Effective construction of canonical Hom-diagrams for equations over torsion-free hyperbolic groups

Effective construction of canonical Hom-diagrams
for equations over torsion-free hyperbolic groups

Olga Kharlampovich, Alexei Myasnikov, Alexander Taam
Abstract

We show that, given a finitely generated group as the coordinate group of a finite system of equations over a torsion-free hyperbolic group , there is an algorithm which constructs a canonical solution diagram by constructing canonical (corrective extensions of) -NTQ-groups. These groups are toral relatively hyperbolic -limit groups. The diagram encodes all homomorphisms from to as compositions of factorizations through -limit quotients (constructed by defining their generators inside canonical NTQ groups) and canonical automorphisms induced on the freely indecomposable factors of these quotients by canonical automorphisms of the corresponding NTQ-subgroups. Additionally, we show that a group is a -limit group if and only if it is an iterated generalized double over .

1 Introduction

Given a group , another group is said to be fully residually (or discriminated by ) if, given any finite subset , , there exists a homomorphism , such that for all . The study of groups discriminated by various classes of groups has been ongoing since the early twentieth century, with increased activity often occuring as other equivalent characterizations of such groups, and applications in different fields, are found. For example, the development of algebraic geometry over groups allowed finitely generated groups discriminated by a free group to be understood as coordinate groups of irreducible systems of equations over . This helped enable solutions to Tarski’s problems on the elementary theory of free groups, given independently by Kharlampovich-Myasnikov and Sela ([17] and [26]).

Throughout, we let be an arbitrary fixed torsion-free hyperbolic group. While some of the theory of fully residually free groups can be generalized easily to groups discriminated by , in general groups discriminated by are more difficult to work with for several reasons. Unlike finitely generated fully residually free groups, finitely generated fully residually groups may be not finitely presented and can contain finitely generated subgroups with no finite presentation, so algorithmic results are often be more difficult to obtain.

For a group which is equationally Noetherian (i.e. every system of equations over with variables is equivalent to a finite sub-system), there are many descriptions of the class of finitely generated fully residually groups. Let be a finitely generated group and an equationally Noetherian group. The following are equivalent:

  1. is a -limit group (in the space of marked groups, see [9] for definition)

  2. is fully residually

  3. embeds in an ultrapower of

  4. ( is universal first order theory)

  5. ( is existential first order theory)

  6. is the coordinate group (see Section 2.1) of an irreducible (in the Zariski topology) algebraic set over defined by a system of coefficient-free equations (call such a system an irreducible system)

There are analagous equivalent characterizations for the case with coefficients (see [18] Theorem B).

Given a group , a free rank one extension of centralizer over is a group , where is finite. Note that . A free rank extension of centralizer is defined similarly as . A group is called an iterated extension of centralizer over if it may be obtained from by a finite sequence of extension of centralizers (each centralizer in the previous group).

Proposition 1.

(Theorem E in [18]) Given a fixed toral relatively hyperbolic group , a group is a -limit group if and only if it embeds into an iterated extension of centralizer over .

In fact, the embeddings into iterated extentions of centralizers are obtained via embeddings into -NTQ groups, which are coordinate groups of nicely structured systems of equations (see Section 2.1.2). Some relevant details of such embeddings are described in Section 5, along with other related constructions.

Our first main theorem shows that, for a fixed torsion-free hyperbolic group , the class of iterated generalized doubles over (terminology introduced by Champetier and Guirardel in [3] for free groups) is also equivalent to the class of -limit groups.

Definition 1.

A group is a generalized double over a -limit group if it splits as , or , where and are finitely generated such that:

  1. is a non-trivial abelian group whose images are maximal abelian subgroups in the vertex groups.

  2. there is an epimorphism which is injective on each vertex group

Note that each vertex group is discriminated by .

Definition 2.

A group is an iterated generalized double over if it belongs to the smallest class of groups that contains groups isomorphic to subgroups of , and is stable under free products and the construction of generalized doubles over groups in .

Champetier and Guirardel showed in [3] that is the class of -limit groups. Our first main theorem shows that the same holds for -limit groups.

Theorem 1.

is a -limit group if and only if it is an iterated generalized double over

Theorem 1 is proved in Section 3. The proof uses Bass-Serre theory and the fact that -limit groups are exactly the groups which embed into iterated extensions of centralizers over .

