertex v_{0} to its neighbor v_{i}, we associate a base point b_{i}. If the specialization of the (distinguished) vertex group V_{0} is abelian, we set the base point b_{i} to be the point in the axis of the (specialization of the) edge group E_{i}, associated with the edge e_{i}, that is the closest to the identity vertex in X, the Cayley graph of F_{k}. If the specialization of the vertex group V_{0} is not abelian, there exists a point u_{0} in the Cayley graph X of F_{k} that is moved less than any other point in X by the fixed set of generators of the group V_{0}. In this case we set the base point b_{i} to be the point in the axis of the (specialization of the) edge group E_{i}, associated with the edge e_{i}, that is the closest to u_{0} in X, the Cayley graph of F_{k}. At this point we assume that the specializations of the fixed set of generators of the vertex group V_{i}, associated with the vertex v_{i} adjacent to the edge e_{i}, moves the base point b_{i} the smallest possible distance under the action of the graded modular group of Sld(x, p, a), GMod(Sld(x, p, a)).

We repeat this construction for the vertices adjacent to the vertices v_{i}, i.e., with each edge e_{j} adjacent to a vertex v_{i} that is adjacent to v_{0} in T we associate a base point b_{j}. If the specialization of the vertex group V_{i} is abelian, we set the base point b_{j} to be the base point b_{i}. If the specialization of the vertex group V_{i} is not abelian, there is a point u_{i} in X that is moved less by the fixed set of generators of V_{i}. In this case we set b_{j} to be closest point to u_{i} on the axis of the edge group E_{j} in X. Given the base point b_{j} we assume that the specialization of the fixed set of generators of the vertex group V_{j}, associated with the vertex group v_{j} that is adjacent to the edge e_{j}, moves the base point b_{j} the smallest possible distance under the action of all the elements of the graded modular group GMod(Sld(x, p, a)) that fix the vertex groups adjacent to the distinguished vertex v_{0} in the tree of groups T. We repeat this construction until we exhaust all the vertices in the tree of groups T, and then set the specializations of the fixed set of Bass-Serre generators to be the shortest possible under the action of all the elements of GMod(Sld(x, p, a)) that fix the subgroup of Sld(x, p, a) which is the fundamental group of the tree of groups T (cf. section 3 in [Se4]).

In theorem 1.7 we have shown that there exists a global constant R, so that each strictly solid homomorphism h : Sld(x, p, a) → F_{k} that is among the shortest in its solid family of homomorphisms is R-AP -covered. The argument used to prove theorem 1.7 works without any changes to almost shortest homomorphisms, hence, there exists some constant R′ so that every strictly solid almost shortest homomorphism h : Sld(x, p, a) → F_{k} is R′-AP -covered.

Suppose there is no bound on the number of strictly solid families for all specializations p_{0} of the defining parameters p. Then there must exist some sequence {p_{n}} of specializations of the defining parameters p, so that for every index n, the solid graded limit group Sld(x, p, a) admits n distinct strictly solid families of specializations. For each index n, and each strictly solid family associated with the specialization p_{n}, we associate an almost shortest specialization (x_{i_{n}}^{n}, p_{n}, a). Given the sequence of parameters {p_{n}}, and their associated collections of almost shortest specializations, we apply the argument used to prove theorems 2.1 and 2.3 to obtain an action of a maximal limit group MLim(u, p̂, a) associated with the system of equations Σ(u, p̂, a) on some real tree Y. We may assume that this limit group is not rigid with respect to the parameter subgroup < p̂ >, that the fluctuations associated with a subsequence, still denoted {(x_{i_{n}}^{n}, p_{n}, a) | 1 ≤ i_{n} ≤ n}, factor through MLim(u, p̂, a), and that for any index n and any fluctuation u_{i_{n}}, the length of the maximal fluctuation among the u_{i_{n}}’s is at least n times longer than the maximal length of the fractions p̂_{n}. With the limit group MLim(u, p̂, a) we associate a limit group MP Lim(u, p̂, p′, a) generated by the limit group MLim(u, p̂, a) and the fractions p′ that were ”uncovered” through the process presented in the proof of theorems 2.1 and 2.3. By construction there is a natural map η : Sld(x, p, a) → MP Lim(u, p̂, p′, a). Let L_{∞}(x, p, a) be the image of Sld(x, p, a) in MP Lim(u, p̂, p′, a), L_{∞}(x, p, a) = η(Sld(x, p, a)). Let Θ_{L_{∞}} be the abelian decomposition inherited by L_{∞} from the graded abelian decomposition of MP Lim(u, p̂, p′, a) (with respect to the parameter subgroup < p̂, p′, a >).

The limit groups MLim(u, p̂, a), MP Lim(u, p̂, p′, a), L_{∞}(x, p, a) are obtained from a sequence of almost shortest specializations, still denoted {(x_{i_{n}}^{n}, p_{n}, a)}, which are all assumed to be strictly solid. Since the specializations {(x_{i_{n}}^{n}, p_{n}, a)} are all almost shortest specializations, it can not be that all the vertex groups and all the Bass-Serre generators in the graded abelian decomposition Λ of Sld(x, p, a) are mapped by the quotient map η into non-abelian, non-QH vertex groups or into edge groups in the abelian decomposition Θ_{L_{∞}}. Now, the argument used to prove theorem 1.7, implies that if there exists a vertex group in Λ that is not mapped by the quotient map η into a non-abelian, non-QH vertex group nor into an edge group in the abelian decomposition Θ_{L_{∞}}, then for large enough n, the specializations (x_{i_{n}}^{n}, p_{n}, a) are all not strictly solid, which clearly contradicts our assumptions.

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Proving the existence of a bound on the number of strictly solid families of specializations of a solid limit group with no QH vertex group in its associated graded abelian decomposition, we further generalize the argument used to prove the existence of a bound on the number of rigid solutions of a rigid limit group to get a bound on the number of strictly solid families of specializations of a general solid limit group.

Theorem 2.7. Let Sld(x, p, a) be a solid graded limit group. For any particular value p_{0} of the defining parameters p, there are at most b_{S} distinct strictly solid families of homomorphisms h : Sld(x, p, a) → F_{k} satisfying h(p) = p_{0}.

Proof: The argument we use is based on the arguments used to prove theorems 2.3 and 2.5. Suppose there is no bound on the number of strictly solid families for all specializations p_{0} of the defining parameters p. Then there must exist some sequence {p_{n}} of specializations of the defining parameters p, so that for every index n, the solid graded limit group Sld(x, p, a) admits n distinct strictly solid families of specializations. For each index n, and each strictly solid family associated with the specialization p_{n}, we associate an almost shortest specialization (x_{i_{n}}^{n}, p_{n}, a). Given the sequence of parameters {p_{n}}, and their associated collections of almost shortest specializations, we apply the argument used to prove theorems 2.1 and 2.3 to obtain an action of a maximal limit group MLim(u, p̂, a) associated with the system of equations Σ(u, p̂, a) on some real tree Y. We may assume that this limit group is not rigid with respect to the parameter subgroup < p̂ >, that the fluctuations associated with a subsequence, still denoted {(x_{i_{n}}^{n}, p_{n}, a) | 1 ≤ i_{n} ≤ n}, factor through MLim(u, p̂, a), and that for any index n and any fluctuation u_{i_{n}}, the length of the maximal fluctuation among the u_{i_{n}}’s is at least n times longer than the maximal length of the fractions p̂_{n}. With the limit group MLim(u, p̂, a) we associate a limit group MP Lim(u, p̂, p′, a) generated by the limit group MLim(u, p̂, a) and the fractions p′ that were ”uncovered” through the process presented in the proof of theorems 2.1 and 2.3. By construction, there exists a natural map η : Sld(x, p, a) → MP Lim(u, p̂, p′, a). Since the limit group MLim(u, p̂, a) is not rigid, either it admits a non-trivial graded free decomposition, i.e., a non-trivial free decomposition in which the subgroup < p̂, a > can be conjugated into one of the factors, or it admits a non-trivial graded abelian JSJ decomposition, which implies that there is a non-trivial graded modular group associated with it.

The limit group MLim(u, p̂, a) was obtained as a limit group from a sequence of specializations {(u_{i_{m}}, p̂_{n_{m}}, a)}, that were derived from a sequence {(x_{i_{jm}}^{n_{m}} , p_{n_{m}}, a)} of strictly solid, almost shortest specializations of the solid limit group Sld(x, p, a) we have started with. If the limit group MLim(u, p̂, a) admits a non-trivial graded free decomposition, then its associated limit group MP Lim(u, p̂, p′, a) admits a non-trivial graded free decomposition (with respect to the parameter subgroup < p̂, p′, a >), which implies that all the specializations in the sequence {(x_{i_{jm}}^{n_{m}} , p_{n_{m}}, a)} split through epimorphisms: λ_{m} : Sld(x, p, a) → F_{k} ∗ F_{t}, where F_{t} is a non-trivial free group, and so that the epimorphisms {λ_{m}} map the subgroup < p, a > onto the free group F_{k}. This clearly contradicts the specializations {(x_{i_{j}}^{n_{m}}, p_{n_{m}}, a)} being strictly solid. Hence, we may assume that the limit group MLim(u, p̂, a) admits no non-trivial graded free decomposition. The solid limit group Sld(x, p, a) is mapped into the limit group MP Lim(u, p̂, p′, a) by the homomorphism η : Sld(x, p, a) → MP Lim(u, p̂, p′, a). Let Λ be the graded abelian JSJ decomposition associated with Sld(x, p, a), and let Θ be the graded abelian JSJ decomposition associated with MP Lim(u, p̂, p′, a). By the construction of the limit group MP Lim(u, p̂, a), the image of the original solid limit group Sld(x, p, a), η(Sld(x, p, a)), can not be conjugated into the fundamental group of a proper subgraph of the graph of groups Θ. If there exists a non-QH vertex group or an edge group in Λ that is not mapped by the homomorphism η into a non-abelian, non-QH nor into an edge group in the graded abelian decomposition Θ, the argument used to prove theorem 1.7 implies that for large enough n, the specializations {(x_{i_{j}}^{n_{m}}, p_{n_{m}}, a)} are not strictly solid. Hence, we may assume that the homomorphism η : Sld(x, p, a) → MP Lim(u, p̂, p′, a) maps every non-QH vertex group and every edge group in the graded abelian decomposition Λ, associated with Sld(x, p, a), into a non-abelian, non-QH or into an edge group in the graded abelian decomposition Θ associated with MP Lim(u, p̂, p′, a).

