Compatibility JSJ decomposition of graphs of free abelian groups

Compatibility JSJ decomposition of graphs of free abelian groups

Benjamin Beeker
July 15, 2019
Abstract

A group is a group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We describe the compatibility JSJ decomposition over abelian groups. We prove that in general this decomposition is not algorithmically computable.

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1 Introduction

The theory of JSJ splittings starts with the work of Jaco-Shalen and Johansson on orientable irreducible closed 3-manifolds giving a canonical family of 2-dimensional tori. Kropholler first introduced the notion into group theory giving a JSJ decomposition for some Poincaré duality groups [Krop]. Then Sela gave a construction for torsion-free hyperbolic groups [Sela]. This notion has been more generally developed by Rips and Sela [RiSe], Dunwoody and Sageev [DunSa2], Fujiwara and Papasoglu [FuPa] for various classes of groups.

However, in general, given a group and a class of subgroups, there is not a unique JSJ splitting of over these subgroups. Guirardel and Levitt define in [Gl3a] the JSJ deformation spacse which generalize the previous notions. They also introduce in [Gl3b] a new object called the compatibility JSJ splitting of over these subgroups. Unlike the usual JSJ splittings, the compatibility JSJ splitting is unique, and so is invariant under is automorphisms. However, it is often harder to construct it, the purpose of this article is to give some positive and negative results about the constructibility of this object.

In this paper we focus on the construction of the compatibility JSJ tree over abelian groups for the following class of groups. Let be a group acting on a simplicial tree with free abelian vertex (and edge) stabilizers. We call such a group a Generalized Baumslag-Solitar group of variable rank, or group, and such a tree a (-)tree. We call the collection of groups which admits an action on a simplicial tree with vertex stabilizers with a fixed integer. This two classes of groups generalize the one of groups (or groups) introduced by Forester in [for03] as examples of groups for which we do not have a canonical (usual) JSJ tree.

Recall that an element of is elliptic if it fixes a vertex of , and hyperbolic otherwise. Given a hyperbolic element , it acts by translation on a line of called the axis of or its characteristic space. The characteristic space of an elliptic element is the set of its fixed points. A subgroup of is elliptic, if it is included in the stabilizer of a vertex. A subgroup of is universally elliptic if it is elliptic in every -tree. With no restriction on the -trees, almost any tree can be universally elliptic, we thus consider -trees whose edge groups are included in a set of subgroups of . We then talk about -trees over , and a subgroup is -universally elliptic if it is elliptic in every -tree over .

Given two -trees and , the tree refines if may be obtained from by equivariantly collapsing edges. The tree dominates if every elliptic group of is elliptic in . In particular, if refines then it also dominates it. We say that and are compatible if there exists a -tree which refines both and . A -tree is -universally compatible if it is compatible with every -tree (over ).

A -tree over is a JSJ tree over if it is -universally elliptic and dominates every universally elliptic -tree over . A -tree over is a compatibility JSJ tree over if it is -universally compatible and dominates every universally compatible -tree over .

The present paper splits into two parts. We first describe the compatibility JSJ tree over groups of the groups. In the second part, we describe the compatibility JSJ tree over abelian groups of the groups.

Let be a group. We propose to describe the compatibility JSJ tree over groups in the following sense. Starting from a JSJ tree of over groups (except for some degenerated cases, any tree is a JSJ tree [Moi1]), we explicit a set of edges and a set of vertices such that the compatibility JSJ tree is obtained by expanding (in a precise way) these vertices and collapsing these edges in .

The compatibility relation is very restrictive. For example, two -trees with an isomorphic quotient graph of groups may be not compatible. Take the group . Its JSJ deformation space contains infinitely many reduced trees and any reduced JSJ tree of this deformation space have same isomorphic quotient graphs of groups. However, each of these reduced JSJ tree is compatible with exactly two others reduced JSJ trees.

Here are the two main examples of edges to collapse. The labelled graphs represent groups, and must be understood in the following manner: all vertices and edges carry the infinite cyclic group , the number at the end of each edge indicates that the injection of the edge group into the group of its endpoint is .

