LIMIT SET INTERSECTION THEOREM FOR GRAPH OF RELATIVELY HYPERBOLIC GROUPS

Limit Set Intersection Theorem for Graph of Relatively Hyperbolic Groups

Swathi Krishna Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar, Punjab 140306, India swathi280491@gmail.com
August 17, 2019
Abstract.

Let be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isomterically (qi) embedded condition. We prove that the set of conjugates of all the vertex and edge groups satisfy the limit set intersection property for conical limit points (refer to 2.1 and 2.4 for the definitions of conical limit points and limit set intersection property respectively). This result is motivated by the work of Sardar for graph of hyperbolic groups [16].

Key words and phrases:
2010 Mathematics Subject Classification:
(Primary); 20F65 (Secondary) 20E06

1. Introduction

Limit set intersection theorem first appears in the work of Susskind [19], in the context of geometrically finite subgroups of Kleinian groups. Further, Susskind and Swarup [18] prove it for geometrically finite hyperbolic subgroups of hyperbolic groups. This is followed by the work of J.W. Anderson in [1], [2] and [3] for general subgroups of Kleinian groups. Susskind conjectured that if is a non-elementary Kleinian group acting on for some , and are non-elementary subgroups of , then , where and denote the comical limit sets of and respectively, and Anderson shows that it holds most of the time in [4]. But in [7], Das and Simmons construct a non-elementary Fuchsian group that admits two non-elementary subgroups such that but , thus providing a negative answer to Susskind’s conjecture.

However, this prompts the following question in hyperbolic and relatively hyperbolic groups:

Suppose is a hyperbolic (resp. relatively hyperbolic) group and are hyperbolic (resp. relatively hyperbolic) subgroups of , then is true?

In 2012, Yang [20] proved a limit intersection theorem for relatively quasiconvex subgroups of relatively hyperbolic groups. Limit intersection theorem is not true for general subgroups of hyperbolic groups, and it was known to hold only for quasiconvex subgroups until the recent work of Sardar [16], where he proves a limit set intersection theorem for conical limit sets of vertex and edge subgroups of a graph of hyperbolic groups. In the paper he claims that limit set intersection theorem holds for general limit sets, however, in a communication with Sardar it has been pointed out that this only holds for conical limit sets, i.e, if is a hyperbolic group that admits a decomposition into a finite graph of hyperbolic groups structure with quasi-isomterically (qi) embedded condition, and if is a collection of conjugates of vertex and edge groups, then for any , . This motivates the question of what happens for the graph of relatively hyperbolic groups.

Our starting point is the following theorem:

Theorem 1.1.

Strong Combination Theorem[14] Let be a graph of strongly relatively hyperbolic groups satisfying

  1. the qi embedded condition

  2. the strictly type-preserving condition

  3. the qi-preserving electrocution condition

  4. the induced tree of coned-off spaces satisfied the hallways flare condtion

  5. the cone-bounded hallways strictly flare condition.

Then the fundamental group of is strongly hyperbolic relative to a family of maximal parabolic subgroups.

We prove the following theorem:

Theorem 1.2.

Suppose is a group admitting a decomposition into a graph of relatively hyperbolic groups satisfying the following:

  1. the qi-embedded condition

  2. the strictly type-preserving condition

  3. the qi-preserving electrocution condition

  4. the induced tree of coned-off spaces satisfies the hallways flare condition

  5. the cone-bounded hallways strictly flare condition.

Then the set of conjugates of vertex and edge groups of satisfy limit intersection property for conical limit points.

The conditions in Theorem 1.2 come from Theorem 1.1 which guarantee that the fundamental group of the corresponding graph of relatively hyperbolic groups is a relatively hyperbolic group.

The proof relies heavily on the ladder construction by Mj and Pal in [13].

Outline of the paper:

  • First we recall the construction of the tree of relatively hyperbolic metric spaces associated to a graph of relatively hyperbolic groups.

  • Then we recall the construction of the ladder from [13], but for geodesic rays, along with another important concept, vertical quasigeodesic rays.

  • In Proposition 5.13, we prove the following: Suppose we have two conical limit points from two distinct vertex spaces that map to the same point under the respective Cannon-Thurston maps, then we find a third vertex space, such that each of the conical limit points can be “flowed to”(for definition 5.11) the boundary of this new vertex space. In fact, their flows are same.

  • Finally, we prove Theorem 1.2.

2. Preliminaries

For definitions and basic properties of hyperbolic metric spaces, Gromov hyperbolic groups and its boundary one may refer to [6] and [11]. For a quick review of limit points and results pertaining to it, one may refer to [16].