We associate with the limit group MP Lim(u, p̂, p′, a) its (canonical) graded Makanin-Razborov diagram with respect to the parameter subgroup < p̂, p′ >. The entire sequence of specializations {(u_{i_{m}}, p̂_{n_{m}}, p′_{n_{m}}, a) | 1 ≤ i_{m} ≤ m} factors through the limit group MP Lim(u, p̂, p′, a), hence, we may assume that a subsequence, still denoted {(u_{i_{m}}, p̂_{n_{m}}, p′_{n_{m}}, a) | 1 ≤ i_{m} ≤ m} factors through a graded resolution GRes(u, p̂, p′, a) in the taut graded Makanin-Razborov resolution of MP Lim(u, p̂, p′, a). With the i-th level of graded resolution GRes(u, p̂, p′, a) there is an associated graded limit group, a quotient of MP Lim(u, p̂, p′, a), which we denote MP Lim_{i}(u, p̂, p′, a). The natural map η : Sld(x, p, a) → MP Lim(u, p̂, p′, a) composed with the quotient map τ_{i} : MP Lim(u, p̂, p′, a) → MP Lim_{i}(u, p̂, p′, a) gives rise to a natural map: η_{i} : Sld(x, p, a) → MP Lim_{i}(u, p̂, p′, a). If the graded limit groups MP Lim_{i}(u, p̂, a) admits a non-trivial graded free decomposition, or if any of the maps η_{i} : Sld(x, p, a) → MP Lim_{i}(u, p̂, p′, a) does not map a non-QH vertex group or an edge group in the graded abelian decomposition Λ associated with Sld(x, p, a) into a non-QH, non-abelian vertex group nor into an edge in the graded abelian JSJ decomposition Θ_{i} associated with MP Lim_{i}(u, p̂, p′, a), then for large enough m, the specializations (x_{i_{m}}, p_{n_{m}}, a) are not strictly solid, a contradiction to our assumptions. Furthermore, by construction, the image of the original limit group Sld(x, p, a) into a limit group MP Lim_{i}(u, p̂, a) can not be conjugated into the fundamental group of a proper subgraph of the graphs of groups Θ_{i}.

The terminal limit group of the graded resolution GRes(u, p̂, p′, a), MP Lim_{f}(u, p̂, p′, a), is by definition, either rigid or solid. If the terminal limit group MP Lim_{f}(u, p̂, p′, a) is rigid, then by construction there is no bound on its rigid specializations, a contradiction to theorem 2.3. Hence, the terminal limit group of the graded resolution GRes(u, p̂, p′, a), MP Lim_{f}(u, p̂, p′, a), has to be solid. The sequence of specializations: {(x_{i_{m}}, p_{n_{m}}, a) | 1 ≤ i_{m} ≤ m} of the solid limit group Sld(x, p, a) we have started with, are assumed to be strictly solid and almost shortest in their corresponding solid families. The corresponding sequence of specializations {(u_{i_{m}}, p̂_{n_{m}}, p′_{n_{m}}, a) | 1 ≤ i_{m} ≤ m} of the terminal graded limit group of the graded resolution GRes(u, p̂, p′, a), MP lim_{f}(u, p̂, p′, a), are necessarily solid specializations. If there is no subsequence of specializations (still denoted {(u_{i_{m}}, p̂_{n_{m}}, p′_{n_{m}}, a) | 1 ≤ i_{m} ≤ m}) which are strictly solid specializations of the terminal solid limit group MP Lim_{f}(u, p̂, p′, a) of the graded resolution GRes(u, p̂, p′, a), we extend the graded resolution GRes(u, p̂, p′, a), by continuing from the terminal graded limit group MP Lim_{f}(u, p̂, p′, a) to a flexible quotient that is associated with a subsequence of non-strictly solid specializations (still denoted {(u_{i_{m}}, p̂_{n_{m}}, p′_{n_{m}}, a) | 1 ≤ i_{m} ≤ m}). By repeatedly extending the graded resolution GRes(u, p̂, p′, a) in case a terminal solid limit group does not admit a subsequence of strictly solid specializations, we obtain an extended graded resolution (still denoted GRes(u, p̂, p′, a)), for which the associated sequence of specializations of its terminal solid limit group (still denoted MP Lim_{f}(u, p̂, p′, a)), are all strictly solid. Clearly all the properties of the graded limit groups and the maps {η_{i}} associated with the various levels of the graded resolution GRes(u, p̂, p′, a), remain valid for the graded limit groups in the extended graded resolution, and the maps from the original solid limit group Sld(x, p, a) into these limit groups, which we still denote {eta_{i}}.

The properties of the maps η_{i} from the solid limit group Sld(x, p, a) we have started with into the graded limit groups MP Lim_{i}(u, p̂, p′, a), allow us to bound the reduced complexities of the graded abelian decompositions Θ_{i} of MP Lim_{i}(u, p̂, p′, a) in terms of the reduced complexity of the graded abelian decomposition Λ of Sld(x, p, a).

Theorem 2.8. Let Glim(y, p, a) be a graded limit group, with respect to the subgroup P =< p >, and suppose Glim(y, p, a) admits no non-trivial graded free decomposition. Let Δ be its graded abelian JSJ decomposition.

Let Q_{1}, …, Q_{n} be the QH vertex groups in the graded abelian decomposition Δ, and let S_{1}, …, S_{n} be their corresponding surfaces. We set the complexity of a QH vertex group Q_{j}, to be the ordered couple (|χ(S_{j})|, genus(S_{j})). We set the reduced complexity of the graded abelian decomposition Δ, denoted Rcomp(Δ), to be the tuple:

((χ(S_{1})|, genus(S_{1})), …, (|χ(S_{n})|, genus(S_{n})), d)

where the sequence of couples (χ(S_{1})|, genus(S_{1})), …, (|χ(S_{n})|, genus(S_{n})) is ordered in a decreasing lexicographical order, and the integer d is set to be d = e − a, where e is the number of edges between non-QH vertex groups in Δ, and a is the number of abelian vertex groups adjacent to the collection of these edges.

Let Δ_{1} and Δ_{2} be two graded abelian decompositions of the graded limit groups Glim_{1}(y, p, a) and Glim_{2}(y, p, a) in correspondence, and Glim_{1}(y, p, a), Glim_{2}(y, p, a) admit no non-trivial graded free decompositions. We say that Rcomp(Δ_{1}) ≤ Rcomp(Δ_{2}) if the sequence associated with Δ_{1} is not greater than the sequence associated with Δ_{2} in the natural lexicographical order on the above sequences of (non-negative) integers. Clearly, the reduced complexity of any two such graded decompositions can be compared.

At this point the properties of the maps η_{i} enable us to strictly bound the reduced complexity of the graded abelian JSJ decomposition of the terminating solid limit group of the graded resolution GRes(u, p̂, a) in terms of the reduced complexity of the graded abelian JSJ decomposition of the original solid graded limit group Sld(x, p, a).

Proposition 2.9. Let MP Lim_{f}(u, p̂, p′, a) be the terminating solid limit group of the (extended) graded resolution GRes(u, p̂, p′, a). Let Θ_{f} be the graded abelian JSJ decomposition of the solid limit group MP Lim_{f}(u, p̂, p′, a), and let Λ be the graded abelian JSJ decomposition of the original solid limit group Sld(x, p, a). Then:

Rcomp(Θ_{f}) < Rcomp(Λ).

Proof: By theorem 2.5 we may assume that the graded abelian JSJ decomposition of the original solid limit group Sld(x, p, a) contains a QH vertex group. The map η_{i} : Sld(x, p, a) → MP Lim_{i}(u, p̂, p′, a) maps every non-QH vertex group and every edge group in the graded abelian JSJ decomposition Λ of Sld(x, p, a), into a non-QH vertex group or an edge group in T heta_{i}, the graded abelian JSJ decomposition of MP Lim_{i}(u, p̂, p′, a). Furthermore, by construction, the image of Sld(x, p, a), η_{i}(Sld(x, p, a)), can not be conjugated into a proper subgraph of the graph of groups Θ_{i}. Hence, every QH vertex group in the abelian decomposition inherits a small splitting from its image into the limit group MP Lim(u, p̂, p′, a), i.e., from the graded abelian decomposition Θ_{i}, and the pre-image of every QH vertex group in Θ_{i} contains conjugates of subgroups of QH vertex groups that correspond to subsurfaces of the surfaces associated with the QH vertex groups in the graded abelian decomposition Λ, and each such subgroup of a QH vertex group in Λ is mapped into a finite index subgroup of the corresponding QH vertex group in Θ_{i}. Since every QH vertex group in Θ_{i} is ”covered” by a subsurface of the surfaces associated with the QH vertex groups in Λ, since the maps η_{i} map every non-QH vertex group and every edge group in Λ into a non-QH vertex group or into an edge group in Θ_{i}, and since η_{i}(Sld(x, p, a)) can not be conjugated into a proper subgraph of the graph of groups Θ_{i}, the reduced complexity of each of the graded abelian decompositions Θ_{i} is bounded above by the reduced complexity of the graded abelian decomposition Λ, i.e.:

Rcomp(Θ_{i}) ≤ Rcomp(Λ).

Suppose that for some level i, Rcomp(Θ_{i}) = Rcomp(Λ). In this case our aim is to show that equality holds also for the graded abelian decomposition associated with the top level of the graded resolution GRes(u, p̂, p′, a): Rcomp(Θ_{1}) = Rcomp(Λ). Hence, in this part of the argument we assume that i > 1, and i is the minimal index for which: Rcomp(Θ_{i}) = Rcomp(Λ). Since Rcomp(Θ_{i}) = Rcomp(Λ), every QH vertex group in Λ is mapped by η_{i} isomorphically onto a QH vertex group in the graded abelian decomposition Θ_{i}, and the non-QH vertex group in Λ are mapped by η_{i} into distinct non-QH vertex groups in Θ_{i}. Hence, the edge groups in Λ are mapped by η_{i} into distinct edge groups in Θ_{i}, and every edge group in Θ_{i} contains the image of one of the edge groups in Λ.

