Abelian JSJ decomposition of graphs of free abelian groups

Abelian JSJ decomposition of graphs of free abelian groups

Benjamin Beeker
July 17, 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 construct the JSJ decomposition of a  group over abelian groups. We prove that this decomposition is explicitly computable, and may be obtained by local changes on the initial graph of groups.

1 Introduction

The theory of JSJ decomposition starts with the work of Jaco-Shalen and Johansson on orientable irreducible closed -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 [6]. Then Sela gave a construction for torsion-free hyperbolic groups [8]. This notion has been more generally developed by Rips and Sela [7], Dunwoody and Sageev [2], Fujiwara and Papasoglu [4] for various classes of groups. In [5], Guirardel and Levitt generalize the object by introducing the definition of JSJ deformation space, proving the existence of this space for finitely presented groups.

In this paper we consider the following class of groups.

Let be a finite graph of groups with all vertex groups finitely generated free abelian. Let be the fundamental group of . If the rank of all edge and vertex groups is equal to a fixed integer , we call such a group a GBS group, standing for Generalized Baumslag-Solitar groups of rank . When the rank is variable, we call such a group a  group. The goal of this paper is to describe the JSJ decomposition of over abelian groups and to give a way to construct it.

A decomposition of a group is a graph of groups with fundamental group . To define what a JSJ decomposition is, we need the notion of universally elliptic subgroups. Given a group and a decomposition of , a subgroup is elliptic if the group is conjugate to a subgroup of a vertex group of . Given a class of subgroups of , a subgroup is universally elliptic, if is elliptic in every decomposition of as a graph of groups with edge groups in . A decomposition is universally elliptic if all edge groups are universally elliptic.

A decomposition dominates another decomposition , if every elliptic group of is elliptic in . A decomposition is a JSJ decomposition if it is universally elliptic, and it dominates every other universally elliptic decomposition. Then given a vertex in a JSJ decomposition, either , the vertex group of , is universally elliptic, we say is rigid, or is not then we say is flexible.

For instance, looking at the JSJ decomposition of a torsion-free hyperbolic group over cyclic groups described by Sela in [8] and Bowditch [1], the flexible vertex groups are exactly the surface groups.

The defining decomposition of a  group is a good approximation of the JSJ. For example, for a GBS group, the graph of groups is a JSJ decomposition whenever the associated Bass-Serre tree (which is locally finite in this case) is not a line ([3],[5]) or equivalently whenever is not polycyclic.

In the general case three kinds of local changes may be needed to obtain the JSJ from .

  • When the quotient of a vertex group of the decomposition by the group generated by all adjacent edge groups is virtually cyclic, then the vertex is blown into a loop (see figure 1 for the simplest example).

    if
    Figure 1: Expansion of the vertex

    To be more specific, in figure 1, if , then the edge group has finite index in the vertex group of and is rigid. If , there exists a unique non-trivial splitting of which leaves the part elliptic. The vertex must be blown into a loop as in the figure, and the new vertex is then rigid. If then is flexible.

  • Conversely some loops must be collapsed.

    A loop based at a vertex is called a loop if both inclusion maps of its edge group into are bijections. The fundamental group of the subgraph of groups composed of and is a semi-direct product . Some of the loops are collapsed, depending on and on the other edges adjacent to .

    For example, in figure 2, the loop must be collapsed if and only if : if , there are many decompositions of as an HNN extension leaving in the vertex group. If , there is exactly one, and its edge group is universally elliptic.

    Figure 2: Collapse of a loop
  • A similar phenomenon occurs for edges which are not loops and whose group has index in each adjacent vertex group (see figure 3). We call an edge of this type a edge.

    Figure 3: The group is a Klein bottle group with trivial intersection with the adjacent edge groups.

Given an edge of , call the subgraph consisting of the single edge (and its vertices) and its fundamental group.

To obtain the JSJ decomposition of a  group over abelian groups, we show that it suffices to expand vertex groups such that is virtually cyclic as in figure 1, and collapse edges which are not universally elliptic. These edges have polycyclic groups, and so are loops and edges as in figures 2 and 3.

Theorem 1.1.

Let be a  group. For a vertex, let be the subgroup of generated by groups of edges adjacent to . A JSJ decomposition of over abelian groups can be obtained by

  • expanding the groups such that is virtually cyclic,

  • collapsing edges and loops which are not universally elliptic.