In Figure 1, the two graphs of groups have isomorphic fundamental group, and are related by a slide along the edge denoted by . In this case, the edge is called slippery (see Section 2 for the complete definition) and is collapsed in the compatibility JSJ. In fact, after collapsing the edge , we exactly obtain a compatibility JSJ over group .

Figure 1: The two graphs of groups are related by a slide.

In Figure 2, the two graphs have isomorphic fundamental groups, and are related by a deformation called induction along (see definition in Section 2). In this case, the edge is called strictly ascending. Collapsing , we obtain a compatibility JSJ.

Figure 2: The two graphs of groups are related by an induction.

Given a usual abelian JSJ tree of a group, we obtain a universally compatible tree by collapsing four types of edges. The definitions of these edges is technical and described in Section 2. .

Proposition 1.1.

Let be a group. Let be a reduced abelian JSJ tree of . Let be the set of edges containing the vanishing edges of , the potentially strictly ascending edges of , the non-ascending slippery edges of , the toric -slippery edges of .

  1. The tree obtained from by collapsing the edges of is compatible with every -tree over groups.

  2. Let be a collapse of in which an edge of is not collapsed. Then is not compatible with every -tree over groups.

However the tree is not always the compatibility JSJ tree: some vertices could have to be expanded. Roughly speaking, some vertices could act as dead end: no edge ”arriving” at this vertex by a slide may continue to slide further. These vertices must be expanded in the compatibility JSJ tree in a precise way called a blow up (see Section 2). The vertex of Figure 3 is an example. The top edge may slide along or along , but after performing one of these slides, no new slide is allowed (except the converse one). The compatibility JSJ is obtained by collapsing and (which are slippery) in the graph of groups on the right. The complete description of dead ends is given in Section 2.

Figure 3: The edges and are slippery and is a dead end.
Theorem 1.2.

Let be a group. Let be a reduced abelian JSJ tree of . Call the -tree obtained from by blowing up up the dead ends and collapsing the vanishing edges of , the p.s.a. edges of , the non-ascending slippery edges of , the toric -slippery edges of .

Then is a compatibility JSJ tree over groups.

This construction is algorithmic whenever no vertex group is conjugated to one of its proper subgroup. In the general case, the decidability of the construction is unknown.

The groups also admit splittings over groups. If for the usual JSJ tree, this does not have any incidence - the abelian JSJ tree and the JSJ tree over are isomorphic -, it has one for the compatibility JSJ tree. For example, applying Theorem 1.2, we may easily see that the compatibility JSJ tree over groups of is the one associated to the HNN extension, but this tree is not compatible with the tree associated to the amalgamated product

The incompatibility comes from the fact that the element is hyperbolic in the first splitting but stabilizes an edge in the second. Such an element is said to be potentially bi-elliptic.

In the case of groups, to obtain universally compatible tree over abelian groups, it suffices to collapse in any JSJ tree the four types of edges described previously, and every edge in the axis of a potentially bi-elliptic element. Again, the obtained tree is not always the abelian compatibility JSJ tree, even if we blow up the dead ends. Some other edges have to be expanded:

In some very specific cases (see description in Section LABEL:sectioninert), the axes of potentially bi-elliptic elements must be separated from the rest of the tree, and then the axes must be collapsed (see Figure 4).

Figure 4: The axis of a potentially bi-elliptic element must be ”taken away” from the rest of the tree.

These separations are called expansions (of inert edges). We obtain the following theorem.

Theorem 1.3.

Let be a group. Let be a reduced abelian JSJ tree of . Call the set of non-ascending slippery, potentially strictly ascending and toric -slippery edges.

We define as the -tree obtained by expanding inert edges, blowing up dead ends, then collapsing every edge of and the axes of potentially bi-elliptic elements.

Then is an abelian compatibility JSJ tree of .

This gives a description of the abelian compatibility JSJ tree. However this construction is not and cannot be algorithmic.