For every index j, the pre-image of a QH vertex group in the graded abelian decomposition Θ_{j} contains subgroups corresponding to subsurfaces of the surfaces associated with the QH vertex groups in the graded abelian decomposition Λ, and the fundamental group of each such subsurface is mapped into a finite index subgroup of the corresponding QH vertex group in Θ_{j}. The map η_{i} : Sld(x, p, a) → MP Lim_{i}(u, p̂, p′, a) factors as: η_{i} = ν_{i−1} ∘ η_{i−1}, where nu_{i−1} is the canonical quotient map: ν_{i−1} : MP Lim_{i−1}(u, p̂, p′, a) → MP Lim_{i}(u, p̂, p′, a). Since the map η_{i} maps the QH vertex groups in Λ isomorphically onto the QH vertex groups in Θ_{i}, the map η_{i−1} maps distinct subsurfaces of the surfaces associated with the QH vertex groups in Λ isomorphically onto the QH vertex groups in Θ_{i−1}, and the quotient map ν_{i−1} maps the QH vertex groups in Θ_{i−1} isomorphically onto subsurfaces of the surfaces associated with the QH vertex groups in the graded abelian decomposition Θ_{i}. Since the image of Sld(x, p, a) in MP Lim(u, p̂, p′, a), η_{i−1}(Sld(x, p, a)), can be conjugated into the fundamental group of a proper subgraph of the graph of groups Θ_{i−1}, and for each QH vertex group in Θ_{i−1} there exists a subsurface of a surface associated with a QH vertex group in Λ that is mapped isomorphically onto it by η_{i−1}, each edge group in Θ_{i−1} either contains the image of an edge group in Λ or it contains the image of a non-boundary parallel s.c.c. in one of the surfaces associated with a QH vertex group in Λ.

Since every edge group in the graph of groups Θ_{i} contains the image of an edge group in Λ, and since every edge group in Θ_{i−1} contains the image of either an edge group or a non-boundary parallel s.c.c. on one of the surface associated with a QH vertex groups in Λ, the graph of groups Θ_{i} can be further refined by cutting some of the surfaces associated with its QH vertex groups along non-boundary s.c.c. to give a graph of groups Θ_{i}^{ref} the graded limit group MP Lim(u, p̂, p′, a), so that the quotient map ν_{i−1} maps every edge group in the graph of groups Θ_{i−1} into elliptic subgroups in the refined graph of groups Θ_{i}^{ref}.

Since the (extended) graded resolution GRes(u, p̂, p′, a) is assumed to be taut, every non-QH, non-abelian vertex group in Θ_{i−1} is mapped monomorphically by the quotient map ν_{i−1}. If the reduced complexity of the graph of groups Θ_{i−1}, Rcomp(Θ_{i−1}), is strictly smaller than the reduced complexity of Θ_{i}, Rcomp(Θ_{i}), then the image of at least one of the non-QH, non-abelian vertex groups in Θ_{i−1} under the map ν_{i−1}, say ν_{i−1}(V_{j}), inherits a non-trivial abelian decomposition from the refined graph of groups Θ_{i}^{ref}. Since every edge group in Θ_{i−1} is mapped by ν_{i−1} into an elliptic subgroup in Θ_{i}^{ref}, the graph of groups inherited by ν_{i−1}(V_{j}) from Θ_{i}^{ref} is compatible with the graph of groups Θ_{i−1}. Hence, the graph of groups inherited by the non-QH, non-abelian vertex group V_{j} from Θ_{i}^{ref} can be used to further refine the graph of groups Θ_{i−1}, so that the subgroup < p̂, p′, a) remains elliptic. Since the graph of groups Θ_{i−1} is assumed to be the graded abelian JSJ decomposition of MP Lim_{i−1}(u, p̂, p′, a), a further refinement certainly contradicts its canonical properties. Hence, necessarily, Rcomp(Θ_{i−1}) = RComp(Λ), and by finite induction Rcomp(Θ_{1}) = RComp(Λ).

Suppose the reduced complexity of the graded abelian decomposition Θ_{1} of the graded limit group MP Lim_{1}(u, p̂, p′, a) associated with the top level of the graded resolution GRes(u, p̂, p′, a), is equal to the reduced complexity of the graded abelian decomposition Λ associated with the original solid limit group Sld(x, p, a). Since the natural map: η_{1} : Sld(x, p, a) → MP Lim_{1}(u, p̂, p′, a) maps every non-QH vertex group, and every edge group in the abelian decomposition Λ, into either a non-QH, non-abelian vertex group or an edge group in the abelian decomposition Θ_{1}, the pre-image of every QH vertex group in the graded abelian decomposition Θ_{1}, consists of conjugates of subgroups of QH vertex groups, which correspond to subsurfaces of the surfaces associated with that QH vertex groups, in the abelian decomposition Λ. Hence, in case of equality in the reduced complexities of the graded abelian decompositions Λ and Θ_{1}, every QH vertex group in Λ is mapped by η_{1} isomorphically onto a QH vertex group in the abelian decomposition Θ_{1}.

The graded limit group MP Lim_{1}(u, p̂, p′, a) is generated by the limit groups MLim_{1}(u, p̂, a) and the fractions {p′} of the defining parameters. Since η_{1} maps each QH vertex group in the abelian decomposition Λ isomorphically onto a QH vertex group in the graded abelian decomposition Θ_{1}, the map η_{1} maps every QH vertex group in Λ into the subgroup MLim_{1}(u, p̂, a) of MP Lim_{1}(u, p̂, p′, a). The graded limit group MLim_{1}(u, p̂, a) was constructed from a sequence of specializations: {(u_{i_{n}}^{n}, p̂_{n}, a) | 1 ≤ i_{n} ≤ n} that converge to an action of the limit group MLim_{1}(u, p̂, a) on some real tree Y. If any of the boundary elements of a QH vertex group in Λ is mapped by η_{1} to an element in MLim_{1}(u, p̂, a) that does not fix a point in the real tree Y, then this boundary element is not mapped into a non-QH, non-abelian vertex group nor into an edge group in the abelian decomposition Θ_{1} of MP Lim_{1}(u, p̂, p′, a), hence, by the argument used to prove theorem 1.7, for large enough n, the specializations {(x_{i_{n}}^{n}, p_{n}, a) | 1 ≤ i_{n} ≤ n} are not strictly solid, in contradiction to our assumptions. Since every boundary element of a QH vertex group in Λ is mapped by η_{1} to an element that fixes a point in Y, and since the specialization {(x_{i_{n}}^{n}, p_{n}, a) | 1 ≤ i_{n} ≤ n} are almost shortest, every QH vertex group in Λ fixes a point in the real tree Y.

The limit groups MLim_{1}(u, p̂, a) and MP Lim_{1}(u, p̂, p′, a) are obtained from a sequence of almost shortest specializations, which are all assumed to be strictly solid. Since the specializations {(x_{i_{n}}^{n}, p_{n}, a)} are all almost shortest specializations, and since every QH vertex group in Λ is mapped by η_{1} onto a subgroup that fixes a point in the real tree Y, it can not be that all the non-QH vertex groups and all the edge groups in the graded abelian decomposition Λ of Sld(x, p, a) are mapped by the quotient map η_{1} into non-abelian, non-QH vertex groups or into edge groups in the abelian decomposition Θ_{1} of MP Lim_{1}(u, p̂, p′, a). Now, the argument used to prove theorem 1.7, implies that if there exists a non-QH vertex group or an edge group in Λ that is not mapped by the quotient map η_{1} into a non-abelian, non-QH vertex group nor into an edge group in the abelian decomposition Θ_{1}, then for large enough n, the specializations (x_{i_{n}}^{n}, p_{n}, a) are all not strictly solid, which clearly contradicts our assumptions.

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By proposition 2.9 the reduced complexity of the graded abelian JSJ decomposition Θ_{f}, associated with the terminal solid limit group MP Lim_{f}(u, p̂, p′, a), is strictly smaller than the reduced complexity of the graded abelian JSJ decomposition Λ associated with the solid limit group Sld(x, p, a) we have started with. The solid limit group MP Lim_{f}(u, p̂, p′, a) was constructed from (shortenings) of a sequence of strictly solid specializations: {(x_{i_{n}}^{n}, p_{n}, a) | 1 ≤ i_{n} ≤ n} of the solid limit group Sld(x, p, a) we have started with, that give rise to a sequence of strictly solid specializations: {(u_{i_{n}}^{n}, p̂_{n}, p′_{n}, a) | 1 ≤ i_{n} ≤ n} of the terminal solid limit group of the (extended) graded resolution GRes(u, p̂, p′, a), MP Lim_{f}(u, p̂, p′, a).

Therefore, if there is no bound on the number of strictly solid families of specializations of a solid limit group Sld(x, p, a), then there exists another solid limit group, MP Lim_{f}(u, p̂, p′, a), for which its associated graded abelian JSJ decomposition has strictly smaller reduced complexity than the reduced complexity of the graded abelian JSJ decomposition associated with the original solid limit group Sld(x, p, a). Since a strictly decreasing sequence of reduced complexities of graded abelian decompositions must be finite, a finite induction argument (on the reduced complexity of the graded abelian JSJ decompositions associated with the corresponding solid limit groups) finally completes the proof of theorem 2.7.

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The same argument used to get a global bound on the number of strictly solid families of specializations of a general solid limit group, can be applied to get a global bound on the number of Nielsen equivalence classes of solutions of quadratic equations. The adaptation of the argument used to prove theorem 2.7 to give a bound on the number of Nielsen equivalence classes is rather straightforward and left for the reader. This bound is not used in our work on Diophantine geometry over groups, but is likely to find applications elsewhere.

Theorem 2.10. Let F_{k} be a free group and let Q(x, p) be a quadratic graded limit group defined by either the relation:

[x_{1}, x_{2}] … [x_{2g−1}, x_{2g}] = p_{1} t_{2}p_{2}t_{2}^{−1} … t_{n}p_{n}t_{n}^{−1}

or by the relation:

x_{1}^{2} … x_{g+1}^{2} = p_{1} t_{2}p_{2}t_{2}^{−1} … t_{n}p_{n}t_{n}^{−1}.

With the quadratic limit group Q(x, p) we can naturally associate a surface which we denote S_{Q}. We say that a homomorphism h : Q(x, p) → F_{k} is taut, if no s.c.c. on the surface S_{Q} is mapped to the identity by the homomorphism h.

There exists a constant c = c(g, n), so that for any specialization p_{0} of the defining parameters p, the set of taut homomorphisms h : Q(x, p) → F_{k} belong to at most c(g, n) distinct Nielsen equivalence classes.