As for hyperbolic groups, in a JSJ of a  group, the rigid vertices are also easily identifiable. A vertex is rigid if and only if its vertex group is abelian and virtually generated by the adjacent edge groups.

In order to make theorem 1.1 more explicit, we shall now describe the edges which are not universally elliptic.

We first describe the edges for which has a trivial JSJ decomposition. This is the same as giving the JSJ decomposition of polycyclic GBS groups. As (reduced) decompositions of polycyclic groups must have a line (or a point) as Bass-Serre tree, these decompositions have at most one edge, which may be of two types.

The first are the edges, that is, non-loop edges whose group is of index in the groups of the two adjacent vertices. Then is isomorphic to a direct product of a by either the Klein bottle group , or the twisted Klein bottle group .

The second are loops (whose inclusion maps into the vertex group are bijections), and their groups can be seen as semi-direct product :

Theorem 1.2.
  1. If is an edge of type then has a trivial JSJ decomposition over abelian groups.

  2. Let us take in .

    The group has a trivial JSJ decomposition if and only if can be written in a well-chosen basis in one of the following ways:

    1. with an matrix of finite order and in ,

    2. with in .

    In every other case the semi-direct product is a JSJ decomposition.

This theorem is proved in section 4 and 5.

By [5, lemma 4.10], we know that the edges which are not universally elliptic are those for which has a trivial JSJ decomposition relative to adjacent edge groups. We therefore have to understand how is embedded in the whole group. More precisely, we look at the way the adjacent edge groups inject into . We obtain the following theorem which makes theorem 1.1 explicit.

Define a hyperplane of as the kernel of a morphism from to .

Theorem 1.3.

Let be a  group. For a vertex, let be the subgroup of generated by groups of edges adjacent to . A JSJ decomposition of over abelian groups can be obtained by

  • expanding the groups such that is virtually cyclic,

  • collapsing edges such that may be decomposed as in theorem 1.2 with all adjacent edge groups included in the hyperplane .

  • collapsing edges with vertices and , such that there is a hyperplane of which is also a hyperplane of and and which contains all groups of adjacent edges, excepted.

We prove this proposition in sections 6 and 7. We prove in section 8 that the construction of the JSJ is algorithmic.

From theorem 1.3, we obtain the JSJ decomposition of  groups over abelian groups with bounded rank:

Theorem 1.4.

Let a  group, and . Suppose that is a JSJ decomposition over free abelian groups. Then a JSJ decomposition over free abelian groups of rank may be obtained by collapsing every edge of with group of rank .

2 Preliminaries

Let be a finitely generated group. We denote by the Bass-Serre tree associated to a finite graph of groups decomposition of . For a vertex (resp. an edge ) of , we denote (resp ) its stabilizer. Most of the time will be omitted.

For an (unoriented) edge of with vertices and , we call the type of the couple of potentially infinite numbers with such that is of index in and in . For example, an edge whose stabilizer equals the stabilizer of its vertices, is of type . This type only depends on the orbit of the edge, so we define the type of an edge of the graph of groups as the type of an edge in representing it. We say that an edge of is a loop if its vertices are in the same orbit under the action of . A graph of groups is reduced if all edges of type are loops in the graph. Every decomposition may be reduced by contracting successively the edges of type which are not loops.

Given a Bass-Serre tree , an element is elliptic if it fixes a vertex. Otherwise is said to be hyperbolic. The characteristic space of is the minimal subtree of containing the vertices such that the distance between and is minimal (seeing as a metric space with all edges of lenght ). When is elliptic, it is the set of all fixed edges and vertices. When is hyperbolic this is the only line on which acts by a translation. In this case we call it the axis of . A subgroup of is elliptic if it fixes a vertex or equivalently when is finitely generated if all of its elements are elliptic. In the case of finitely generated abelian groups, the ellipticity of a generating set implies the ellipticity of the whole group.

From now on and for the rest of the paper, the decompositions we consider will be over free abelian groups, meaning that every edge stabilizer is finitely generated free abelian.

An element or a subgroup of is universally elliptic if it is elliptic in the Bass-Serre trees of all graph of groups decompositions. An edge is universally elliptic if it carry a universally elliptic group. A decomposition is universally elliptic if all edge are universally elliptic. A graph of group decomposition dominates an other decomposition if every elliptic subgroups of is elliptic in . A JSJ decomposition is a universally elliptic decomposition which dominates every other universally elliptic decomposition ([5]).