Recall that the Grushko decomposition of a finitely generated group is the free product decomposition , where is a free group of finite rank, and each is non-trivial, freely indecomposable, and not infinite cyclic (see [12]). This decomposition is unique up to permutation of the conjugacy classes of the in . In section 4.2, we describe a canonical decomposition of a certain type, called a primary JSJ decomposition, for a freely indecomposable -limit group, which encodes all of its abelian splittings. Primary JSJ decompositions were shown to exist for -limit groups in [27].

Our second main theorem gives an algorithm that, given a finite system of equations over , constructs a finite diagram with associated groups which has certain canonical properties. Such a canonical Hom-diagram (which is described more precisely in Section 5.5) encodes all homomorphisms from the coordinate group to as compositions of factorizations through -limit quotients, and certain canonical automorphisms which correspond to JSJ decompositions of freely indecomposable factors of those quotients. Furthermore certain canonical NTQ groups are associated to such a diagram (through a process also described in Section 5.5).

Theorem 2.

Let be a finite system of equations over . There is an algorithm to construct a complete set of canonical (corrective extensions of) -NTQ groups that are, in particular, toral relatively hyperbolic -limit groups, and associated completed canonical Hom-diagram and canonical Hom-diagram for . All solutions of in factor through the strict fundamental sequences corresponding to these (corrective extensions of) -NTQ groups.

NTQ -limit groups in the completed diagram with branches (7) are toral relatively hyperbolic and constructed by their finite presentations, and canonical groups of automorphisms are given by their generators (and we know their presentation).

-limit groups in the branch (6) of the Hom-diagram are constructed by defining their generators inside corresponding -groups and canonical automorphisms of these -limit groups are induced on their freely indecomposable factors by canonical automorphisms of .

Theorem 2 is proved in Section 6.

We thank D. Groves and H. Wilton for constructing “cautionary examples” [11] that helped us to improve the exposition of this work, through necessary clarifications and error corrections.

2 Preliminaries

2.1 Algebraic geometry over groups

We start with some basic theory of algebraic geometry over groups. For more background, see [1]. Let be a group generated by a finite set and the free group on . For , the expression is called a system of equations over , and a solution of in , is a -homomorphism such that (a -homomorphism is determined by ; the notation means that corresponds to a solution of ).

Denote the set of all solutions of in by , the algebraic set defined by . Define the Zariski topology on by taking algebraic sets as a pre-basis of closed sets. Note that the algebraic set uniquely corresponds to the normal subgroup in , called the radical of . Call the coordinate group of . Every solution of in can be described as a -homomorphism .

2.1.1 Quadratic equations

A system of equations is said to be (strictly) quadratic if each that appears in , appears at most (exactly) twice, where also counts as an appearance. There are four standard quadratic equations:

(1)
(2)
(3)
(4)

where and are non-trivial elements of . Note that for any strictly quadratic word , there is a -automorphism of that takes to a standard quadratic word. Also, there is a surface with boundary associated to each standard quadratic equation, specifically an orientable surface of genus with zero punctures for (1), an orientable surface of genus with punctures for (2), a non-orientable surface of genus and zero punctures for (3), and a non-orientable surface of genus with punctures for (4). For a standard quadratic equation , let be the Euler characteristic of the associated surface.

2.1.2 -NTQ groups

General systems of equations can exhibit some properties similar to those of quadratic systems. This can be seen when systems are in a particular quasi-quadratic form.

Definition 3.

A system of equations over a group generated by , is called triangular quasi-quadratic over or -TQ, if it can be partitioned into subsystems:

where:

  1. is a partition of

  2. for

  3. (where is a free group of finite rank) or a subgroup of this group that is a free product of , and conjugates of by some generators of .

  4. for and .

Furthermore, for each the subsystems must have one of the following forms:

  1. is quadratic in

  2. where is a maximal cyclic subgroup of

  3. is empty

The number is called the depth of the system.

Notice that it may be assumed that every subsystem of form (I) is actually a single quadratic equation in standard form. Also, it can be checked directly that . is called non-degenerate triangular quasi-quadratic over or -NTQ if it is -TQ and for every , the system has a solution in , and if is of form (II) the set generates a centralizer in . A regular -NTQ system is a -NTQ system in which each non-empty quadratic equation is in standard form, and either and the quadratic equation has a non-commutative solution in , or it is an equation of the form or .