§3. AE sets and the Singular Locus

Given a parameterized system of equations Σ(x, p, a), we have used the graded Makanin-Razborov diagram of it to obtain a ”weak stratification” of the corresponding family of varieties in section 12 of [Se1]. In sections 1 and 2 of this paper we studied the basic combinatorial structure of rigid and solid solutions and finally obtained a global bound on the number of rigid solutions for any possible value of the defining parameters (theorem 2.3) and families of solid solutions (theorem 2.7). To further study some of the basic properties of the graded Makanin-Razborov diagram in order to study elementary predicates and sets in the next papers, we need to define Diophantine, co-Diophantine, universal and co-universal (existential), as well as AE and EA sets.

Definition 3.1. A subset D ⊂ (F_{k})^{u} is called Diophanitine if for some fixed system of equations Σ, D is defined by:

D = { (d_{1}, …, d_{u}) | ∃y = (y_{1}, …, y_{n}) Σ(y, d, a) = 1}

A subset CD ⊂ (F_{k})^{u} is called co-Diophantine if CD is the complement of a Diophantine set, i.e., CD = (F_{k})^{u} \ D for some Diophantine set D.

Diophantine and co-Diophantine sets behave ”nicely” under elementary boolean operations.

Lemma 3.2.

(i) The union and the intersection of finitely many Diophantine sets in (F_{k})^{u} is a Diophantine set.

(ii) The union and the intersection of finitely many co-Diophantine sets in (F_{k})^{u}

is a co-Diophantine set. (iii) Let A be a subset of (F_{k})^{u} and suppose A is obtained by a finite sequence of

binary operations on a finite set of Diophantine and co-Diophantine sets in (F_{k})^{u}. Then A can be expressed as the disjoint (finite) union of Diophantine sets, co-Diophantine sets, and intersections of a Diophantine and a co-Diophantine set.

Proof: Clearly, the intersection of finitely many Diophantine sets is Diophantine. By a theorem of A. I. Malcev (see [Ma],3) the conjunction of the equations x_{1} = 1 and x_{2} = 1 is equivalent to the equation x_{1}^{2}a_{1}x_{1}^{2}a_{1}^{−1} = (x_{2}a_{2}x_{2}a_{2}^{−1})^{2}. By a theorem of G. A. Gurevich (see [Ma],4) the disjunction of the equations x_{1} = 1 and x_{2} = 1 is equivalent to the conjunction of the four equations:

(x_{1}a_{1}^{ϵ}x_{1}a_{1}^{−ϵ})(x_{2}a_{2}^{δ}x_{2}a_{2}^{−δ}) = (x_{2}a_{2}^{δ}x_{2}a_{2}^{−δ})(x_{1}a_{1}^{ϵ}x_{1}a_{1}^{−ϵ})

for ϵ, δ = ±1. Hence, also the union of finitely many Diophantine sets is a Diophantine set and we get part (i) of the lemma. Since the union of finitely many co-Diophantine sets is the complement of the intersection of finitely many Diophantine sets, the union of finitely many co-Diophantine sets is a co-Diophantine set, and similarly the intersection of finitely many co-Diophantine sets is a co-Diophantine set, and we get part (ii). Part (iii) clearly follows from parts (i) and (ii).

□

Definition 3.3. A subset CU ⊂ (F_{k})^{s} is called co-universal (or existential) if for some fixed systems of equations Σ_{1}, …, Σ_{r} and Ψ_{1}, …, Ψ_{r}, CU is defined by:

U = { (p_{1}, …, p_{s}) | ∃y = (y_{1}, …, y_{n}) (Σ_{1}(y, p, a) = 1 ∧ Ψ_{1}(y, p, a) ≠ 1) ∨ …

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

A subset U ⊂ (F_{k})^{s} is called universal if U is the complement of a co-universal set, i.e., U = (F_{k})^{s} \ CU for some co-universal set CU.

Lemma 3.4.

(i) The union and the intersection of finitely many co-universal sets in (F_{k})^{s} is a co-universal set.

(ii) The union and the intersection of finitely many universal sets in (F_{k})^{s} is a

universal set. (iii) Let A be a subset of (F_{k})^{s} and suppose A is obtained by a finite sequence

of binary operations on a finite set of universal sets in (F_{k})^{s}. Then A can be expressed as a disjoint (finite) union of universal sets, co-universal sets, and intersections of a universal and a co-universal set.

Proof: parts (i) and (ii) are immediate from the definition. Part (iii) clearly follows from arts (i) and (ii).

□

Definition 3.5. A subset V ⊂ (F_{k})^{s} is called AE if for some fixed systems of equations Σ_{1}, …, Σ_{r} and Ψ_{1}, …, Ψ_{r}, V is defined by:

V = { (p_{1}, …, p_{s}) | ∀y ∃x (Σ_{1}(x, y, p, a) = 1 ∧ Ψ_{1}(x, y, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, p, a) = 1 ∧ Ψ_{r}(x, y, p, a) ≠ 1)}

A subset T ⊂ (F_{k})^{s} is called EA if T is the complement of an AE set, i.e., T = (F_{k})^{s} \ V for some AE-set V. Alternatively an EA set T is defined by a predicate of the form:

T = { (p_{1}, …, p_{s}) | ∃y ∀x (Σ_{1}(x, y, p, a) = 1 ∧ Ψ_{1}(x, y, p, a) ≠ 1) ∨ …
… ∨ (Σ_{r}(x, y, p, a) = 1 ∧ Ψ_{r}(x, y, p, a) ≠ 1)}

Lemma 3.6.

(i) The union and the intersection of finitely many AE sets in (F_{k})^{s} is an AE set.

(ii) The union and the intersection of finitely many EA sets in (F_{k})^{s} is an EA

set. (iii) Let B be a subset of (F_{k})^{s} and suppose B is obtained by a finite sequence

of binary operations on a finite set of AE sets in (F_{k})^{s}. Then B can be expressed as a disjoint (finite) union of AE sets, EA sets, and intersections of an AE and an EA set.

Proof: parts (i) and (ii) are immediate from the definition. Part (iii) clearly follows from arts (i) and (ii).

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The following proposition follows immediately from the construction of the singular locus in section 12 of [Se1], and the global bound on rigid solutions obtained in section 2. It is essential in our approach to the elimination of quantifiers obtained in the next papers.

Proposition 3.7. Let Glim(x, p, a) be a graded limit group, and let RgdGlim(x, p, a) be a rigid graded limit group in the graded Makanin-Razborov diagram of Glim(x, p, a), and suppose the graded resolution GRes_{i} of Glim(x, p, a) terminates in the rigid graded limit group RgdGlim(x, p, a). Then:

(i) the set of the defining parameters p for which RgdGlim(x, p, a) admits a rigid solution is a co-universal set.

(ii) the set of the defining parameters p for which RgdGlim(x, p, a) admits at

least s rigid solutions is a co-universal set. (iii) the set of the defining parameters p for which RgdGlim(x, p, a) admits

precisely s rigid solutions is the intersection of a universal set and a couniversal set.

(iv) the set of the defining parameters p for which the resolution GRes_{i} is nondegenerate for some rigid solution is a co-universal set. The set of p’s for which a particular part of GRes_{i} is non-degenerate for some rigid solution is a co-universal set.

By section 2 there exists a global bound on the number of rigid solutions of a rigid graded limit group for every given value of the parameters p, and there also exists a global bound on the number of strictly solid families of solutions of a solid graded limit group. Given a rigid graded limit group, proposition 3.7 shows that the set of parameters for which there exists precisely s rigid solutions is the intersection of a universal and co-universal set. To get an analogous statement for solid graded limit groups we need to present strictly solid families of solutions of a solid graded limit group.

Definition 3.8. Let Sld(x, p, a) be a solid graded limit group. On the set of solid families of solutions of Sld(x, p, a) that contain strictly solid solutions we define an equivalence relation as follows. With each such family of strictly solid solutions we can naturally associate a (well-structured) ungraded resolution, and with each such ungraded resolution we can naturally associate its completion. We say that two such families are equivalent if strictly solid solutions that belong to the first family factor through the completion of the resolution associated with the second family and vice versa. This relation on solid families that contain strictly solid solutions is symmetric by definition, and easily seen to be transitive, hence, it is an equivalence relation. We call each equivalence class of such families a strictly solid family of solutions of the solid graded limit group Sld(x, p, a).

Proposition 3.7 studies rigid graded limit groups and graded resolutions passes through them. An analogous statement is valid for graded resolutions terminating in a solid graded limit group. The proof of the following proposition is immediate from the definitions and left for the interested reader.

Proposition 3.9. Let Glim(x, p, a) be a graded limit group, and let Sld(x, p, a) be a solid graded limit group in the graded Makanin-Razborov diagram of Glim(x, p, a). Suppose the graded resolution GRes_{i} of Glim(x, p, a) terminates in a strictly solid family of the solid graded limit group SldGlim(x, p, a). Then:

(i) the set of the defining parameters p for which Sld(x, p, a) admits a strictly solid solution is an EA-set.

(ii) the set of the defining parameters p for which Sld(x, p, a) admits at least s

strictly solid families of solutions is an EA-set. (iii) the set of the defining parameters p for which Sld(x, p, a) admits precisely

s strictly solid families of solutions is the intersection of an EA and an AE-sets.

(iv) the set of the defining parameters p for which the resolution GRes_{i} is nondegenerate for some strictly solid solution is an EA-set. The set of p’s for which a particular part of GRes_{i} is non-degenerate for some strictly solid solution is an EA-set.

Proposition 3.9 shows that the set of values of the parameters p for which there are at least s strictly solid families of solutions is an EA set, and the set of values of the parameters p for which there are precisely s strictly solid families of solutions is the intersection of an EA and an AE sets. However, the goal of our quantifier elimination procedure is to show that every elementary set defined over a free group is in the Boolean algebra of universal sets. To get that, it is clearly essential to show that all the sets defined in proposition 3.10 are indeed in this Boolean algebra. To get that we need to construct solid configuration limit groups associated with a given solid limit group Sld(x, p, a).

§4. Solid Configuration Limit Groups

To show that the set of parameters p, for which a solid limit group Sld(x, p, a) has precisely s (strictly solid) families of solutions, is in the Boolean algebra of universal sets, we need to construct solid configuration limit groups associated with the given solid limit group Sld(x, p, a). To present the construction of these configuration groups we start with solid limit groups with no flexible quotients that have a graded JSJ decomposition with two vertices and a single edge.