Theorem 1.4.

There is no algorithm that constructs the abelian compatibility JSJ tree of groups.

This inconstructibility follows from the fact that in trees, we may not dectect whether an edge is slippery or not. We describe this problem in the last section.

2 Preliminaries

Let be a finitely generated group. A -tree is a simplicial tree equipped with an action of that we suppose without inversion (the stabilizer of an edge is included in the stabilizer of its endpoints) and minimal. Given an (oriented) edge , the opposite edge is denoted by . We denote by the stabilizer of . We have . The stabilizer of a vertex is denoted by. The orbit of a vertex and an edge is denoted by Gothic font and . Similarly the opposite orbit of is denoted by . The valency of a vertex (or an orbit of vertices ) is the number of orbits of edges with initial vertex in .

An edge is reduced if its endpoints are in the same orbit under the action of or its stabilizer is strictly included in both stabilizers of its endpoints. An orbit of edges is reduced if one (or equivalently every) representative is reduced. A tree is reduced if all its edges are reduced. The deformation space of is the set of -trees having the same set of elliptic subgroups as . The reduced deformation space of is the subset of reduced trees of .

To collapse an orbit of edges consists in collapsing every connected component of edges in the orbit to a point. This operation produces a new -tree. The converse of collapsing is called expanding. Given a -tree and an orbit of edges , the tree obtained by collapsing is in the deformation space of if and only if is not reduced. We may thus construct a reduced -tree in the deformation space of by collapsing one by one orbits of non-reduced edges until the tree is reduced (we assume here that number of orbits of edges is finite). Note that if we collapse at the same time all non reduced edges, the obtained tree may be in a different deformation space.

Whitehead moves for -trees

In [ClayFor09], Clay and Forester describe three deformation moves on -trees such that any two reduced -trees of a given deformation space are related by a finite sequence of these three moves, and every intermediate tree in this sequence is reduced.

The first deformation move is the slide of an edge along an edge : if two edges and are such that

  • is not in the orbit of or ,

  • the terminal vertex of is equal to the initial vertex of ,

  • the stabilizer of is included in the stabilizer of ,

then may slide along . Call the terminal vertex of . The new -tree is obtained by changing the terminal vertex of every vertex in the orbit of from to (see Figure 5 for the changes on the associated graph of groups). The case were is not excluded. This move does not change the stabilizers of vertices and edges. This move is fully determined by the data of and . Note that the converse move is also a slide.

The slide of along will be denoted by .

Figure 5: Slide of along

The second deformation move is the induction. Let be an edge with initial and terminal vertices and and a subgroup of such that

  • and are in the same orbit,

  • the stabilizers and are equal,

  • the group contains .

Then we may perform an induction on with group . We first add an edge with terminal vertex , a new initial vertex which is also the new terminal vertex of and with . Every edge of another orbit with initial vertex keeps as initial vertex. We then collapse (see Figure 6). This description is given around the edge but must be made equivariantly.

This move does not change the underlying graph, but the edge and the vertex have been replaced by an edge and a vertex with a distinct stabilizer . It is fully determined by the data of and the group .

The induction on the edge with group will be denoted or if we do not precise the group.

Figure 6: Induction on

If we have an inclusion of groups , we may first perform an induction on and then an induction on . The composition of these two inductions is equal to the induction on : (where is the edge appearing in the induction on ).

The induction with has no effect on the -tree. We call this move a trivial induction. At the opposite we may take . In this case performing the induction is the same as sliding along every edge (not in the orbit of or ) with terminal vertex .

The converse move of an induction with any group is just an induction with group and slides along . Indeed from the first remark, we have . And by the second remark, the move is the same as a finite sequence of slides.

By extension, performing the converse of an induction with group consists in performing an induction with group and then sliding every edge around along . This move is possible whenever, for every edge with initial vertex not in the orbit of , the group contains .