Finally a group is called a (regular) -NTQ group if it is isomorphic to the coordinate group of a (regular) -NTQ system of equations. Note that in Sela’s work [26], -residually free towers provide an analagous structure to -NTQ groups, and in the work of Casals-Ruiz and Kazachkov [2], graph towers are, in a certain sense, higher dimensional analogues of NTQ systems for working with right angle Artin groups.

Suppose is a toral relatively hyperbolic group. Every -NTQ group is also toral relatively hyperbolic [18]. In certain circumstances, when the bottom level is clear from the context we will be talking (abusing the language) about groups, not specifying .

2.2 Graphs of groups

A graph of groups is a connected graph labelled with a group for each vertex , and a group with monomorphisms , for each edge ( and denote the initial and terminal vertices of respectively). Note that is considered to be a non-oriented graph, (i.e. there is an involution with for each ), so the monomorphisms of a graph of groups must also satisfy .

Let be a maximal subtree of . The fundamental group of with respect to is the group which is generated by with relations . Any choice of maximal subtree gives an isomorphic fundamental group of the graph of groups. A splitting of a group over some class of group is an isomorphism from to where each is in . Splittings are discussed in further detail in Section 4.

Graphs of groups are closely related to the Bass-Serre theory of groups acting without inversion on trees. We note here one significant consequence which will be used later.

Proposition 2.

(see [4], Theorem 3.7 in [24]) Given the fundamental group of a graph of groups, and . Then is the fundamental group of the induced graph of groups with vertex and edge groups given by intersections of with conjugates of vertex and edge groups of , respectively.

3 Iterated generalized doubles

We can now prove Theorem 1. We start with the following useful elementary example of a generalized double.

Lemma 1.

A free rank one extension of centralizer over , where , is a generalized double over .

Proof.

Given , let

is injective on and by definition a centralizer is maximal abelian in . ∎

With Proposition 1, this is enough to prove one direction of Theorem 1.

Proposition 3.

Let be a generalized double over a -limit group . Then is a -limit group.

Proof.

First consider the case where is an amalgamated product. Let be the map from to as in the definition of generalized double. Denote the images of under by , , and respectively. Let be the maximal abelian subgroup of containing . Consider the group . Since embeds into an iterated centralizer extension of , so does . Let ; clearly . We claim that .

This can be showed using normal forms. In particular, the maps and defined by composing the restrictions of to and respectively with the natural embedding of and into (and in the case of , composing with conjugation by the stable letter in between), are injective (only the conjugation need be checked, and this follows since is stable letter for ) by Theorem 1.6 of [24]. So define by for each word in in normal form (). Since if then and so , we have in normal form. Since every element of has normal form of , is an isomorphism. Then embeds into and so by Proposition 1, is a -limit group. The case for is similar, since (and using the fact that if images of don’t coincide then they must be conjugate by commutative transitivity) again embeds into . ∎

To prove the converse, we use the Bass-Serre tree.

Proposition 4.

Every -limit group can be constructed by iterated generalized doubles over .

                                       

                                                

                                                           

                                          

                                             

                          

               

                                                                      

                                                                                  

                                                                                      

                                                                                         

                                                                                                         

                                                                                                 

                                                                                   

                                                                          

                                                                                                        

                                                                                              

                                                                                                   

                                                                                                      

                                        

                                                                 

                  

                                                                     

                                              

                         

                                                      

                                                 

Figure 1: Bass-Serre tree
Proof.

Given a freely indecomposable -limit group, by Proposition 1, it may be embedded into an iterated extension of centralizers over . is an iterated generalized double over . We claim that every subgroup of is also an iterated generalized double.

We proceed by induction. Let for some -limit group and . Let . So is an HNN extension . Now by Proposition 2, acts (without inversions) on a tree with vertices given by the cosets (stabilizers are the groups ), and edges (corresponding to groups ), for each coset representative . In particular the initial vertex of the edge is and terminal vertex of is .