If is an edge of a Bass Serre tree, with vertices and , of type representing a loop in the graph of groups, let be a hyperbolic element such that . We call modulus of the linear map such that for all in we have . As , we can see as an element of . Up to conjugacy, the modulus does not depend neither on the choice of nor on the choice of representing but is switched to its inverse if we change the orientation of .

3 Universally elliptic edges

Proposition 3.1.

Let be a graph of groups and its Bass-Serre tree. Let be a hyperbolic element of . Then the centralizer of in is a semi-direct product with a subgroup of an edge stabilizer of and a cyclic subgroup of generated by a hyperbolic element.

When all edge groups of are finitely generated free abelian, the centralizer of a hyperbolic element is a polycyclic group .

Proof.

The group acts on the axis of by translation. This action defines a morphism from to . The kernel of this morphism fixes the axis pointwise and so belongs to the stabilizer of the axis. ∎

The centralizers of hyperbolic elements have a very specific structure, which is not the case for elliptic ones. This forces most of edge groups of  group to be universally elliptic:

Corollary 3.2.

Let be a  group with reduced. Let be an edge of .

  1. If is not a loop of and is not a edge then is universally elliptic.

  2. If is a loop of but is not then is universally elliptic.

Proof.

Call a representative of in the Bass-Serre tree. An abelian group generated by finitely many elliptic elements is elliptic. We just have to show that each element of the edge group is universally elliptic.

Let and be the endpoints of . Then is contained in the centralizer of . If is not of index in both and , the amalgam contains a free group. So the centralizer of cannot be a polycyclic group. By proposition 3.1, every element of the edge stabilizer is elliptic and is universally elliptic.

There remain two cases: when is a loop, and when is a loop with . In both cases, let be such that .

If is a loop, let be a square in . Then the centralizer of contains , and and so the group . But this amalgamated product is of type, the same argument as before works. And so is universally elliptic. Every element of has a universally elliptic square, and is universally elliptic.

If is a loop with , then for all , the group is a subgroup of which is abelian. So its centralizer contains , a strictly increasing union of groups. The centralizer contains a non-finitely generated group and cannot be polycyclic, contradicting proposition 3.1. So is universally elliptic. ∎

Graphs of groups consisting of an HNN-extension of type or an amalgamation of type are exactly the reduced ones whose Bass-Serre trees are lines. They are the polycyclic  groups. The goal of the next two paragraphs is to study the different decompositions of these groups in order to determine their JSJ.

4 Edges of type

In this part, one shows that, for a given n, there are exactly two  groups whose graphs of groups have two vertices with vertex group linked by an edge of type , and that these two groups can be seen as semi-direct products of by . So these groups have both an action by translation and a dihedral action on . We shall prove in next section that they both have a trivial JSJ decomposition.

The first one is the direct product of the Klein bottle group by . One will call the untwisted Klein bottle group.

The second one is a twisted version of the first, it can be described as the product of and the group with presentation

One will call the twisted Klein bottle group.

Graph of groups decompositions of and are as in figure 4.

Figure 4:

The two sets of groups and will be called extended (untwisted or twisted) Klein bottle groups. Like and , they have a decomposition in a amalgam of type .

Proposition 4.1.

Let be a  group and . The following are equivalent:

  1. The group is an extended Klein bottle group or .

  2. The group is a semi-direct product with in a suitable basis of .

  3. The group admits a graph of group decomposition with two vertices carrying groups and an edge of type .

We shall prove successively 1 2 and 3 1

Lemma 4.2.

The group can be seen as a semi-direct product , and as with .

Proof.

The proof consists in giving for and a change of presentation, such that the second one is a semi-direct product.

In the case of , the change of presentation is well known:

In the second case, the presentation

of can be changed to

via the map

Corollary 4.3.

Each group and can be identified with a semi-direct product where has matrix in a well-chosen basis .

Moreover, with this identification:

  • ,

  • for all in , the element is hyperbolic in the decomposition in amalgam of type ,

  • in the case of ,

  • in the case of .

In particular, in the semi-direct product decomposition, seen as an HNN extension, the groups , and the element stay elliptic, the element is elliptic too. On the opposite and are hyperbolic.

One can notice that, in the case of untwisted Klein bottle groups, the element is trivial.

Proof.

