From local to global conjugacy

From local to global conjugacy of subgroups of relatively hyperbolic groups

Abstract.

Suppose that a finitely generated group is hyperbolic relative to a collection of subgroups . Let be subgroups of such that is relatively quasiconvex with respect to and is not parabolic. Suppose that is elementwise conjugate into . Then there exists a finite index subgroup of which is conjugate into . The minimal length of the conjugator can be estimated.

In the case where is a limit group, it is sufficient to assume only that is a finitely generated and is an arbitrary subgroup of .

1991 Mathematics Subject Classification:
Primary 20F65, 20F67. Secondary 20E45
This research was partially supported by SFB 701, “Spectral Structures and Topological Methods in Mathematics”, at Bielefeld University.

1. Introduction

Serre [15, Corollary 3, page 65] famously observed that a finitely generated group acting on a tree has a global fixed point provided each of its elements fixes a point. If the tree corresponds to an HNN-extension , it follows that a finitely generated subgroup of conjugates into if each of its elements conjugates into . If the tree corresponds to a free product , the situation is similar: if each element of conjugates into , then the whole subgroup conjugates into . With amalgamation, however, the sutiation is not as clear cut: if each element of a finitely generated subgroup conjugates into , the whole group conjugates into or into . In some situations, one is able to use additional information to conclude that is conjugate into . This happened to the authors in the proof of [3, Theorem 7.4], which in turn is a central step in the argument of that paper that surface groups are subgroup conjugacy seperable (SCS). Trying to generalize, it is therefore natural to search for sufficient conditions that imply conjugacy of a subgroup into a target given that all elements conjugate into the target. Serre’s observation also hints at the importance of negative curvature for this phenomenon.

In [4], we made the first step in this direction. At the heart of the argument there lies the following:

Theorem A of [4]. Let be a hyperbolic group, let be a quasiconvex subgroup of and let be an arbitrary subgroup of . Suppose that is elementwise conjugate into . Then there exists a finite index subgroup of which is conjugate into .

We recently learned that Theorem A was already contained in [11, Theorem 1]. We used Theorem A to extend subgroup conjugacy separability to a class of groups including surface groups (see [4]). Meanwhile, Chagas and Zalesskii [6] had generalized our result about surface groups in a different direction showing that limit groups are subgroup conjugacy separable. They use different methods.

Here, we extend Theorem A to relatively hyperbolic groups. From a geometric point of view, a finitely generated group is hyperbolic if it acts properly discontinously and cocompactly by isometries on a hyperbolic space. In particular, point stabilizers of the action are finite. If we have an action with infinite point stabilizers, non-negative curvature might be hidden there. We would like to regard a group acting cocompactly on a hyperbolic space still as hyperbolic modulo the pieces of the group acting trivially. The notion of a group hyperbolic relative to a collection of subgroups (think of a set of representatives for point stabilizers) is one way to make this intuition precise. We recall a definition suitable for out needs and some further results in Section 2.

Our main result is:

Theorem 3.1. Suppose that a finitely generated group is hyperbolic relative to a collection of subgroups . Let be subgroups of such that is relatively quasiconvex with respect to and is elementwise conjugate into . Then one of the following holds:

  1. The subgroup is parabolic, i.e., conjugate into some element of .

  2. Some finite index subgroup of is conjugate into .

If is infinite and nonparabolic, then the length of the conjugator with respect to a finite generating system of can be bounded in terms of , the quasiconvexity constant of , and the minimal -length of loxodromic elements of .

If all peripheral subgroups are virtually abelian, then the second option always occurs:

Corollary 4.5. Suppose that a finitely generated group is hyperbolic relative to a collection of virtually abelian subgroups . Let be subgroups of such that is relatively quasiconvex with respect to and is elementwise conjugate into . Then some finite index subgroup of is conjugate into .

Moreover, the following holds:

  1. If is infinite and nonparabolic, then the length of the conjugator with respect to a finite generating system of can be bounded in terms of , the quasiconvexity constant of , and the minimal -length of loxodromic elements of .

  2. If is infinite and parabolic, then a conjugator may be chosen whose -length is bounded a priory in terms of and the minimal -length of elements of infinite order in .

  3. If is finite then the finite index subgroup and the conjugator can be taken to be trivial.

Remark 1.1.