Every coset has a representative of the form where the are representatives for cosets of in , and the are non-zero except possibly and . Now for each edge , the terminal vertex is and the initial vertex is if , and if . See Figure 1 (vertex cosets are in red, edge cosets in blue). We want to show that the corresponding edge group in the quotient graph of groups is maximal abelian in one of the corresponding vertex groups. Now and .

Clearly , and since we have .
Finally, .

A conjugate of a centralizer is again a centralizer in the corresponding conjugate of the ambient group (this can be easily checked directly), and centralizers are maximal abelian. So intersecting a maximal abelian subgroup and it’s ambient group with preserves maximality of the abelian subgroup (again, this can be easily checked directly).

Also, clearly every conjugate is isomorphic to , and so each can be mapped monomorphically to . Since any free factor of a -limit group is a -limit group, the proposition is proved.

Theorem 1 follows (since free products of limit groups are also -limit groups).

4 Further background on splittings

In this section we describe properties of JSJ decompositions of groups. We refer to [23], [7], [8], and [13] for further background on other notions of JSJ decompositions for groups.

Recall that an abelian splitting of a group is an isomorphism to the fundamental group of a graph of groups with all abelian edge groups. It is often convenient to slightly abuse terminology and allow a splitting to refer to the graph of groups itself. An elementary splitting is one in which the graph of groups has exactly one edge. A splitting is reduced if the image of each edge group is a proper subgroup of the corresponding vertex group. A splitting of a group is essential if it is reduced, all edge groups are abelian and for each edge group , if for some , then .

A subgroup is said to be elliptic with respect to a given splitting, if it is conjugate to a subgroup of some vertex group of the splitting. If is not elliptic, then it is called hyperbolic with respect to the splitting.

There are certain elementary transformations of graphs of groups which preserve the fundamental group.

An unfolding of an elementary splitting is another splitting where is a proper subgroup of and . An unfolding of an elementary splitting is another splitting where the image of in is a proper subgroup of , and where is the stable letter of . We say a splitting is unfolded if there is no unfolding of any induced elementary splitting.

4.1 Classification of vertex groups

Vertex groups may be divided into three classes.

  1. Abelian vertex groups

  2. Quadratically hanging vertex groups. A vertex group is quadratically hanging (or QH), if it admits either of the presentations (i.e. it is the fundamental group of a surface with finitely many punctures):

    1. where , and if then

    2. where

    and furthermore:

    • for each edge group with , is conjugate to in for some

    • and for each there is some edge with and conjugate to in .

  3. Vertex groups which are non-abelian and non-QH are called rigid vertex groups.

Any subgroup which is a QH vertex group in some splitting of is called a QH subgroup of . A QH subgroup is a maximal quadratically hanging subgroup (or MQH subgroup), if given any elementary splitting of with edge group , either is elliptic in that splitting, or can be conjugated into and that elementary splitting is induced by splitting along the image of .

4.2 JSJ decompositions

There are various definitions of splittings which are called JSJ decompositions (after the topological notion of decomposing 3-manifolds along essential tori due to Jaco, Shalen, and Jacobsen) for various classes of groups. All of them are canonical in that they encode all other splittings (of a certain type) in some sense. We will use some particular qualifications for the canonical decompositions of -limit groups.

Definition 4.

Given a reduced unfolded abelian splitting of a freely indecomposable -limit group , call a JSJ decomposition of , if the following properties are satisfied:

  1. Every MQH subgroup of is conjugate to a vertex group of the , (so every QH subgroup of can be conjugated into a vertex group of the JSJ), and every vertex group of the which is not conjugate to a MQH subgroup of L is elliptic in any abelian subgroup of .

  2. Any elementary abelian splitting of which is hyperbolic in another elementary abelian splitting, can be obtained from by the splitting of an MQH subgroup, which is induced by cutting the corresponding surface along an essential simple closed curve, and collapsing all other edges.

  3. Any elementary abelian splitting of which is elliptic with respect to every other elementary elementary abelian splitting of , can be obtained from by a sequence of collapsings, foldings, and conjugations.

  4. Any two reduced unfolded abelian splittings of satisfying the above three properties can be obtained from one another by a sequence of slidings, conjugations, and modification of boundary monomorphisms by conjugations.

  5. All non-cyclic abelian subgroups of are elliptic in .

In [27] (Theorem 1.10), every freely indecomposable -limit group is shown to have such a JSJ decomposition.