The third move is the -move. We first describe the -move. Suppose that is an edge with initial and terminal vertices and , and that is an edge with terminal vertex and initial vertex not in , such that

  • and are in the same orbit,

  • the stabilizers and are equal,

  • for every edge with initial vertex not in the orbit of , the stabilizer of contains .

We may then perform an -move on by first performing the converse of an induction on with group and then collapsing which is now non reduced (see Figure 7 for the changes on the underlying graph of groups). We will say that (and ) is a vanishing vertex and (and ) a vanishing edge.

For some technical reasons, the -move is not exactly the same as the one described in [ClayFor09]: we allow here the vertex to be of valence more than .

However the definition of an -move remains the same: Let be an edge with initial and terminal vertices and and such that

  • (so and belong to the same orbit of edges),

  • we have .

Call the set of the edges with initial vertex not in the orbit of or . Performing a -move consists in performing the two following deformations. First expand in an edge with initial and terminal vertex and , with and and such that the initial vertex of and the new terminal vertex of are and the initial vertices of the edges of are . Then perform an induction on with group .

The -move is determined by the data of the orbits of and , however the -move is determined by the data of two edges and in the same orbit such that the terminal vertex of is the initial edge of , and .

A -move on is denoted or .

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Figure 7: -move

A move is admissible if we may perform it, if the obtained tree it is reduced, and if it is not a trivial induction. A sequence of moves is admissible if after performing the first moves of the sequence the th is admissible.

If is an admissible sequence of moves on the -tree , the tree obtained by applying is denoted by .

The number of orbits of vertices and edges in the -trees of a reduced deformation space is not fixed. However, the only way to make a vertex or an edge disappear is to perform an -move. Thus if two -trees and are related by a single move, we may identify the orbits of vertices and edges of with the ones of , except for the one that vanish. Moreover if the move is not an induction or an -move, the edge and vertex stabilizers does not change, we may then identify not only the orbits of edges but each edge. To be more precise, let be a -tree, an edge of and m an admissible move on such that

  • the move m is not an induction on the orbit ,

  • the move m is not an -move on the orbit ,

  • if m is a -move, its vanishing orbit of edges is not .

We may then identify with an edge of and this identification is equivariant.

By extension, an orbit of vertices or edges is vanishing if we may perform an admissible sequence of moves ending by a -move in which it vanishes. A sequence of moves preserves an orbit if no move of the sequence is an -move in which vanishes.

As two reduced -trees in the same reduced deformation space are related by a finite admissible sequence of deformation moves, given two reduced -trees and in the same deformation space, we may identify any non-vanishing orbit of vertices or edges of to one of .

Edges properties

Let be a reduced -tree. An orbit of edges of is slippery if there exists an admissible sequence of moves on , preserving and ending by a slide along . The orbit is -slippery if there exist an admissible sequence of moves on , preserving and two distinct orbits of edges and of such that the slides and are simultaneously admissible in . Note that is allowed.

An edge of is ascending if its endpoints are in the same orbit, and if its stabilizer is equal to the stabilizer of its initial vertex. It is strictly ascending if moreover its stabilizer is strictly contained in the stabilizer of its terminal vertex. An edge is toric if its stabilizer is equal to the stabilizer of both its endpoints. The opposite edge of a (strictly) ascending edge is (strictly) descending.

An edge is pre-ascending in if its initial and terminal vertices and are in the same orbit and if there exists such that and . The opposite edge of a pre-ascending edge is said to be pre-descending. The edge is potentially strictly ascending (or p.s.a.) if is pre-ascending after a finite admissible sequence of moves preserving . Note that a strictly ascending edge is also pre-ascending. If an edge is such that or is p.s.a. then is said to be p.s.a.d..

An orbit of edges is slippery, -slippery, pre-ascending, pre-descending, (potentially) strictly ascending (or descending) or toric if one of its representative is.

The inductions and -moves may only be performed on strictly ascending (or descending) edges. An -move may only be performed on the pre-ascending edges which are not ascending. Note that an -move changes a pre-ascending edge into a strictly ascending edge.