In Theorem 3.1, passage to a finite index subgroup of cannot be avoided even in case that is a free group. Indeed, let be the alternating group of degree 4, the Klein subgroup of , and a subgroup of of order 2. Let be the free group of rank 2 and an epimorphism. We set and . Then is locally, but not globally conjugate into .

For a subset of a relatively hyperbolic group , let denote the subset of all loxodromic elements of .

Limit groups are hyperbolic relative to a collection of representatives of conjugacy classes of maximal noncyclic abelian subgroups [8, Theorem 4.5], see also [1, Corollary 3.5]. This allows us to apply the main result in this case. In this situation, we also prove that the index depends only on :

Corollary 5.2. Let be a limit group and let and be subgroups of , where is finitely generated. Suppose that is elementwise conjugate into . Then there exists a finite index subgroup of which is conjugate into .

The index depends only on . The length of the conjugator with respect to a fixed generating system of depends only on and , where

As already mentioned, our main result generalizes Theorem A of [4]. Analogously to [4], our main result enables us to prove that a large class of relatively hyperbolic groups are quasiconvex-SCS and quasiconvex-SICS. In particular, we obtain a new proof that limit groups are SCS. We will explain this in a forthcoming paper.

2. Relatively hyperbolic groups

The following discussion is based on [13, Chapter 2]. We shall choose among the many equivalent characterizations of relatively hyperbolic groups a definition that is well suited to our needs. The first paragraphs are taken almost verbatim from [13, Chapter 2], but notation is slightly changed to suit our current needs.

Let be a group, a collection of subgroups of , a subset of . We say that is a relative generating set of with respect to if is generated by the set . (We always assume that is symmetrized, i.e. .) In this situation the group can be regarded as the quotient group of the free product

where is the free group with the basis . If the kernel of the natural homomorphism coincides with the normal closure of a subset , we say that has relative presentation

If and are finite, this relative presentation is called finite and the group is called finitely presented with respect to .

Theorem 2.1.

([13, Theorem 1.1]) Let be a finitely generated group, a collection of subgroups of . Suppose that is finitely presented with respect to . Then is finite and each subgroup is finitely generated.

Suppose that (2.2) is a finite relative presentation. Let Given a word in the alphabet such that represents 1 in , there exists an expression

with the equality in the group , where and for any . The smallest possible number in a representation of type (2.3) is denoted .

A function is called a relative isoperimetric function of (2.2) if for any and for any word over of length representing the trivial element of the group , we have . The smallest relative isoperimetric function of (2.2) is called the relative Dehn function of with respect to and is denoted by (or simply by when the group and the collection of subgroups are fixed).

There are simple examples showing that is not always well-defined, i.e. it can be infinite for certain values of the argument. However if is well-defined, it is independent of the choice of the finite relative presentation up to the following equivalence relation. Two functions are called equivalent if there are positive constants such that and .

Definition 2.2.

[13, Definition 2.35 and Corollary 2.54] We call a group hyperbolic relative to a collection of subgroups if is finitely presented with respect to , the corresponding Dehn function is well-defined, and the Cayley graph is a hyperbolic metric space. We call a peripheral structure for .A subgroup is called parabolic if it is conjugate into some .

In particular, a group is hyperbolic (in the ordinary non-relative sense) if and only if it is hyperbolic relative to the trivial subgroup.

Here we meet the main difficulty: The space is hyperbolic, but is not locally finite if is infinite. Suppose that generates . Then there are two distance functions on , namely and . For brevity, we denote . Clearly, .

For reference, we collect a few statements from [2, 9, 13, 14] that will allow us to deal with the relatively hyperbolic case. From now on and to the end of this section, we will assume the following.

Assumption. The group is generated by a finite set and is hyperbolic relative to a collection of subgroups .

Theorem 2.3.

([13, Theorem 3.26]) There exists a constant having the following property. Let be a triangle whose sides are geodesics in . Then for any vertex on , there exists a vertex on the union such that

Recall that an element is called parabolic if it is conjugate to an element of one of the subgroups , . An element is called hyperbolic if it is not parabolic. An element is called loxodromic if it is hyperbolic and has infinite order.

Lemma 2.4.

([13, Corollary 4.20]) For any loxodromic element , there exist , such that

for any .