5 Hom-diagrams and -NTQ groups

We present here some constructions from [18] and [14], along with some similar background from [17] in the free group case. Building on those foundations, we define and prove the existence of canonical Hom-diagrams and associated canonical NTQ groups.

Unless otherwise stated, for systems with coefficients in , all homomorphisms in Hom-diagrams for the system are assumed to be -homomorphisms.

5.1 Canonical automorphisms

Consider an elementary abelian splitting of a group . If we have , for define an automorphism such that for and for . If instead we have then for we define such that for and .

Call the a Dehn twist obtained from the corresponding elementary abelian splitting of . If is an -group, where is a subgroup of one of the factors or , then Dehn twists that fix elements of the group are called canonical Dehn twists. Similarly, one can define canonical Dehn twists with respect to an arbitrary fixed subgroup of . Let [resp. ] be the set of all abelian splittings of [resp. the set of all abelian splittings such that is elliptic].

Definition 5.

Let [] be an abelian splitting of a group and be either a QH or an abelian vertex of . Then an automorphism is called a canonical automorphism corresponding to the vertex if satisfies the following conditions:

  1. fixes (up to conjugation) element-wise all vertex group in , other than . If then also fixes each element of . Note that also fixes (up to conjugation) all the edge groups.

  2. If is a QH vertex in , then is a Dehn twist (canonical Dehn twist) corresponding to some essential -splitting of along a cyclic subgroup of .

  3. If is an abelian subgroup then acts as an automorphism on which fixes all the edge subgroups of .

Definition 6.

Let be an edge in an abelian splitting of a group . Let . Call a canonical automorphism corresponding to the edge if is a Dehn twist of with respect to the elementary splitting of along the edge , induced from the splitting . If , then must fix element wise.

Definition 7.

The group of canonical automorphisms of a closed surface group with respect to a trivial splitting (ie. is the only vertex group) is (the mapping class group of the surface).

Otherwise, the group of canonical automorphisms of a freely indecomposable group with respect to an abelian splitting is the subgroup , generated by all canonical automorphisms of corresponding to all edges, all vertices, and all abelian vertices of . If is not a then include conjugation as well. The group of canonical automorphisms of a free product is the direct product of the canonical automorphism groups of factors.

5.2 Solution tree

In this section we describe how to encode solutions to equations over using a -diagram. We begin by describing a -diagram for generalized equations [16] over free groups. There is an algorithm described in [16] that, given a generalized equation over the free group , constructs a diagram, which encodes the solutions of . Let be the coordinate group of considered as a system of equations over . Specifically, the algorithm constructs a directed, finite, rooted tree that has the following properties:

  1. Each vertex of is labelled by a pair , where is an -quotient of and the subgroup of canonical automorphisms in corresponding to a splitting of as a fundamental group of a grapg of groups, that we find from the Elimination process of . The root is labelled by and every leaf is labelled by where is some finite set (called free variables). Each , except possibly , is fully residually .

  2. Every (directed) edge is labelled by a proper surjective
    -homomorphism .

  3. For every , that is a solution of (that must be non-cancellable in ) there is a path , where is a leaf labelled by , elements , and a -homomorphism such that

    (5)

    Considering all such -homomorphisms , the family of the above sequences of homomorphisms is called the fundamental sequence over corresponding to .

    Considering all such -homomorphisms that produce solutions of the family of the solutions of factoring through the above fundamental sequence is called the fundamental sequence for the generalized equation over corresponding to .

  4. The splitting of each fully residually free group is its Grushko decomposition followed by the abelian splittings of the factors that are found by the Elimination process. If is such a factor, then the splitting is not necessarily the JSJ decomposition of , but it is maximal in a sense that it encodes all elementary abelian splittings of that can be found by the Elimination process, and has maximal QH and abelian vertex groups that can be found by the elimination process for .

Notice that not all the homomorphisms that factor through (5) are solutions of , but they all are homomorphisms from to .

In [19] it is shown, given a finite system of equations over , how to construct a similar diagram encoding . We briefly review the construction here.