Let Sld(x, p, a) be a solid limit group with no flexible quotients and with graded JSJ decomposition containing two vertices and a single edge. Suppose Sld(x, p, a) = < y, p, a > ∗_{A} < w >, where the subgroup < w > is not abelian. Since < w > is not abelian and the graded JSJ decomposition of Sld(x, p, a) consists of a single edge and two vertices, the abelian edge group A is necessarily cyclic. Since By theorem 2.7 for any specific specialization of the parameters p, there is a global bound on the number of strictly solid families of specializations of the subgroup Sld(x, p, a). Hence, by the definition of strictly solid families and the specific structure of the solid limit group Sld(x, p, a), we may assume that for any specific specialization p_{0} of the parameters p, there are at most d different specializations of the subgroup (distinguished vertex group) < y, p, a > for which p = p_{0}, and for each specialization (y_{0}, p_{0}, a) there are at most e specializations w_{0} of the subgroup < w > up to conjugation by an element commuting with the specialization of the subgroup (edge group) A_{0}. To construct configuration limit groups we need to define configuration homomorphisms.

Definition 4.1. A configuration homomorphism of the solid limit group Sld(x, p, a) is a homomorphism:

h : < cy_{1}, s_{1}, cw_{1}^{1}, …, cw_{e}^{1}, …, cy_{d}, s_{d}, cw_{1}^{d}, …, cw_{e}^{d}, p, a > → F_{k}

for which:

(1) h(cy_{i}, cw_{j}^{i}, p, a) is a strictly solid specialization of the solid limit group Sld(x, p, a) = Sld(y, w, p, a).

(2) h(cy_{i}, cw_{j}^{i}, p, a) is a non-degenerate specialization of Sld(x, p, a).

(3) h(s_{i}) is a generator of the subgroup h(a_{i}), i.e., h(s_{i}) is a generator of the cyclic subgroup in F_{k} that contains the image of the edge group h(A).

(4) for each i, 1 ≤ i ≤ d, if we fix h(cw_{1}^{i}), then h(cw_{1}^{i}, …, cw_{e}^{i}) is the shortest possible under the (separate) action of conjugation by powers of the primitive root h(s_{i}) on each of the variables cw_{2}^{i}, …, cw_{e}^{i}.

A limit group obtained as the limit of a sequence of configuration homomorphisms is called a configuration limit group. By construction, there are only finitely many maximal configuration limit groups. We call each of this canonical collection of maximal limit groups a solid configuration limit group and denote it Conf(Sld)(s, cy, cw, p, a). By construction, a solid configuration limit group of the solid limit group Sld(x, p, a) that has a graded JSJ decomposition with a unique edge and two vertices, admits the following graph of groups:

Having constructed solid configuration limit groups, we analyze the structure of their subgroups V_{i} =< s_{i}, cw_{1}^{i}, …, cw_{e}^{i} > for i = 1, …, d. Clearly, the subgroup V_{i} is in particular an ungraded limit group, so it admits a canonical free decomposition and each factor admits a canonical (ungraded) abelian JSJ decomposition.

Lemma 4.2. Let Conf(Sld)(s, cy, cw, p, a) be a solid configuration limit group associated with the solid limit group Sld(x, p, a), let V_{i} =< s_{i}, cw_{1}^{i}, …, cw_{e}^{i} > for some index i, 1 ≤ i ≤ d, and let Λ_{V_{i}} be the abelian decomposition of V_{i} obtained by first factoring V_{i} into the most refined free decomposition and then taking the abelian JSJ decomposition of each of the factors. If V_{i} is not a surface group, i.e., if Λ_{V_{i}} is not the trivial decomposition, then every non-trivial element in the abelian subgroup < s_{i}, A_{i} >, is not elliptic in Λ_{V_{i}}. In particular, the subgroup < s_{i}, A_{i} > has to be a cyclic group.

Proof: Suppose A_{i} is elliptic in the decomposition Λ_{V_{i}}. For each index j, 1 ≤ j ≤ e, the subgroup CW_{j}^{i} =< cw_{j}^{i} > being a subgroup of V_{i} inherits a decomposition Δ_{j}^{i} from the graph of groups Λ_{V_{i}}. Clearly, A_{i} is elliptic in the graph of groups Δ_{j}^{i}. The subgroup CW_{j}^{i} was constructed from a sequence of (strictly solid) configuration homomorphisms {h_{n} : Conf(Sld)(s, cy, cw, p, a) → F_{k}}. If the graph of groups Δ_{j}^{i} is not the trivial graph of groups, then for large enough n the restriction of the homomorphism h_{n} to the subgroup < CY_{i}, CW_{j}^{i} > does not belong to a solid family of homomorphisms of Sld(x, p, a). Hence, in the graph of groups Λ_{V_{i}} each of the subgroups CW_{j}^{i} fixes a vertex. But the strictly solid homomorphisms {h_{n}} were assumed to shorten the specializations of the various subgroups V_{i} for i = 1, …, d, a contradiction. Therefore, A_{i} can not be elliptic in the graph of groups Λ_{V_{i}}. Now, if a non-trivial element of the subgroup < s_{i}, A_{i} > is elliptic in the decomposition Λ_{V_{i}}, then the subgroup < s_{i}, A_{i} > is free abelian of rank 2, so the entire subgroup < s_{i}, A_{i} > is elliptic in Λ_{V_{i}}, a contradiction. Therefore, every non-trivial element from of the (abelian) subgroup < s_{i}, A_{i} > acts hyperbolically on the Bass-Serre tree corresponding to the graph of groups Λ_{V_{i}}, and the subgroup < s_{i}, A_{i} > is cyclic.

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Lemma 4.3. With the notation of lemma 4.2 suppose that V_{i} is not a surface group and let Δ_{j}^{i} be the graph of groups inherited by the subgroup CW_{j}^{i} from the graph of groups Λ_{V_{i}}. Then the edge group A_{i} is hyperbolic in the graph of groups Δ_{j}^{i}, and A_{i} can not be conjugated into the fundamental group of any proper subgraph of Λ_{V_{i}} and any proper subgraph of Δ_{j}^{i}.

Proof: A_{i} is hyperbolic in Δ_{j}^{i} since it is hyperbolic in Λ_{V_{i}}. If the subgroup A_{i} can be conjugated into a proper subgraph of Λ_{V_{i}} or Δ_{j}^{i}, then for large enough n, the homomorphisms {h_{n}} from which the configuration solid limit group V_{i} was constructed are not solid homomorphisms.

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Proposition 4.4. With the notation of lemma 4.2, let Γ_{V_{i}} be an arbitrary nontrivial essential abelian decomposition of the subgroup V_{i}. Then the subgroup < s_{i}, A_{i} > is not elliptic in Γ_{V_{i}}.

Proof: Suppose the subgroup < s_{i}, A_{i} > is elliptic in Γ_{V_{i}}. W.l.o.g. we may assume that Γ_{V_{i}} contains only one edge. If the edge group in Γ_{V_{i}} is trivial, i.e., Γ_{V_{i}} corresponds to a free decomposition of the group V_{i} in which the subgroup A_{i} can be conjugated into one of the factors, then the restrictions of the configuration homomorphisms {h_{n}} from which the configuration limit group Conf(Sld)(s, cy, cw, p, a) was constructed to at least one of the subgroups < CW_{j}^{i} > are flexible, a contradiction to the homomorphisms {h_{n}} being configuration homomorphisms. Hence, we may assume that the edge group in the graph of groups Γ_{V_{i}} is a non-trivial abelian group. By exactly the same argument, the edge group in Γ_{V_{i}} has to include the cyclic subgroup A_{i}, and since by lemma 4.2 the subgroup A_{i} is not elliptic in the canonical (JSJ) decomposition Λ_{V_{i}} of V_{i} in case V_{i} is not a surface group, the maximal abelian group containing A_{i} is cyclic. Hence, the edge group in Γ_{V_{i}} is the maximal cyclic subgroup containing the subgroup < s_{i}, A_{i} > in V_{i}.

Since A_{i} is contained in the edge group of Γ_{V_{i}}, and in particular A_{i} is elliptic in Γ_{V_{i}}, each of the subgroups CW_{j}^{i} is elliptic in Γ_{V_{i}}. If V_{i} is a surface group, then by lemma 1.5 of [Se3] for every index j each of the configuration homomorphisms {h_{n}} (from which Conf(Sld)(s, cy, cw, p, a) was constructed) restricted to the subgroup CW_{j}^{i} are flexible homomorphisms, a contradiction to the homomorphisms {h_{n}} being configuration homomorphisms. Hence, we may assume that V_{i} is not a surface group. By lemma 4.3, if V_{i} is not a surface group then each of the non-cyclic freely indecomposable factors of V_{i} are elliptic in Γ_{V_{i}}. To conclude the proof of the proposition we need the following proposition. The proposition is proved for a f.g. free group in [Be-Fe2]. The proof for a general free product is a straightforward generalization of the argument presented in [Be-Fe2].

Proposition 4.5 ([Be-Fe2], ). Let G be a f.g. group and suppose G = B_{1} ∗ … ∗ B_{n} ∗ F_{s} is the maximal (most refined) free decomposition of G, where B_{i} are noninfinite-cyclic freely indecomposable and F_{s} is free. If G = D_{1} ∗_{C} D_{2}, C is infinite cyclic, and each of the subgroups B_{1}, …, B_{n} can be conjugated into either D_{1} or D_{2}, then C is a factor in a free decomposition of either D_{1} or D_{2}. If G = D∗_{C}, C is infinite cyclic and each of the subgroups B_{1}, …, B_{n} can be conjugated into D, then either C or tCt^{−1} (where t is the corresponding Bass-Serre generator) is a factor in a free decomposition of D.

According to the above proposition the subgroup A_{i} is a factor in one of the vertex groups it is connected to. Hence, at least for one index j the configuration homomorphisms {h_{n}} (from which Conf(Sld)(s, cy, cw, p, a) was constructed) restricted to the subgroup CW_{j}^{i} are flexible homomorphisms, a contradiction to the homomorphisms {h_{n}} being configuration homomorphisms.