Lemma 2.5.

([13, Corollary 4.21]) Let be a loxodromic element in . If for some and , then .

Recall that a subgroup of a group is called elementary if it contains a cyclic subgroup of finite index.

Theorem 2.6.

([14, Theorem 4.3]) Every loxodromic element is contained in a unique maximal elementary subgroup, namely in

Lemma 2.7.

([2, Lemma 3.5]) Let be a group hyperbolic relative to the collection of subgroups . For any and , there exists a finite subset such that if , then is loxodromic.

Lemma 2.8.

([9, Lemma 2.13]) Let be a subgroup of a relatively hyperbolic group . If is infinite and torsion, then is parabolic.

Lemma 2.9.

Let be a subgroup of a relatively hyperbolic group . Then contains a loxodromic element if and only if is infinite and nonparabolic.

Proof. Suppose that is infinite and nonparabolic. By Lemma 2.8, contains an element of infinite order. Suppose that is not loxodromic. Then for some and . Since is not parabolic, there exists such that . By Lemma 2.7, there exists a finite set such that is loxodromic for any . Since has infinite order, some power lies outside of ; hence the element is loxodromic. Thus, is a loxodromic element in . The converse direction is obvious.

Definition 2.10.

([13, Definitions 4.9, 4.11])

Let be a group generated by a finite set , a collection of subgroups of .

(a) A subgroup of is called relatively quasiconvex with respect to (or simply relatively quasiconvex when the collection is fixed) if there exists such that the following condition holds. Let be two elements of and an arbitrary geodesic path from to in . Then for any vertex , there exists a vertex such that

(b) A relatively quasiconvex subgroup of is called strongly relatively quasiconvex if the intersection is finite for all , .

In the case of a finitely generated relatively hyperbolic group these notions do not depend on a choice of a finite generating set:

Proposition 2.11.

([13, Proposition 4.10]) Let be a group hyperbolic relative to a collection of subgroups and let be a subgroup of . Suppose that are two finite generating sets of . Then is (strongly) relatively quasiconvex with respect to if and only if it is (strongly) relatively quasiconvex with respect to .

Theorem 2.12.

([13, Theorem 4.19]) Let be a finitely generated group hyperbolic relative to a collection of subgroups and let be a hyperbolic element of . Then the centralizer of in is a strongly relatively quasiconvex subgroup in .

Lemma 2.13.

For every loxodromic element , there exists such that the following holds. Let be a natural number and a geodesic segment in connecting and . Then the Hausdorff distance (induced by the -metric) between the sets and is at most .

Proof. A path in is called an -path if its edges are labelled by elements of . Let be a path of minimal length among all -paths from 1 to . We set for and . The paths and have the same endpoints; the path is a geodesic and the path is a quasi-geodesic (with respect to ) with uniform constants (i.e., independent of the exponent ) by Lemma 2.4.

The paths and do not have backtracking [13, Definition 3.9] and every vertex of and is a phase vertex [13, Definition 3.10]: in the case of , this is because the path has only labels from ; and for this follows from the path being geodesic. Thus, the technical hypotheses of [13, Proposition 3.15] are satisfied. By this proposition, there is a uniform constant such that the Hausdorff distance between and (with respect to ) is at most . Then the Hausdorff distance betweeen and is at most .

We use the following result of B.H. Neumann.

Theorem 2.14.

[12, Lemma 4.1] Suppose that is a finite family of subgroups of a group and is a finite family of elements of with the property . Then there exists such that has a finite index in .

3. The main theorem

For this section, we let be a group generated by a finite set that is hyperbolic relative to a fixed peripheral structure . We put . In the following proof we will fix a loxodromic element . To not interrupt the proof, we fix some notation and constants in advance.

  1. For an element , we denote . Clearly, .

  2. Let be the constant defined in Theorem 2.3.

  3. Let and be the constants defined in Lemma 2.4 and Lemma 2.13 for the loxodromic element .

Enlarging and , we may assume that all of them are integers.

Theorem 3.1.

Suppose that a finitely generated group is hyperbolic relative to a collection of subgroups . Let be subgroups of such that is relatively quasiconvex with respect to and is elementwise conjugate into . Then one of the following holds:

  1. The subgroup is parabolic, i.e., conjugate into some element of .

  2. Some finite index subgroup of is conjugate into .

If is infinite and nonparabolic, then the length of the conjugator with respect to a finite generating system of can be bounded in terms of , the quasiconvexity constant of , and the minimal -length of loxodromic elements of .