The changes of presentation described in the previous proof can be extended to or by noticing that . The four points are easy to check. ∎

This proves 1 2 in proposition 4.1.
The automorphism is conjugate to if all coordinates of are even, to if one is odd. So the semi-direct product is isomorphic to one of or . This implies the 2 1 part.

1 3 has already been done.

Let us prove 3 1.

Let and be two copies of generated by the sets and respectively. By assumption, has a presentation

where is an isomorphism between and .

If is a square in , let be its square root. Let us take the family as a basis of . The presentation of is then changed to

which is a presentation of an extended Klein bottle group.

If is not a square in , then may be written , and there exists for which is odd. We may assume . The element is such that its image by is a square in with out of the image of . As can be written , the family is a basis of . Taking as generators for the elements , and for , the group admits as presentation

And is isomorphic to .

5 Loops of type

The groups we consider here are semi-direct products of by . The goal of the section is to determine their JSJ decompositions.

For an element of we write the group . We will call hyperplane of the kernel of a linear map .

Proposition 5.1.

If cannot be written (up to conjugation) in one of the following ways:

  1. with an matrix of finite order and in ,

  2. with in .

then the group has a unique non-trivial (reduced) graph of groups decomposition.

Its JSJ decomposition is the HNN-extension .

Lemma 5.2.

Let be an element of the part of . Assume that there exists a graph of groups decomposition of for which is hyperbolic. Then there exists a hyperplane elliptic in the decomposition , stable under the action of , and such that has finite order. Moreover, the set of elliptic elements of in is exactly , and if does not belong to then . In particular can be written in one of the forms described in proposition 5.1.

Proof.

The group is polycyclic, so the Bass-Serre tree associated to is a line. The set of elliptic elements of in is the kernel of a non trivial homomorphism from onto . It is therefore a hyperplane in not containing . Elements of act as the identity on . Moreover is also an elliptic subgroup of in . We obtain the inclusion , and so stabilizes .

If , then the lemma holds. Let us assume and let us show that has finite order and that belongs to .

Write . If the stable letter is elliptic in , then it commutes with and so . The letter must therefore be hyperbolic. There exist two non-zero integers and such that is elliptic. Then must commute with , but also commutes with , so commutes with . One has .

There exists an element for which with . But and act by translation implying that and that belongs to . ∎

Proof of proposition 5.1.

By lemma 5.2, when is not as in 1. and 2. the vertex group of (seen as an HNN extension) is universally elliptic. By [5, lemma 4.6], the JSJ decomposition of is the HNN extension . ∎

Every other group has at least one other action:

Lemma 5.3.

Let be a basis of in which can be written as in 1. or 2. of proposition 5.1. Then the semi-direct product can be decomposed as a graph of groups in which is elliptic and is hyperbolic for all in .

Proof.

The second case has already been done (see proposition 4.1 and corollary 4.3). It remains the first case.

We produce another decomposition of as a semi-direct product. Call the order of .

Define by , , and for all . As and with in , this is well defined.

Let act on the line by translations via . The kernel of the action is generated by and which commute. It is therefore an abelian group . ∎

Proposition 5.4.

If can be written in one of the following ways

  1. with an matrix of finite order and in ,

  2. with in

then the group has trivial JSJ decomposition.

Proof.

Let , and be as in lemma 5.3. Following lemma 5.3, no element of is universally elliptic. In every JSJ decomposition each of these elements is hyperbolic or fixes a unique vertex of the Bass-Serre tree of this decomposition.

Assume every element of is elliptic. Since is abelian, they all fix the same vertex. So the group is elliptic and fixes a unique vertex . But is normal, so the decomposition is trivial.

Let us now assume that there exists a hyperbolic element . Either is elliptic and fixes the whole tree, or is hyperbolic and acts by a translation. In particular, there exist in and in such that is elliptic and belongs to an edge stabilizer. This element, hyperbolic in the initial graph of groups, is not universally elliptic, which is a contradiction. ∎

6 Interactions between subgraphs

For an edge of a graph of group , call the subgraph consisting of the single edge (and its vertices) and its fundamental group.

Until now, we have shown that the non-universally elliptic edges must be loops or edges (Corollary 3.2). In this section, we characterize the non-universally elliptic edges . We show that necessarily has trivial JSJ decomposition, and that adjacent edge groups are included in a specific hyperplane of .

We first prove that two adjacent edges cannot be both non-universally elliptic.

Lemma 6.1.