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According to proposition 3.10 the sets containing all values of the parameters p for which there are at least s strictly solid families are EA sets, the sets for which there are at most s strictly solid families are AE sets, and the sets for which there are exactly s strictly solid families are the intersection of an EA and an AE sets. To show that these sets are in fact all in the Boolean algebra of universal sets, we need to be able to bound the possible conjugacies of the solid specialization of vertex groups by elements that commute with the corresponding specialization of edge groups. This was the purpose of constructing solid configuration limit groups. To obtain a desired bound on the possible required conjugacies we need to define the configuration scalar of a solid configuration limit group.

Definition 4.6. Let Conf(Sld)(s, cy, cw, p, a) be a solid configuration limit group. We say that an integer C_{conf} is a configuration scalar of Conf(Sld)(s, cy, cw, p, a) if for every sequence of configuration homomorphisms: {h_{n} : Conf(Sld)(s, cy, cw, p, a) → F_{k}} that converges to the solid configuration limit group Conf(Sld)(cy, cw, p, a), and for which for every index n, h_{n}(s_{i}) is the generator of the cyclic subgroup of F_{k} that contains h_{n}(A_{i}), there exists some index n_{0} (depending on the sequence of the configuration homomorphisms), so that for every n > n_{0} if for some index i, 1 ≤ i ≤ d, and couple of indices j_{1}, j_{2}, 1 ≤ j_{1} < j_{2} ≤ e, there exists some element t_{n} ∈ F_{k} for which t_{n} commutes with h_{n}(A_{i}) and t_{n}h_{n}(cw_{j_{1}}^{i})t_{n}^{−1} = h_{n}(cw_{j_{2}}^{i}), then there exists some integers m_{n}, −C_{conf} ≤ m_{n} ≤ C_{conf}, for which: h_{n}(s_{i})^{m_{n}}h_{n}(cw_{j_{1}}^{i})h_{n}(s_{i})^{−m_{n}} = h_{n}(cw_{j_{2}}^{i}).

Proposition 4.7. For every solid configuration limit group Conf(Sld)(s, cy, cw, p, a) of a solid limit group Sld(x, p, a) with no flexible quotients and graded JSJ decomposition consists of a single edge, there exists a configuration scalar C_{conf}.

Proof: Suppose there was no configuration scalar for some configuration limit group Conf(Sld)(s, cy, cw, p, a) associated with a solid limit group Sld(x, p, a) with the properties given above. Then there exists a sequence of configuration homomorphisms: h_{n} : Conf(Sld)(s, cy, cw, p, a) → F_{k} and an index i, 1 ≤ i ≤ d, and couple of indices j_{1}, j_{2}, 1 ≤ j_{1} < j_{2} ≤ e, so that for every index n there exists an element t_{n} ∈ F_{k} for which t_{n} commutes with h_{n}(A_{i}) and t_{n}h_{n}(cw_{j_{1}}^{i})t_{n}^{−1} = h_{n}(cw_{j_{2}}^{i}), but for every integer m, −n ≤ m ≤ n: h_{n}(s_{i})^{m}h_{n}(cw_{j_{1}}^{i})h_{n}(s_{i})^{−m} ≠ h_{n}(cw_{j_{2}}^{i}). By passing to a subsequence (still denoted {h_{n}}), we may assume that the sequence of homomorphisms {h_{n}|_{V_{i}}} converges into a (stable) action of the subgroup V_{i} of the configuration limit group Conf(Sld)(s, cy, cw, p, a) on some real tree Y. By [Be-Fe1], with the action of V_{i} on the real tree Y, one canonically associates a graph of groups Γ_{Y} with fundamental group Y and abelian edge groups, and by proposition 4.4 the element s_{i} is hyperbolic in the graph of groups Γ_{Y}, so s_{i} acts hyperbolically on the real tree Y. Hence, by the Gromov-Hausdorff convergence of the homomorphisms {h_{n}} into the action of V_{i} on the real tree Y, the sequence of configuration homomorphisms {h_{n}} itself does have a configuration scalar, i.e., there exists some integers ℓ_{0} so that for every index n if there exists an element t_{n} for which t_{n} commutes with h_{n}(A_{i}) and t_{n}h_{n}(cw_{j_{1}}^{i})t_{n}^{−1} = h_{n}(cw_{j_{2}}^{i}), then for some integer m, −ℓ_{0} ≤ m ≤ ℓ_{0}: h_{n}(s_{i})^{m}h_{n}(cw_{j_{1}}^{i})h_{n}(s_{i})^{−m} = h_{n}(cw_{j_{2}}^{i}), a contradiction to the properties of the sequence {h_{n}}, and the proposition follows.

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The homomorphisms that factor through Conf(Sld)(s, cy, cw, p, a) and can not be ”scaled” by a bounded power of the generator h(s_{i}) of the image of the edge group h(A_{i}) are called unscaled homomorphisms.

Definition 4.8. Let C_{conf} be a configuration scalar of the solid configuration limit group Conf(Sld)(s, cy, cw, p, a). A configuration homomorphism μ : Conf(Sld)(s, cy, cw, p, a) → F_{k} is said to be an unscaled configuration homomorphism if for some index i, 1 ≤ i ≤ d, and couple of indices j_{1}, j_{2}, 1 ≤ j_{1} < j_{2} ≤ e, there exists some element t ∈ F_{k} for which t commutes with μ(A_{i}), tμ(cw_{j_{1}}^{i})t^{−1} = μ(cw_{j_{2}}^{i}), and there is no integer m, −C_{conf} ≤ m ≤ C_{conf}, for which: μ(s_{i})^{m}μ(cw_{j_{1}}^{i})μ(s_{i})^{−m} = μ(cw_{j_{2}}^{i}).

Proposition 4.9. Let UnScConf(s, cy, cw, p, a) be a limit group obtained from a converging sequence of unscaled homomorphisms {μ_{n} : Conf(Sld)(s, cy, cw, p, a) → F_{k}}. Then UnScConf(s, cy, cw, p, a) is a proper quotient of the configuration limit group Conf(Sld)(s, cy, cw, p, a).

Proof: If UnScConf(s, cy, cw, p, a) is not a proper quotient of the configuration limit group Conf(s, cy, cw, p, a) then there exists a sequence of configuration homomorphisms h_{n} : Conf(s, cy, cw, p, a) → F_{k} converging to Conf(s, cy, cw, p, a) for which C_{conf} in not a configuration scalar, a contradiction to the properties of the configuration scalar given in definition 4.6.

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Definition 4.10. Let C_{conf} be a configuration scalar of the solid configuration limit group Conf(Sld)(s, cy, cw, p, a). Clearly, the collection of unscaled configuration homomorphisms μ : Conf(Sld)(s, cy, cw, p, a) → F_{k} factors through a finite (canonical) collection of maximal limit groups. We call each of these maximal limit group an unscaled configuration quotient of the solid configuration limit group Conf(Sld)(s, cy, cw, p, a). By proposition 4.9 every unscaling configuration quotient of a solid configuration limit group is a proper quotient of it.

The construction of solid configuration limit groups and their canonical finite collection of unscaled configuration quotients, are the objects needed in order to ”count” strictly solid families. So far we have analyzed solid limit groups with no flexible quotients whose graded JSJ decomposition consists of a single edge. The construction presented for solid limit groups with a single edge graded JSJ decomposition generalizes in a straightforward way to a solid limit group with no flexible quotients whose graded JSJ decomposition is a tree of groups containing no QH vertex groups with more than a single boundary component. We start the generalization of the construction of configuration limit groups to arbitrary solid limit groups by describing the construction for solid limit groups with no flexible quotients whose graded JSJ decomposition consists of two (non-QH) vertex groups and two edges connecting them.

Let Sld(x, p, a) be a solid limit group with no flexible quotients whose graded abelian decomposition has the following form:

To fix notation we will set Sld(x, p, a) = Sld(y, w, t, p, a) where < y, p, a > is the distinguished vertex group, W =< w > is the second vertex group, and t is the Bass-Serre generator. Let V = t^{−1}W t. By theorem 2.7 for every specific specialization of the parameters p, there is a global bound on the strictly solid families of specializations of the subgroup Sld(x, p, a). Hence, by the definition of strictly solid families and the specific structure of the solid limit group Sld(x, p, a), we may assume that for any specific specialization p_{0} of the parameters p, there are at most d different specializations of the (distinguished vertex) subgroup < y, p, a >, and fixing a specialization of the subgroup < y, p, a >, up to the action of the modular group of Sld(x, p, a) there are at most e specializations of the other vertex group < w >.

Definition 4.11. A configuration homomorphism of the solid limit group Sld(x, p, a) is a homomorphism:

h : < s_{i}, r_{i}, t_{i}, cy_{i}, u_{j}^{i}, cw_{j}^{i}, p, a > → F_{k}

for which:

(1) h(cy_{i}, cw_{j}^{i}, t_{j}^{i}, p, a) is a strictly solid specialization of the solid limit group Sld(x, p, a) = Sld(y, w, t, p, a).

(2) h(t_{1}^{i}) = h(t_{i}) and h(t_{j}^{i}) = h(t_{i}u_{j}^{i}).

(3) h(cy_{i}, cw_{j}^{i}, t_{j}^{i}, p, a) is a non-degenerate specialization of Sld(x, p, a).

(4) h(s_{i}) is a generator of the maximal cyclic subgroup containing the image under h of the first edge group in the free group F_{k}, and h(r_{i}) is the generator of the maximal cyclic subgroup containing the image under h of the second edge group in the free group F_{k}.

(5) for each i, 1 ≤ i ≤ d, if we fix h(cw_{1}^{i}), then h(u_{2}^{i}, cw_{2}^{i}, …, u_{e}^{i}, cw_{e}^{i}) is the shortest possible under conjugation by a power of h(s_{i}) and multiplying h(t_{j}^{i}) by a power of h(r_{i}).

A limit group obtained as the limit of a sequence of configuration homomorphisms is called a configuration limit group. By construction, there are only finitely many maximal configuration limit groups. We call each of this canonical collection of maximal limit groups a solid configuration limit group and denote it Conf(Sld)(s, r, t, cy, u, cw, p, a). By construction, a solid configuration limit group admits the following graph of groups:

Having constructed the solid configuration limit groups, we analyze the structure of their subgroups W_{i} =< s_{i}, t_{i}r_{i}(t_{i})^{−1}, cw_{1}^{i}, u_{2}^{i}, cw_{2}^{i}, …, u_{e}^{i}, cw_{e}^{i} > and V_{i} = (t_{i})^{−1}W_{i}t_{i} for i = 1, …, d. Clearly, the subgroup W_{i} is in particular an ungraded limit group, so it admits canonical free decomposition and each factor admits a canonical abelian JSJ decomposition.