Proof. We may assume that is infinite. Suppose that is nonparabolic. By Lemma 2.9, contains a loxodromic element .

Ultimately, we obtain the finite index subgroup of by means of Theorem 2.14. We shall cover by finitely many cosets of subgroups. One of the subgroups is the intersection . In order to cover the other elements, let us consider an arbitrary element . Then for every , there exists such that . Let be the shortest element with respect to the metric among those elements which satisfy the following property: There exist such that and

We consider the path starting at 1 with label . The path ends at a point . We also consider the geodesic 6-gone in , where , , , , , and (see Fig. 1).

Figure 1.

Let be the quasiconvexity constant for . Let be the number of words of length up to in the alphabet . We choose

where and

Without loss of generality, we may assume that . Then .

For each , we consider the vertex . By Lemma 2.13, there exists a vertex such that

By Theorem 2.3, there exists a point such that

In particular,

We prove that if is sufficiently large, then there are many consecutive values of such that . We will consider four cases, where and show that each case can occur only for a restricted number of values of , which we shall bound explicitly.

Case 1. Suppose that .

We set . Then

By minimality of and using Lemma 2.4, we get:

Therefore . In particular, the number of such is at most .

Case 2. Suppose that .

Then

On the other side, since and , we get by Lemma 2.4 that

Thus . In particular, the number of such is at most .

Case 3. Suppose that .

Recall that . By Lemma 2.13, there exists a vertex such that . Then . We write , where .

Since and , we have

We claim that this case can occur for at most values of . Indeed, otherwise there would exist different and such that Then . Lemma 2.5 implies that . By Theorem 2.6, we obtain that contradicts our assumption.

Case 4. Suppose that .

We set . Then . Hence, by minimality of , we have .

Note that is the label of a path from to . Therefore

Then .

We claim that this case can occur for at most values of . Indeed, since lies on the geodesic and , there are at most possibilities for . Since , there are at most possibilities for , and hence for .

By the choice of , there exists a subset consisting of consecutive integers such that for every . Since the subgroup is -quasiconvex, there exists with . We choose an -geodesic for each . We have

Since is the number of words of length up to in the alphabet , there exist different such that and have the same labels. Let be this label. Then the label of the path is . We set . Thus, there exist and such that

, and .

Moreover, the label of the path is for some , and we have

Recall that

From (3.2) and (3.3), it follows that there are exponents with , and using (3.1), we can in addition arrange for . Thus,

Since is an arbitrary element in and , we have

where

Since , we have

Since the set is finite, we deduce from Theorem 2.14 that either is of finite index in , or there exists such that is of finite index in . In the first case, has finite index in and we are done. In the second case, a finite index subgroup of is conjugate into .

4. Toral relatively hyperbolic groups

In this section we specialize Theorem 3.1 for the case, where the peripheral subgroups of the relatively hyperbolic group are virtually abelian.

Theorem 4.1.

([13, Theorem 1.4]) Let be a group, a collection of subgroups of . Suppose that is finitely presented with respect to and the relative Dehn function of with respect to is well-defined, i.e., it takes finite values for each . Then the following conditions hold.

  1. For any the intersection is finite whenever .

  2. The intersection is finite for any .

A quasiconvex subgroup of a relatively hyperbolic group is hyperbolic relative to an induced peripheral structure:

Theorem 4.2.

([10, Theorem 9.1]) Let be a finitely generated group hyperbolic relative to a collection of subgroups and let be a relatively quasiconvex subgroup. Consider the following collection of subgroups of :

Then the elements of lie in only finitely many conjugacy classes of . Furthemore, if is a set of representatives of these classes, then is hyperbolic relative to . We call the induced peripheral structure of .

Lemma 4.3.

Let be a finitely generated group hyperbolic relative to a collection of subgroups . Let be subgroups of such that is relatively quasiconvex with respect to and is a subgroup of a peripheral subgroup .
Suppose that is elementwise conjugate into . Then there exists a finite collection of elements such that each element of infinite order of is conjugate (in ) into the union of . The elements can be chosen so that they depend only on , but not on .