Definition 2.1.

[8] Suppose a group acts on a hyperbolic metric space . Then is called a conical limit point if for a geodesic ray asymptotic to and any , there exists a constant such that and .

We denote the subset of the limit set of consisting of conical limit points by .

Lemma 2.2.

Suppose is a Gromov hyperbolic group and let be a subset of . Then for every ,

  1. ;

  2. .

General version of Lemma 2.2 is given in [16].

Definition 2.3.

Let be proper hyperbolic metric spaces and let be a proper embedding. A Cannon-Thurston(CT) map is a continuous extension of , .

Here, and i.e. their respective visual compactifications. We denote by .

Definition 2.4.

Suppose is a Gromov hyperbolic group. Let be a collection of subgroups of . is said to have limit intersection property if for every , , .

Theorem 2.5 (Arzela Ascoli Theorem).

[6] Let and be points in a proper geodesic metric space . Suppose that there is a unique geodesic segment joining to in ; let be a linear parameterization of this segment. Let be linearly reparameterized geodesics in , and suppose that the sequences of points and . Then uniformly in compact-open topology.

3. Relative hyperbolicity

The notion of relatively hyperbolic groups was introduced by Gromov in his article [10] on hyperbolic groups. [10],[9] and [5] provide a good reference to the various notions of relative hyperbolicity. Mj and Reeves provide a modification of electrocution in Farb’s strong relative hyperbolicity in [14].

Notations:

Suppose is a geodesic metric space hyperbolic relative to .

  • The coned-off metric space is denoted by or . The coned-off metric is denoted by .

  • The space obtained by attaching hyperbolic cones, as in Gromov’s definition is denoted by or , with the hyperbolic metric .

  • For a group , its Cayley graph is denoted by

Below, we list some results that will be used in the paper.

.

We have the following theorem due to Bowditch [5].

Theorem 3.1.

The following are equivalent:

  1. X is hyperbolic relative to the collection of uniformly separated subsets in .

  2. X is hyperbolic relative to the collection of uniformly separated subsets in in the sense of Gromov.

  3. is hyperbolic relative to the collection .

Definition 3.2.

Suppose is hyperbolic relative to a subgroup . Then the relative hyperbolic boundary of with respect to is the boundary of , and it will be denoted by .

The subgroup and its conjugates are called parabolic subgroups. In , each hyperbolic cone has a single limit point in and it is called a parabolic limit point.

Lemma 3.3.

[[15], Lemma 1.2.19]

Let be a geodesic metric space hyperbolic relative to a collection of uniformly -separated, uniformly properly embedded closed subsets, in the sense of Gromov. Then is properly embedded in i.e, for all , there exists such that implies , for every . Here is the inclusion map.

Lemma 3.4.

([14], Lemma 2.10) Let be a geodesic metric space and let be a collection of subsets of such that is relatively hyperbolic. Let be a collection of -hyperbolic metric spaces and be a collection of uniformly coarse Lipschitz maps . (Indexing sets of are the same.) Let be the corresponding partially electrocuted space. Then is a hyperbolic metric space and are uniformly quasiconvex subsets.

Lemma 3.5.

([13], Lemma 1.12, Lemma 1.21)

  1. An electric geodesic in and a relative geodesic in joining the same pair of points have similar intersection patterns with for all , i.e. they track each other off horosphere-like sets.

  2. An electric geodesic in (after identification with ) and a geodesic in joining the same pair of points in have similar intersection patterns with for all , i.e. they track each other off horoball-like sets.

    Same holds if we replace geodesic by quasigeodesic.

  3. Let be a -quasigeodesic in and be a - partially electrocuted quasigeodesic in joining the same pair of points. Then there exists such that lies in a -neighbourhood of in . Also, they track each other outside a -neighbourhood of horoball-like sets that meets.

Lemma 3.6.

([15], Lemma 1.2.31) Let , , , . Suppose are geodesic spaces and are collections of -separated and intrinsically geodesic closed subspaces of respectively. Let be a -quasi-isometry such that for each , there exists such that in and in . Then induces

  1. a -quasi-isometry , for some , ;

  2. a -quasi-isometry , for some , .

4. Graphs of Groups

For details of graph of groups one may refer to [17]

Definition 4.1.

A graph is an ordered pair of sets with , the set of vertices of and a set , the set of edges of , and a pair of maps

and

satisfying the following conditions: , and for all .