Lemma 4.12. Let Conf(Sld)(s, r, t, cy, u, cw, p, a) be a solid configuration limit group associated with the solid limit group Sld(x, p, a), and for some index i, 1 ≤ i ≤ d, let W_{i} be the subgroup defined above, and V_{i} = (t_{i})^{−1}W_{i}t_{i}. Let Λ_{W_{i}} and Λ_{V_{i}} be the canonical graphs of groups associated with the (ungraded) limit groups W_{i} and V_{i}. Let A^{1} be the first edge group and A^{2} be the second edge group connected to the vertex stabilized by W in the graded abelian decomposition of Sld(x, p, a), and let A_{i}^{1} and A_{i}^{2} be the corresponding subgroups of CW_{i}.

Then either the subgroup < s_{i}, A_{i} > or the subgroup < t_{i}r_{i}(t_{i})^{−1}, A_{i}^{2} > act hyperbolically on the Bass-Serre tree corresponding to the graph of groups Λ_{W_{i}}. In particular, each of these two subgroups that act hyperbolically on the graph of groups Λ_{W_{i}} is infinite cyclic.

Proof: The proof is similar to the proof of lemma 4.2. Suppose both A_{i}^{1} and A_{i}^{2} are elliptic in the decomposition Λ_{W_{i}}. For each index j, 1 ≤ j ≤ e, the subgroup CW_{j}^{i} =< cw_{j}^{i} > being a subgroup of W_{i} inherits a decomposition Δ_{j}^{i} from the graph of groups Λ_{W_{i}}. Clearly, A_{i}^{1} and A_{i}^{2} are both elliptic in the graph of groups Δ_{j}^{i}. The subgroup CW_{j}^{i} was constructed from a sequence of (strictly solid) configuration homomorphisms {h_{n} : Conf(Sld)(s, r, t, cy, u, cw, p, a) → F_{k}}. If the graph of groups Δ_{j}^{i} is not the trivial graph of groups, then for large enough n the homomorphisms h_{n} can not be a configuration homomorphism, since the corresponding specializations of the subgroup CW_{j}^{i} belong to a flexible quotient of the solid limit group Sld(x, p, a). Hence, in the graph of groups Λ_{W_{i}} each of the subgroups CW_{j}^{i} fixes a vertex. But the strictly solid homomorphisms {h_{n}} were assumed to shorten the specializations of the various subgroups W_{i} for i = 1, …, d, a contradiction. Therefore, A_{i}^{1} and A_{i}^{2} can not both be elliptic in the graph of groups Λ_{W_{i}}. Suppose A_{i}^{1} is not elliptic in the graph of groups Λ_{W_{i}}. Now, if a non-trivial element of the subgroup < s_{i}, A_{i}^{1} > is elliptic in the decomposition Λ_{W_{i}}, then the subgroup < s_{i}, A_{i}^{1} > is free abelian of rank 2, so the entire subgroup < s_{i}, A_{i}^{1} > is elliptic in Λ_{W_{i}}, a contradiction. Therefore, every non-trivial element from of the (abelian) subgroup < s_{i}, A_{i} > acts hyperbolically on the Bass-Serre tree corresponding to the graph of groups Λ_{W_{i}}, and the subgroup < s_{i}, A_{i}^{1} > is cyclic. By an identical argument if A_{i}^{2} is not elliptic in Λ_{W_{i}} then < t_{i}r_{i}t_{i}^{−1}, A_{i}^{2} > is cyclic.

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Lemma 4.13. With the notation of lemma 4.12, suppose that W_{i} is not a surface group and let Δ_{j}^{i} be the graph of groups inherited by the subgroup CW_{j}^{i} from the graph of groups Λ_{W_{i}}. Then either the edge group A_{i}^{1} or the edge group A_{i}^{2} are hyperbolic in the graph of groups Δ_{j}^{i}, and A_{i}^{1} and A_{i}^{2} can not be conjugated into the fundamental group of any proper (not necessarily connected) subgraph of Λ_{W_{i}} and any proper subgraph of Δ_{j}^{i}.

Proof: Either A_{i}^{1} or A_{i}^{2} are hyperbolic in Δ_{j}^{i} since at least one of these subgroups are hyperbolic in Λ_{W_{i}}. If both subgroups A_{i}^{1} and A_{i}^{2} can be conjugated into a proper subgraph of Λ_{W_{i}} or Δ_{j}^{i}, then for large enough n, the homomorphisms {h_{n}} from which the configuration solid limit group W_{i} was constructed are not solid homomorphisms.

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Proposition 4.14. With the notation of lemma 4.12, let Γ_{W_{i}} be an arbitrary nontrivial essential abelian decomposition of the subgroup W_{i}. Then either the subgroup < s_{i}, A_{i}^{1} > or the subgroup < t_{i}r_{i}t_{i}^{−1}, A_{i}^{2} > are not elliptic in Γ_{W_{i}}.

Proof: The proof is similar to the proof of proposition 4.4. Suppose both subgroups A_{i}^{1} and A_{i}^{2} are elliptic in Γ_{V_{i}}. W.l.o.g. we may assume that Γ_{W_{i}} contains only one edge. If the edge group in Γ_{W_{i}} is trivial, i.e., Γ_{W_{i}} corresponds to a free decomposition of the group W_{i} in which both subgroups A_{i}^{1} and A_{i}^{2} can be conjugated into the two factors, then the restrictions of the configuration homomorphisms {h_{n}} from which the configuration limit group Conf(Sld)(s, r, t, cy, u, cw, p, a) was constructed to at least one of the subgroups < CW_{j}^{i} > are flexible, a contradiction to the homomorphisms {h_{n}} being configuration homomorphisms. Hence, we may assume that the edge group in the graph of groups Γ_{W_{i}} is a non-trivial abelian group. By exactly the same argument, both subgroups A_{i}^{1} and A_{i}^{2} can be conjugated into the edge group in Γ_{W_{i}}. Since by lemma 4.12 either W_{i} is a surface group or the two subgroups A_{i}^{1} and A_{i}^{2} can not be both conjugated into a proper subgraph of the canonical (JSJ) decomposition Λ_{W_{i}} of W_{i}, the maximal abelian group containing A_{i}^{1} and A_{i}^{2} is cyclic. Hence, the edge group in Γ_{W_{i}} is the maximal cyclic subgroup containing the subgroup < s_{i}, A_{i}^{1} > in W_{i}, and the subgroup < t_{i}r_{i}t_{i}^{−1}, A_{i}^{2} > can be conjugated into this edge group.

Since both A_{i}^{1} and A_{i}^{2} can be conjugated into the edge group of Γ_{W_{i}}, and in particular A_{i}^{1} and A_{i}^{2} are elliptic in Γ_{W_{i}}, each of the subgroups CW_{j}^{i} is elliptic in Γ_{W_{i}}. If W_{i} is a surface group, then by lemma 1.5 of [Se3] for every index j each of the configuration homomorphisms {h_{n}} (from which Conf(Sld)(s, r, t, cy, u, cw, p, a) was constructed) restricted to the subgroup CW_{j}^{i} are flexible homomorphisms, a contradiction to the homomorphisms {h_{n}} being configuration homomorphisms. Hence, we may assume that W_{i} is not a surface group. By lemma 4.13, if W_{i} is not a surface group then each of the non-cyclic freely indecomposable factors of W_{i} are elliptic in Γ_{W_{i}}.

By proposition 4.5, at least for one index j the configuration homomorphisms {h_{n}} (from which Conf(Sld)(s, r, t, cy, u, cw, p, a) was constructed) restricted to the subgroup CW_{j}^{i} are flexible homomorphisms, a contradiction to the homomorphisms {h_{n}} being configuration homomorphisms.

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We define the configuration scalar in the HNN case in a similar way to the definition in the amalgamated product case.

Definition 4.15. Let Conf(Sld)(s, r, t, cy, u, cw, p, a) be a solid configuration limit group as above. We say that a couple of integers (C_{conf}^{1}, C_{conf}^{2}) is a configuration couple for Conf(Sld)(s, r, t, cy, u, cw, p, a) if for every sequence of configuration homomorphisms: {h_{n} : Conf(Sld)(s, r, t, cy, u, cw, p, a) → F_{k}} that converges to the solid configuration limit group Conf(Sld)(s, r, t, cy, u, cw, p, a), and for which for every index n, h_{n}(s_{i}) is the generator of the cyclic subgroup of F_{k} that contains h_{n}(A_{i}^{1}), and h_{n}(r_{i}) is the generator of the cyclic subgroup of F_{k} that contains h_{n}(t_{i}^{−1}A_{i}^{2}t_{i}), there exists some index n_{0} (depending on the sequence of the configuration homomorphisms), so that for every n > n_{0} if for some index i, 1 ≤ i ≤ d, and couple of indices j_{1}, j_{2}, 1 ≤ j_{1} < j_{2} ≤ e, there exists some element t_{n} ∈ F_{k} for which t_{n} commutes with h_{n}(A_{i}^{1}) and t_{n}h_{n}(cw_{j_{1}}^{i})t_{n}^{−1} = h_{n}(cw_{j_{2}}^{i}), then either there exists some integers m_{n}^{1}, −C_{conf}^{1} ≤ m_{n}^{1} ≤ C_{conf}^{1}, for which:

h_{n}(s_{i})^{m_{n}^{1}}h_{n}(cw_{j_{1}}^{i})h_{n}(s_{i})^{−m_{n}^{1}} = h_{n}(cw_{j_{2}}^{i}) or there exists some integers m_{n}^{2}, −C_{conf}^{2} ≤

m_{n}^{2} ≤ C_{conf}^{2}, for which: h_{n}(r_{i})^{m_{n}^{2}}h_{n}(cv_{j_{1}}^{i})h_{n}(r_{i})^{−m_{n}^{2}} = h_{n}(cv_{j_{2}}^{i})

Proposition 4.16. For every solid configuration limit group Conf(Sld)(s, r, t, cy, u, cw, p, a) of a solid limit group Sld(x, p, a) with no flexible quotients and graded JSJ decomposition consists of two vertices and two edges between them, there exists a configuration couple (C_{conf}^{1}, C_{conf}^{2}).