Let be a  group and its Bass-Serre tree. Let be a vertex of . Let and be loops of type or edges of type , distinct and adjacent to . Then and are universally elliptic.

Proof.

Let , and be preimages in of , and respectively such that and have as common vertex. We call and the second vertices of and respectively. There are three cases to handle.

  1. The edges and are of type .

    Let and such that and Take and suppose is not universally elliptic. Let be a graph of groups in which is hyperbolic. As , and commute, the elements and must stabilize the axis of . So there exist , , and integers such that , , and and are elliptic with characteristic spaces containing the axis of . So and must commute. Yet their projections in the (topological) fundamental group of the graph generate a free group of rank , which is a contradiction.

  2. The edges and are of type .

    We show that every is universally elliptic. Replacing by we may assume that . Let us fix , and . So and are in the centralizer of , since , and are in it. Yet those are hyperbolic elements with distinct axis crossing precisely in , they generate a free group. Applying proposition 3.1 is universally elliptic.

  3. The edge is of type and of type .

    Let be as in the first case. Let us fix and . In the same way as last case, the elements and are in the centralizer of (the set of squares of ) and generate a free group.

Proposition 6.2.

Let be a  group. Let be a loop based at a vertex and the modulus of . Let be the group generated by the groups of adjacent edges, excepted. Then is universally elliptic if and only if there is no decomposition such that

  1. ,

  2. stabilize

    • either and act on with finite order,

    • or and .

Proof.

Assume is not universally elliptic, and let be a decomposition in which is not elliptic. From lemma 5.2, there exist a decomposition satisfying 2 and 3 such that is exactly the set of all elliptic elements of in the decomposition . By corollary 3.2 and lemma 6.1 , the group is universally elliptic, so is included in .

Conversely, suppose such a decomposition exists.

Let be a stable letter of . Take as basis of , completing it by to make a basis of . Then act on by a linear map with , a finite order matrix and if . We can apply lemma 5.3, and so there is a graph of groups of in which is hyperbolic and is elliptic. Call an vertex of with . We can construct a new graph of group decomposition of as follows. The underlying graph is obtained by removing from and gluing by adding an edge between and . We define the vertex groups in the following way:

  • for every vertex of coming from a vertex of , we define ,

  • for every edge of coming from an edge of not adjacent to , we define with the natural inclusions in the adjacent vertices,

  • ,

  • for every edge of coming from an edge of adjacent to , we define , with the inclusion coming from the inclusions coming from the assumption 1,

  • for every vertex of coming from a vertex of including , we define ,

  • for every edge of coming from an edge of , we define with the natural inclusions in the adjacent vertices,

  • for the edge between and , we define , with natural inclusions and coming from .

Using the isomorphism , we easily check that In particular is a decomposition of in which is hyperbolic. The element is not universally elliptic. ∎

The case of a type edge is similar.

Proposition 6.3.

Let be a  group. Let be an edge of type with vertices and . We identify with its images into and . Then is universally elliptic if and only if there is no hyperplane of such that is also a hyperplane of and , and contains all groups of adjacent edges, excepted.

If is not universally elliptic, a decomposition in which is not elliptic may be obtain from by replacing by a loop whose stabilizer is not universally elliptic (see figure 5).

Example.

Let and be two  groups with defining graphs and and , two vertices with groups . We construct a new  group with graph of groups as the union of and and a new edge between and . We define and . We then define the edge group to be identifying with in and with in and the part with and .

The group has an other decomposition , with underlying graph obtained gluing and by identifying and together, and adding a loop over the new vertex. Define . The loop carries the HNN extension with sable letter define by and for all in . To obtain the isomorphism between and , it suffices to identify with , with , and the parts together.

In the group is a extended Klein bottle group. The part of plays the role of in proposition 6.3.

Figure 5: The left graph represent , the right one .
Proof.

We call the group . By proposition 4.1 is an extended Klein bottle group, so we have with a free abelian subgroup, , and the relators of the -twisted or not- Klein bottle group. We have with in the case of the untwisted Klein bottle group.

First, assume there exists a hyperplane of which is also a hyperplane of and and contains all groups of adjacent edges, excepted. Let us prove is not universally elliptic.

We can fix and two elements such that and . As has index 2 in both and , and is included in , we must have . So there exists such that . Up to taking the inverse of , one may assume .

If is a square of an element , replacing by , the group may have the presentation . By lemma