Figure 8 represent a Generalized Baumslag-Solitar group and is constructed in the following way: all vertices and edges carry the infinite cyclic group , the number at the end of each edge indicate that the injection of the edge group into the group of its endpoint is .

In such a representation, pre-ascending edges and admissible slides may be seen as divisibility relation. For example in Figure 8, the edge is pre-descending, the edges and are slippery since may first slide along then along , and is p.s.a. since it is pre-ascending after sliding along and .

Figure 8:

Expansion

Let be a -tree, and a vertex of . Take a set of edges with pairwise distinct orbits and initial vertex , and a subgroup of containing for all . Then the expansion of of group and set is the tree obtained from by expanding in an edge of initial and terminal vertex and , such that , , the edges of have initial vertex , and every edge of initial vertex in and not in the orbit of any edge of has initial vertex (see Figure 9 for the changes in the graph of groups). This construction produces a non-reduced tree.

Figure 9: Expansion of of set .

Dead ends

Let be a reduced -tree. Let be a non-vanishing vertex and an edge with initial vertex . The vertex is a dead end with wall (or ) if

  • ,

  • the edge is slippery or p.s.a.d.,

  • for every edge with initial vertex not in the orbit of , we have the equality ,

  • there exists an edge not in the orbit of , with initial vertex , with , such that for all edges not in the orbit of with initial vertex there exists such that ,

    (we may notice that automatically and do not slide one along the other)

and if for every -tree in the reduced deformation space of and any representative of in , there exists an edge with initial edge which has the listed properties.

The orbit of is a dead end if is a dead end.

If there exists two edges and with distinct orbits such that is a dead end with wall and with wall in , then there are exactly two orbits of edges with initial vertex . However given another -tree in the reduced deformation space of , the orbits of the walls of in and may differ.

For example, Figure 10 represents the two reduced graphs of groups of the deformation space of a decomposition. We may reach the right graph from the left one by a slide of along .

The vertex is a dead end. In the graph on the left, is the wall and is the edge orbit as in the fourth point. In the one on the right, is the wall and plays the role of . The edge orbit may be non-unique since another edge may have the same stabilizer as the one of .

Figure 10: Example of dead end vertex.

To blow up this dead end vertex in consists in expanding with group and set (note that in this case the expansion does not depends of the choice of ). Figure 11 represents the blow up of in the left graph of Figure 10. Performing a blow-up does not change the deformation space. If the dead end has valence , we may choose equivalently one of the two edges to be the wall. However the blow up does not depend on this choice.

Figure 11: Blow up of a dead end vertex

3 Compatibility JSJ tree of groups over subgroups

3.1 Construction

Given a group and any deformation space of , we may canonically construct a -tree associated to , as follows.

Definition 3.1.

Let be a reduced -tree in the deformation space . Define as the -tree obtained from by blowing up the dead ends and collapsing the following edges:

  • the vanishing edges of

  • the p.s.a. edges of ,

  • the non-ascending slippery edges of ,

  • the toric -slippery edges of .

The non-vanishing edge and vertice orbits of a reduced -tree may be identified with another orbit in any -tree in the same reduced deformation space. Being a dead end vertex, being p.s.a., being slippery, and being -slippery does not depend on the chosen tree in the deformation space. But a toric edge may be changed in a non-ascending edge and vice-versa.

Note that as an edge is either non-ascending, strictly ascending (or descending) or toric, all -slippery edges are collapsed in .

Proposition 3.2.

The tree does not depend on the choice of the reduced tree in taken for the construction.

Proof.

By Clay-Forester [ClayFor09, Corollaire 1.2], two reduced -trees of are related by a finite sequence of deformation moves. We just have to show that we obtain the same tree starting from two trees and which differ by a single move.

For each move we proceed in three steps. We first prove that the set of orbits we have to collapse is the same before or after performing the move. We then prove that the move commutes with the collapses. Finally we prove that the move commutes with the blow-ups.

For the first step as p.s.a. edges are collapsed, we have to prove that if a non-ascending slippery edge is turned into a toric edge, then it is -slippery.