Proof: Given proposition 4.14, the proof is identical with the proof of proposition 4.7.

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The homomorphisms that factor through Conf(Sld)(s, r, t, cy, u, cw, p, a) and can not be ”scaled” by bounded powers of the generators h(s_{i}) and h(t_{i})h(r_{i})h(t_{i}^{−1}) of the images of the edge groups h(A_{i}^{1}) and h(A_{i}^{2}) are called unscaled homomorphisms.

Definition 4.17. Let (C_{conf}^{1}, C_{conf}^{2}) be a configuration couple for the configuration limit group Conf(Sld)(s, r, t, cy, u, cw, p, a). A configuration homomorphism μ : Conf(Sld)(s, r, t, cy, u, cw, p, a) → F_{k} is said to be an unscaled configuration homomorphism if for some index i, 1 ≤ i ≤ d, and couple of indices j_{1}, j_{2}, 1 ≤ j_{1} < j_{2} ≤ e, there exists some element t ∈ F_{k} for which t commutes with μ(A_{i}^{1}), tμ(cw_{j_{1}}^{i})t^{−1} = μ(cw_{j_{2}}^{i}), and there are no integers m^{1}, m^{2}, −C_{conf}^{1} ≤ m^{1} ≤

C_{conf}^{1}, −C_{conf}^{2} ≤ m^{2} ≤ C_{conf}^{2}, for which: μ(s_{i})^{m^{1}}μ(cw_{j_{1}}^{i})μ(s_{i})^{−m^{1}} = μ(cw_{j_{2}}^{i}) or:

μ(r_{i})^{m^{2}}μ(cv_{j_{1}}^{i})μ(r_{i})^{−m^{2}} = μ(cv_{j_{2}}^{i}).

Proposition 4.18. Let UnScConf(s, r, t, cy, u, cw, p, a) be a limit group obtained from a converging sequence of unscaled homomorphisms {μ_{n} : Conf(Sld)(s, r, t, cy, u, cw, p, a) → F_{k}}. Then UnScConf(s, r, t, cy, u, cw, p, a) is a proper quotient of the configuration limit group Conf(Sld)(s, r, t, cy, u, cw, p, a).

Proof: Identical to the proof of proposition 4.9.

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Definition 4.19. Let (C_{conf}^{1}, C_{conf}^{2}) be a configuration couple for the configuration limit group Conf(Sld)(s, r, t, cy, u, cw, p, a). Clearly, the collection of unscaled configuration homomorphisms μ : Conf(Sld)(s, r, t, cy, u, cw, p, a) → F_{k} factors through a finite (canonical) collection of maximal limit groups. We call each of these maximal limit group an unscaled configuration quotient of the configuration limit group Conf(Sld)(s, r, t, cy, u, cw, p, a). By proposition 4.18 every unscaling configuration quotient of a configuration limit group is a proper quotient of it.

The construction of the configuration limit groups and their unscaled quotients are the tools needed for analyzing the solid families in case a solid limit group has no flexible quotients and its graded JSJ decomposition contains no QH vertex groups with more than one boundary component.

Proposition 4.20. Let Sld(x, p, a) be a solid graded limit group with no flexible quotients and suppose its graded JSJ decomposition contains a single (distinguished) non-abelian, non-QH vertex groups, possibly few abelian and QH vertex groups with a single boundary components connected to it, and possible few loops based on the distinguished vertex. Then the set of the defining parameters p for which Sld(x, p, a) admits precisely s strictly solid families of solutions is in the Boolean algebra of universal sets.

Proof: In case the JSJ decomposition of the solid limit group Sld(x, p, a) contains a single non-abelian, non-QH vertex group, and (possibly) few abelian vertex groups and QH vertex groups with a single boundary component connected to it, and (possibly) few loops based on the distinguished vertex, and assuming Sld(x, p, a) has no flexible quotients, the number of the strictly solid families of Sld(x, p, a) is precisely the number of different specializations of the distinguished vertex group. Therefore, the analysis of the set of parameters p for which there are precisely s strictly solid families of solutions, is identical with the analysis of the different number of rigid solutions of a rigid limit group with no flexible quotients (proposition 3. ). Hence, the set of parameters p for which there are precisely s strictly solid families of solutions is in the Boolean algebra of universal sets.

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Proposition 4.21. Let Sld(x, p, a) be a solid graded limit group with no flexible quotients and suppose its graded JSJ decomposition contains two non-abelian, non-QH vertex groups connected by one or two edges. Then the set of the defining parameters p for which Sld(x, p, a) admits precisely s strictly solid families of solutions is in the Boolean algebra of universal sets.

Proof: Suppose the graded JSJ decomposition of Sld(x, p, a) contains a single edge. Counting degenerate strictly solid families of solutions of Sld(x, p, a) for a given value of the parameter p is identical to counting the number of strictly solid families of solutions of a solid limit group with a graded JSJ decomposition containing a single distinguished vertex connected to an abelian vertex group. Hence, by proposition 4.20, the set of the defining parameters p for which Sld(x, p, a) admits precisely s_{d} degenerate strictly solid families of solutions is in the Boolean algebra of universal sets, so to complete the proof of proposition 4.21 we need to analyze the set of parameters p for which Sld(x, p, a) admits s non-degenerate strictly solid families of solutions.

To analyze the set of parameters p for which there are precisely s non-degenerate strictly solid families of solutions of Sld(x, p, a) we first look at the set of parameters p for which there are precisely q specializations of the distinguished vertex that can be extended to non-degenerate solutions of Sld(x, p, a). This set is defined in a similar way to the set of parameters for which a rigid limit groups admits a fixed number of rigid solutions, so it is in the Boolean algebra of universal sets. Furthermore, the configuration limit groups Conf(s, cy, cw, p, a) associated with Sld(x, p, a) leave us with only finitely many possibilities for the order of a primitive root of a specialization of the cyclic edge group A. Hence, the set of parameters p for which there are precisely q different specializations of the distinguished vertex group that can be extended into non-degenerate solid specializations of Sld(x, p, a) and for which the corresponding specializations of the edge group A admits primitive roots of orders o_{1}, …, o_{q} (for each possible q-tuple of positive integers) is in the Boolean algebra of universal sets. We will denote this set P_{(q,o_{1},... ,o_{q})}.

At this point we can use the (finitely many) scaling constants and the unscaled quotients of the configuration limit groups to analyze the set of parameters p for which there are precisely s non-degenerate strictly solid families of solutions of Sld(x, p, a). We analyze the sets P_{(q,o_{1},... ,o_{q})} in parallel. If p ∈ P_{(q,o_{1},... ,o_{q})}, there are at least s non-degenerate solid families associated with p, if there exist s non-degenerate specializations of the corresponding configuration limit group, a specialization that does not factor through any of the unscaled quotients and for which no pair of the different specializations of Sld(x, p, a) associated with the specialization of Conf(s, cy, cw, p, a) can be made equal by conjugating by some power, bounded by the corresponding scaling factor, of the specialization of the primitive root of the of the edge group A. There are at most s solutions if the last condition does not hold for any specialization of a configuration limit group associated with m > s non-degenerate strictly solid families of solutions. Since these conditions are universal and co-universal in correspondence, and the set P_{(q,o_{1},... ,o_{q})} is in the Boolean algebra of universal sets, the set of the defining parameters p, for which Sld(x, p, a) admits precisely s non-degenerate strictly solid families of solutions is in the Boolean algebra of universal sets. So far we have analyzed solid limit groups with a graded JSJ decomposition containing one edge and two vertices. The analysis of a solid limit group Sld(x, p, a) having a graded JSJ decomposition with two vertices and two edges is exactly the same given the scaling couples and the unscaled quotients associated with the configuration limit groups associated with such solid limit group.

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Proposition 4.22. Let Sld(x, p, a) be a solid graded limit group with no flexible quotients and suppose its graded JSJ decomposition is a tree of groups with no QH vertex groups containing more than one boundary component. Then the set of the defining parameters p for which Sld(x, p, a) admits precisely s strictly solid families of solutions is in the Boolean algebra of universal sets.

Proof: Analyzing solid limit groups with no flexible quotients and a tree JSJ decomposition is essentially identical to the analysis of a solid limit group with a JSJ decomposition containing two vertices and a single edge. We first look at the set of parameters p for which there are precisely q different specializations for the distinguished vertex that can be extended to solid specializations of Sld(x, p, a). This set is clearly in the Boolean algebra of universal sets. Once we have defined the set of parameters for which the root of the tree JSJ decomposition of Sld(x, p, a) admits q different specializations, and for each of the different specializations the corresponding edge groups admit primitive roots of certain prescribed orders, we continue in the same way by defining the sets for which the vertices connected to the root in the tree JSJ decomposition admit certain prescribed numbers of different specializations and for each of these specializations the edge groups connected to these vertices admit primitive roots of certain prescribed orders. These sets are also in the Boolean algebra of universal sets. By further continuing to the vertices in the next levels of the tree JSJ decomposition, we exhaust the entire tree, and get that each of the sets for which each of the vertices in the tree JSJ decomposition admit certain prescribed number of different specialization and the edge groups connected to each vertex admit primitive roots of certain prescribed orders are in the Boolean algebra of universal sets.

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Theorem 4.23. Let Sld(x, p, a) be a solid graded limit group with no flexible quotients and suppose its graded JSJ decomposition contains no QH vertex groups with more than one component. Then the set of the defining parameters p for which Sld(x, p, a) admits precisely s strictly solid families of solutions is in the Boolean algebra of universal sets.

Proof: To define the set of parameters with a given number of strictly solid families of solutions, we go over all the various possibilities for different specializations of the non-abelian non-QH vertex groups in the JSJ decomposition of Sld(x, p, a), and all the possibilities for the orders of primitive roots of the various edge groups in the JSJ decomposition. There are precisely s strictly solid families of solutions for a given specialization of the parameters p, if there s families which are different when we check possible conjugations by powers bounded by the relevant scaling constants of primitive roots of the corresponding edge groups, non of the possibilities for s+1 or more strictly solid families hold, and the edge groups do not have roots of higher order than the prescribed ones. All these conditions on the defining parameters p can be stated in terms of predicates in the Boolean algebra of universal predicates, so the entire set of the defining parameters p for which there are precisely s strictly solid families of solutions is in the Boolean algebra of universal sets.

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