Proof. We will use notation of Theorem 4.2 applied to . First, we fix a peripheral structure of induced by the peripheral structure of : Then there exists a finite collection of elements of and finite subsets for each such that

  1. consists of representatives of conjugacy classes in of subgroups from and

  2. is hyperbolic relative to .

Recall that for some . Let be an element of infinite order. We show that is conjugate (in ) into for some . By assumption, there exists such that . Thus,

In particular, is infinite. By choice of , there exist , and such that

It follows that is infinite, whence by Theorem 4.1, we have and . Now (4.1) and (4.2) imply , and we are done.

Lemma 4.4.

([13, Lemma 5.8]) Let be a group generated by a finite set . Suppose that is hyperbolic relative to a collection of recursively presented subgroups , each with solvable conjugacy problem. There exists a computable function with the following property: if is conjugate into , then for some with .

Since finitely generated virtually abelian groups are finitely presented and have solvable conjugacy problem, we obtain the following:

Corollary 4.5.

Suppose that a finitely generated group is hyperbolic relative to a collection of virtually abelian subgroups . Let be subgroups of such that is relatively quasiconvex with respect to and is elementwise conjugate into . Then some finite index subgroup of is conjugate into .

Moreover, the following holds:

  1. If is infinite and nonparabolic, then the length of the conjugator with respect to a finite generating system of can be bounded in terms of , the quasiconvexity constant of , and the minimal -length of loxodromic elements of .

  2. If is infinite and parabolic, then a conjugator may be chosen whose -length is bounded a priory in terms of and the minimal -length of elements of infinite order in .

  3. If is finite then the finite index subgroup and the conjugator can be taken to be trivial.

Proof. We assume that is infinite, otherwise the statement is trivial. If is nonparabolic, then the statement follows from Theorem 3.1. Thus, we assume that is infinite and parabolic. Then there exist and such that .

Let be the element of infinite order with the minimal -length. Since , we have for some with (see Lemma 4.4). Then is infinite. By Lemma 4.1 we have . Hence, , where the -length of is bounded as above. Thus, we may assume that lies in .

Let be a maximal torsion free abelean subgroup of . Clearly, is finite. Let be a set of representatives of left cosets of in . We set .

By Lemma 4.3, there exists a finite collection of elements such that is elementwise conjugate (in ) into the union of . Then lies in the union of , and we have

By Theorem 2.14, one of the subgroups has finite index in and hence in . Recall that by Lemma 4.3, the elements depend only on , and the elements depend only on .

5. An application to limit groups

Recall that a subgroup of a group is called a retract of if there exists an epimorphism with . The epimorphism is called a retraction. Equivalently, is retract of if for some subgroup of . A subgroup of is called a virtual retract of if is a retract of a finite index subgroup of . In the following proposition we list some properties of limit groups that we use later.

Proposition 5.1.
  1. Limit groups have the unique root property, i.e., with implies . In particular, limit groups are torsion free.

  2. Retracts of limit groups are closed under taking of roots.

  3. If is a finitely generated subgroup of a limit group , then is a virtual retract of  [17, Theorem B], and hence is quasi-isometrically embedded [17, Corollary 3.12]. In particular, is quasiconvex in .

Proof. (1) It suffices to argue the unique root property for subgroups of limit groups generated by two elements. By [7, Proposition 3.1 (4)], those subgroups are free or free abelian; and those have the unique root property.

(2) Let be a limit group and a retraction. Suppose that is such that for some . Then . Since has the unique root property, we have .

Corollary 5.2.

Let be a limit group and let and be subgroups of , where is finitely generated. Suppose that is elementwise conjugate into . Then there exists a finite index subgroup of which is conjugate into .

The index depends only on . The length of the conjugator with respect to a fixed generating system of depends only on and , where

Proof. Let be a set of representatives for the conjugacy classes of maximal non-cyclic abelian subgroups of . As a limit group, is hyperbolic relative to the finite family ([8, Theorem 4.5], another proof is given in [1, Corollary 3.5]).

By statement (3) of Proposition 5.1, is quasiconvex. By Corollary 4.5, there exists a finite index subgroup of and an element such that and the -length of is bounded as above. It remains to prove that the index only depends on .

By statement (3) of Proposition 5.1, ther