  1. Assume that and differ by a slide of an edge along another edge .

    Call the initial vertex of . The only edge which may possibly become toric is . If is non-ascending slippery, then as is reduced we have , in , thus also in . Then is not toric in . Thus no non-ascending slippery edge may be turned into a toric edge. Thus the edges in and that must be collapsed are the same.

    For the second step, it suffices to to check that either is collapsed in or .

    As slides along , the edge is slippery. If is not collapsed in , then must be toric and not -slippery. Then in the associated graph of group is a loop and is the only edge adjacent to (see Figure 12).This case is exactly the rigidity case describe in [Levitt05, Theorem 1, case 3], that is, the slide leaves unchanged.

    Figure 12: The only orbits of edges adjacent to are , and .

    It remains to show that any blow-up of a dead end vertex commutes with the slide. The only non-trivial cases is when an endpoint of is a dead end. We may assume that the initial vertex of is a dead end. As slides along , neither nor may be a wall. Thus if we blow up , the edges and will be on the same side of the new edge. Hence this is equivalent to blow up a vertex then slide along or slide along and then blow up a vertex.

  2. Assume that and differ by an induction on an orbit of edges .

    Call the orbit of the endpoints of .

    Here (or ) is strictly ascending in both trees. Assume that an edge was non-ascending slippery and becomes toric. Then the endpoints of are in . And as is toric in , the orbit of edges and may slide along in . Then is -slippery. Thus the edges in and that must be collapsed are the same.

    As is strictly ascending in both trees, then it is collapsed in both and , and the trees obtained are the same.. Moreover as is ascending the vertex orbit is not a dead end, thus blowing up dead end vertices and performing the induction commute.

  3. Assume that and differ by an -move of an orbit of edges with collapse of an orbit .

    When we perform the -move on the orbit of edges , if an orbit of edges becomes toric (or stops beeing toric), then , , ans have same terminal vertex orbit, thus and may slide along thus is two slippery.

    In , the orbit is strictly ascending and is vanishing, thus both and are collapsed in . In , the orbit is already collapsed and is pre-ascending, thus collapsed in . Thus the collapses commute with the move.

    It remains to show that the move commutes with the blow up of dead ends. Before performing the -move the vertex of is not a dead end, since is ascending. However the terminal orbit of vertices of (which is also the vertex of after performing the -move) may be a dead end. This is the only non-trivial case of commutativity. Assume is a dead end. Call a wall in . If , then remains the wall in and it is easy to see that the -move and the blow up commute.

    Two distinct orbits may play the role of the wall if and only if the orbit of vertices is of valence . Assume that is the unique wall of , then is of valence at least . Let be as in the definition of the dead end and another orbit of edges with initial vertex in . Take a representative of and , and representatives of , and with initial vertex . By assumption is maximal for inclusion and there exists in the orbit of with initial vertex such that . But in , as is pre-ascending, thus there exists two edges and in the orbit of such that the initial vertex of and the terminal vertex of are and and we still have with and maximal for the inclusion among groups of edges adjacent to . There is no wall to , this is a contradiction we the fact that is a dead end.

We obtain the same tree starting from two reduced tree related by a move. Thus the tree only depends from the deformation space . ∎

We denote by the -tree associated to Guirardel-Levitt’s JSJ deformation space of .

Lemma 3.3.

Let be a JSJ tree. Let be a reduced Guirardel-Levitt’s JSJ tree refined by . Let be an edge of with initial vertex and let be a subgroup of containing such that for every edge with initial vertex we have . Denote by the tree obtained from by performing an expansion of with group and set . Then is compatible with .

Proof.

Let be the lift of in and a lift of such that . Such a exists since and are both JSJ trees. Call the path between the initial vertex of and . Call the vertex of the closest of such that .

If , then the -tree obtained by the expansion of with group and set obviously refines .