First construct the generalized equations over from the system over by taking canonical representatives in for elements of (which exists by Theorem 4.5 of [22]), see Lemma 3 in [19] for more details on the construction of these systems. Construct the free group solution tree as described above, for each generalized equation . We form a new, larger tree by taking a root vertex labelled by , attaching it to the root vertex of each by an edge labelled by , where is the homomorphism induced by canonical representatives. For each leaf of , labelled by , build a new vertex labelled by and an edge labelled by the epimorphism which is induced from by acting as the identity on . We call a path from the root to a leaf a branch of .

Now associate to each branch the set consisting of all homomorphisms of the form

where

is a solution of the generalized equation, and where and . Since is in bijective correspondence with the set of functions , all elements of can be effectively constructed.

Notice, that is not a fundamental sequence over , it is a fundamental sequence of solutions of a generalized equation over , followed by a homomorphism .

5.3 -NTQ to -NTQ reworking process

In this section we recall the construction that first was used in [14] and was developed in [19] that for each constructs a strict fundamental sequence over .

Definition 8.

A fundamental sequence or a fundamental set of homomorphisms over corresponding to the diagram

where

  1. are freely indecomposable groups isomorphic to subgroups of , and are -limit groups.

  2. are fixed proper epimorphisms, is an epimorphism but may not be proper.

  3. The homomorphisms in this sequence are compositions , where is a canonical automorphism of corresponding to a Grushko decomposition of followed by some abelian decompositions of the freely indecomposable factors where all non-cyclic abelian subgroups are elliptic. Canonical automorphisms are identical on the free factor of this Grushko decomposition.

  4. is a homomorphism that maps each monomorphically into a conjugate of a fixed subgroup of (and for each it is a fixed monomorphism followed by a conjugation) and maps into .

Definition 9.
  1. A fundamental sequence defined above is called strict if it has the following properties:

  2. The image of each non-abelian vertex group of under is non-abelian.

  3. For each , is injective on rigid subgroups, edge groups, and subgroups generated by the images of edge groups in abelian vertex groups in .

  4. For each , if is a rigid subgroup in and , , the abelian vertex groups in connected to by edge groups with the maps , then is injective on the subgroup which we will call the envelop of .

  5. The images of different factors in the Grushko decomposition of under are different factors in the free decomposition of .

The construction of the tree of strict fundamental sequences over , relies on a ‘reworking process’, which converts -NTQ systems into appropriate -NTQ systems. In , each strict fundamental sequence corresponds to a -NTQ group (with depth of -NTQ system equal to the length of the branch ) into which maps by a homomorphism . We briefly summarize the construction and important properties of here.

Each branch of has a corresponding -NTQ system that comes from a generalized equation for canonical representatives. Considering the -NTQ system as a system of equations over does give groups through which all homomorphisms from to factor, but the systems may not be in -NTQ form (as a result of the map trivializing some vertex or edge groups). The reworking process fixes those degenerate portions, while maintaining the factoring of all homomorphisms . For each -NTQ system , the reworking process constructs a new -NTQ system of equations over (see [14], section 3.3 and [19]. The following proposition is proved in [19], and it follows from the construction in [14].

Proposition 5.

Each branch of has the following form

where are freely indecomposable groups isomorphic to freely indecomposable factors in the Grushko decomposition of ; are NTQ -limit groups; is a fixed homomorphism; and are fixed proper epimorphisms which are retractions onto for .

There is a strict fundamental sequence assigned to each branch. The homomorphisms in this sequence are compositions where is a canonical automorphism corresponding to , and is a homomorphism that maps each monomorphically onto a conjugate of the corresponding subgroup of (and for each it is a fixed monomorphism followed by a conjugation), and maps into .

Every homomorphism from to factors through one of the fundamental sequences corresponding to the branches of . Factors in the Grushko decomposition of are mapped into different factors in the Grushko decomposition of .

Each branch of can be effectively constructed.

5.4 Quasi-convex closure

Let be a torsion-free non-elementary hyperbolic group, and a subgroup of given by generators. We describe a certain procedure that constructs a group , such that is quasi-convex in and every abelian splitting of induces a splitting of .

If is hyperbolic in every cyclic splitting of , then let .

If is elliptic in a cyclic splitting of , we consider instead of a vertex group containing a conjugate of . Such a subgroup is quasi-convex and so hyperbolic, as it is a vertex group, and the quasi-convexity constants can be found effectively. If is elliptic in a cyclic splitting of , continue this process until is not elliptic in any decomposition of the corresponding vertex group . So let be the corresponding conjugate of such that . This subgroup is a quasi-convex closure of . is relatively quasi-convex, so therefore hyperbolic. Note that hierarchical accessibility for hyperbolic groups was proved by Louder and Touikan in [20].

5.5 A complete set of canonical NTQ groups

In Section 5.2 we constructed a tree of strict fundamental sequences (or a -diagram) encoding all solutions of a finite system of equations over using the tree of fundamental sequences for “covering” systems of equations over . A canonical -diagram is a tree of “canonical” strict fundamental sequences (in [17], Section 7.6, a similar tree was called the (augmented) canonical embedding tree ) which satisfy certain properties.

We define a canonical -diagram iteratively. All the homomorphisms from into factor through a finite number of maximal -limit quotients of . We continue with each maximal -limit quotient (that we denote by ) in parallel. We factor into freely indecomposable factors and the free factor . Homomorphisms from each to in a canonical -diagram are given by compositions of canonical automorphisms corresponding to a JSJ decomposition of with one of the finite number of fixed epimorphisms from onto its maximal standard (proper) -limit quotients and with homomorphisms from these proper -limit quotients to that are constructed iteratively. Namely, we factor again each into a free product of freely indecomposable factors and the free factor, etc. Since each proper sequence of - limit quotients is finite, the construction of -diagram terminates after finitely many steps. It terminates with either a fixed embedding of a -limit group in the diagram into followed by a conjugation in and substitutions of the terminal free groups that appear in the diagram, into .

The canonical -diagram is a tree. For each branch of this tree

(6)

there is a strict fundamental sequence assigned. Here

1) are freely indecomposable groups isomorphic to subgroups of ,

2) are fixed proper epimorphisms, may not be proper.

3) The homomorphisms in this sequence are compositions , where is a canonical automorphism of corresponding to a Grushko decomposition of followed by the JSJ decompositions of the freely indecomposable factors,

4) is a homomorphism that maps each monomorphically onto a conjugate of a fixed subgroup of (and for each it is a fixed monomorphism followed by a conjugation) and maps into .

The existence of such a diagram can be obtained from [27] and [14]. Indeed, the difference between the diagram that we described and the diagram constructed in [27] is that in [27] the homomorphism must be proper and homomorphism can be an arbitrary embedding of groups into . In our diagram, if has a non-trivial abelian splitting, then appears as a factor in the Grushko decomposition of , and in the leaves of the diagram we have a fixed (up to conjugacy) monomorphism of into a conjugate of . Notice that (non-canonical) fundamental sequences from Proposition 5 terminate with fixed conjugacy classes of monomorphisms of into , and we are going to use this fact to prove that canonical fundamental sequences satisfy the same property. Notice also that by [10], Lemma 7.2, has infinitely many conjugacy classes of monomorphisms into if and only if has a non-trivial cyclic splitting. We define two embeddings of into to be equivalent if one is obtained from the other by pre-composing with an automorphism of generated by Dehn twists corresponding to cyclic splittings of and post-composing with a conjugation.

We claim that there is only a finite number of non-equivalent embeddings of each that appear in the diagram, into . We will follow [10]. Suppose to the contrary, that there is an infinite family of non-equivalent monomorphisms. Let be a fixed system of generators of . Consider a sequence of non-equivalent minimal monomorphisms (corresponding to minimal in the equivalence class sum of lengths of the images of ). After passing to a subsequence, the sequence can be taken to be stable, it converges to a stable isometric action of on a real tree. Since one can use Rips machine and the shortening argument to show that has an essential cyclic splitting such that some of the monomorphisms in the sequence can be shorten by pre-composing with Dehn’s twists corresponding to this splitting. Since the sequence consists of minimal monomorphisms, we have a contradiction with the assumption about infinite number of equivalence classes.

Construction of the NTQ group for a strict fundamental sequence. We can assume (combining foldings and slidings) that all JSJ decompositions have the property that each vertex with non-cyclic abelian vertex group that is connected to a rigid subgroup is connected to only one rigid subgroup.

We assign an NTQ system to this branch as follows. First, replace each subgroup that is not a hyperbolic closed surface group by its quasi-convex closure , then is replaced by