Otherwise, denote by the last edge of . We have . Moreover as for every edge of initial vertex in , we have , we obtain . Hence for the lift of in , the edges and are in distinct components of . Let be the tree obtained by performing an expansion of with group and set . Then refines . ∎

Corollary 3.4.

Let be a JSJ tree. The -tree is compatible with . Moreover there exists a common refining tree which is a JSJ tree.

Proof.

For an edge of call the tree obtained by collapsing every edge of except .

By the point of [Gl3b, Proposition 3.22], we have to show that is compatible with for every edge of .

Call a reduced JSJ tree refined by . By construction, refines every tree with an edge of not obtained by a blow-up. Thus also refines all these trees.

If comes from a blow-up, then we may apply Lemma 3.3, taking the JSJ tree, the reduced JSJ tree, the dead end from comes from, the wall of and . We obtain that is compatible with .

If follows that is compatible with . ∎

3.2 Universal compatibility

Proposition 3.5.

Let be a group that admits a JSJ tree over a set of subgroup and such that the edge groups of every minimal -tree over are elliptic in the JSJ deformation space of . Then is universally compatible.

Proof.

From Lemma 5.3 of [Gl3a], every -tree is refined by a JSJ tree. Thus by Corollary 3.4, the tree is universally compatible. ∎

A group is generic if the trivial -tree is not abelian JSJ -tree. The description of the generic groups is given in [Moi1]. In particular, if a group is not isomorphic to a semi-direct product then it is generic.

Lemma 3.6.

Let be a generic group. If is a minimal -tree with an edge group with , then and is universally elliptic over abelian -trees.

Proof.

Since is generic, any tree is a JSJ -tree ([Moi1, Theorem 1.2]). Then all edge stabilizers of a given JSJ tree are universally elliptic and commensurable. Let be one of these stabilizers. Its commensurator is . In every other -tree over groups with , the group is elliptic, thus included in the stabilizer of some vertex . As its commensurator is , it is then virtually contained in every stabilizer of vertex in the orbit of . If we assume is minimal, it implies that is virtually included in every edge stabilizer. Hence every edge stabilizer is a which contains with finite index a universally elliptic group. Thus every edge stabilizer is universally elliptic. ∎

Corollary 3.7.

Let be a group. The tree associated to the JSJ deformation space of is compatible with every -tree over the subgroups of .

Proof.

If is generic, this is a direct consequence of Proposition 3.5 and Lemma 3.6.

If is not generic, then the trivial -tree is a JSJ decomposition. Hence is trivial and universally compatible. ∎

3.3 Maximality

In this section, we prove that in the case of groups (and not only groups), the tree dominates every other universally compatible trees.

For this we need some technical lemmas, that we divide in three categories. The lemmas of the first category give conditions on trees to be compatible. the lemmas of the second category give existence of deformation sequences in the deformation space and the lemmas of the third category give some refinement conditions.

Compatibility lemmas

For a set of elements of and a -tree, call the convex hull of all characteristic spaces of elements of in . Note that if is reduced to one element then is just the characteristic space of this element.

Lemma 3.8.

Let and be two -trees and let , , , and be four elements of .

  • If the intersection is empty, and the intersection contains an edge, then and are not compatible.

  • If and are disjoint in and and are disjoint in , then and are not compatible.

Proof.

First notice that if is a refinement of , then surjects onto via the natural map.

Thus for the first point, the two assumptions are stable by refinement. Hence a common refinement should have both properties. This is impossible.

For the second point, assume there exists a common refinement . As and are disjoint in then and are also disjoint in . Call the bridge in between and . Then is contained in and in . Thus is not empty in , hence is not empty in . ∎

Lemma 3.9.

Let be a group and be an abelian JSJ -tree. Let be a reduced edge with initial vertex in such that . Then there exists that centralizes .

Proof.

If is abelian, the lemma is trivial. Otherwise by [Moi1], the group is a semi-direct product , with a non-trivial automorphism of , and , thus any element in centralizes . ∎

Corollary 3.10.

Let be a -tree. Let be an orbit of reduced edges of . Assume has one of the following properties:

  1. is pre-ascending in and is not an ascending HNN-extension,

  2. is not ascending and there exists an orbit of reduced edges which slides along two consecutive edges of in ,

  3. is a -tree, the orbit is not ascending and there exists an orbit of reduced edges which slides along in .

  4. is toric of endpoint and there exist two distinct orbits of reduced edges and with terminal vertex , distinct from the orbits or .

Let be a collapse of such that is not collapsed in . Then is not universally compatible.

Here we do not need the slides to be admissible.

Proof.
  1. If is pre-ascending in , then is pre-ascending in . By [GL07, Proposition 7.1] the tree is not universally compatible.

  2. Let , and be edges of and an edge of such that may consecutively slide along and in . Let and be the initial and terminal vertices of , call the terminal vertex of and the initial vertex of (note that the terminal vertex of is and the initial vertex of is ).

    As is reduced and non ascending, we may find three elements , and . If is not ascending let be in . If is ascending, let be such that , an call . The sets and are separated by an edge in . As is not collapsed in , the sets and are disjoint in . But making slide along and , we now have a new tree in which (see Figure 13). By Lemma 3.8, the tree is not universally compatible.

    Figure 13:
  3. Assume that is a -tree. Let be an edge which slides along a reduced non ascending edge . Call the terminal vertex of . The set is not empty since is reduced and non-ascending. Applying Lemma 3.9, there exists in this set such that , thus slides consecutively on and . Thus the point is implied by the point .

  4. As is toric then slides along two consecutive edges in . If , collapsing in , the edge is no more ascending (and still reduced) in the new tree , we may apply the point . Note that may perhaps not refines , however it refines the tree obtained from by collapsing , this is enough to obtain the result.

    If is in the orbit of . Let be an edge of . Call its terminal vertex and let be an edge of with terminal vertex . Let be an edge of with initial vertex (on which may slide). Let be the terminal vertex of and be the initial vertex of . Call an element which sends onto and which sends onto .

    As is toric , and slides consecutively along and . Define the four elements , , and (see Figure 14). Then the sets and are disjoint in but after making slide along , and , we have . Thus by Lemma 3.8, the tree cannot be universally compatible. ∎

    Figure 14:
Corollary 3.11.

Let be a -tree. Let be an edge of with initial and terminal vertices and . Call the tree obtained from by collapsing all orbits of edges except .

If there are two reduced non-ascending edges, with initial vertex and with initial vertex , such that , , and , then is not universally compatible.

Here the edge is not necessarily reduced.

Proof.

As and are reduced and non ascending, we may take an element which fixes the terminal vertex of but not and which fixes the terminal vertex of but not . Let be an element of and define and . Define and . We are as in Figure 15. Here and are disjoint. But may slides along and . By equivariance, at the same time slides along and .

After performing these slides, we obtain a tree in which and are disjoint. By Lemma 3.8, the tree is not universally compatible. ∎

Figure 15:
Lemma 3.12.

Let be a -tree. Let be an edge of whose endpoints are in distinct orbits, call the initial vertex of and its terminal vertex. Call the tree obtained from by collapsing all orbits of edges except . If there exists two reduced edges and in the same orbit , such that the initial vertex of is , the terminal vertex of is , and , then is not universally compatible.

Proof.

As and are in the same orbit we may take such that . Let us define and .

First assume . As , we have and . Thus collapsing in , the edge becomes pre-ascending, hence it stays pre-ascending in . By Corollary 3.10, the tree is not universally compatible.

Assume now that or . We have . Take in , and call . As or , as is reduced and , we have .

We may now take an element which fixes the terminal vertex of but not and an element which fixes the terminal vertex of but not .

If , take in (automatically and ). If then . Take . Define and take which fixes the terminal vertex of but not .

Then and are separated at least by . Thus is empty.

Now call the tree obtained by collapsing in and call the initial vertex of in . Perform the expansion of the vertex with group and set . Call the edge obtained in this expansion, and the terminal edge of .

We may note that:

  • the element still stabilizes