Here, is the initial vertex of the edge and is the terminal vertex; is the inverse of , i.e. the edge with the opposite orientation.

Definition 4.2.

A graph of groups consists of a finite graph , group associated to each vertex and group associated to each edge along with the following monomorphisms:

with the extra condition that .

Definition 4.3.

Let be a graph of groups. Let be a maximal subtree of . Then the fundamental group of is defined in terms of generators and relators as:

Generators are the disjoint union of generating sets of the vertex groups and the elements of .

Relators are the following:

  • relators from the vertex groups;

For each , we fix the generating set of to be and , we fix the generating set of to be such that . Then

(4.1)

is a generating set of . .

Now we recall the definition of Bass-Serre tree associated to this graph of groups.

Definition 4.4 ([17], Section 5.3, Section 5.4).

The Bass-Serre tree is a tree with vertex set and edge set . Here, .

So, for an edge , and .

Definition 4.5.

[13], [16]

Let be a graph of groups and be a maximal subtree. Let be the fundamental group and be a generating set of , from 4.1.

A tree of metric spaces is the union of the following vertex and edge graphs connected by the edges from (3).

  1. For all , is a subgraph of with and are connected by an edge if . Let be the induced path metric. is called a vertex space.

  2. For all , is a subgraph of with and are connected by an edge if . Let be the induced path metric. is called an edge space.

  3. For any edge connecting vertices and , for every , we join to and by edges of length .

    These extra edges give us maps and with and . These maps are quasi-isometric embeddings and this is called the qi-embedded condition.

    This is a tree of metric spaces, , from . acts on it by a proper cocompact action of and admits a simplicial Lipschitz -equivariant map .

    Further, is a tree of relatively hyperbolic metric spaces, if the following are also satisfied:

  4. Each group , , is hyperbolic relative to a collection of subgroups . Then for each vertex , is hyperbolic relative to , a collection of subsets that are images of left cosets of in under left multiplication by , i.e,

    .

    These are called horosphere-like sets.

  5. Each group , , is hyperbolic relative to a collection of subgroups . Let . Then for each edge , is hyperbolic relative to a collection of subsets , which are images of left cosets of in under left multiplication by , i.e, . These are the horosphere-like sets.

  6. The coned-off vertex spaces , and the coned-off edge spaces ,) are uniformly hyperbolic metric spaces.

  7. maps are strictly type-preserving i.e.  is either some or empty and for every , there exists and such that .

  8. The maps induced by , ,, are uniform quasi-isometric embeddings. This is called the qi-preserving electrocution condition.

This tree of coned-off metric spaces is called the induced tree of coned-off spaces from the graph of groups . We denote it by .

We recall the following from [16]: Let be fixed. Then, . Let denote the identity element of . By Milnor-Schwarz lemma, the orbit map given by is a quasi-isometry.

Remark 4.6.
  1. There exists a constant such that for every vertex space , .([16], Lemma 3.5)

    For any , . Let denote the identity element in . Suppose be a geodesic joining to in . Then is a path joining to in , for every . We choose .

  2. Let . induces a quasi-isometry . For each , we map to such that . This map is coarsely well-defined.

    By Lemma 3.6, induces a quasi-isometry and induces a quasi-isometry .

Definition 4.7.

The cone locus of is defined as a graph with the vertex set consisting of cone points in the vertex spaces, denoted by and the edge set consists of the cone points in the edge spaces, denoted by . So, we have edges with end points and identified with the appropriate ’s. The incidence relations depend on the incidence relations of .

By the mappings between horosphere-like sets, the cone locus is a forest.

Each connected component of the cone locus is called a maximal cone-subtree denoted by and let be the collection of maximal cone-subtrees.

Corresponding to each , we get a subtree of horosphere-like sets and denote it by . We call it the maximal cone-subtree of horosphere-like spaces. Let be the collection of ’s.

Let be the induced tree of horosphere-like sets and be the collection of ’s.

Thus, we have a partially electrocuted space and denote it by .

Lemma 4.8.

([13], Lemma 1.24) Suppose is hyperbolic relative to the collection . Then is a hyperbolic metric space.

Denote by , the quotient space obtained by attaching hyperbolic cones to by identifying with for all .

Recall from [13] that the inclusion induces a uniform proper embedding i.e. for every , there exists such that for any vertex and , implies that .

Suppose for every , the inclusion map is a proper embedding, then the induced map is also a proper embedding.

5. Proof of the Theorem

One of the most important results we use is the existence of CT map.

Theorem 5.1.

[13] For each , CT map exists for the inclusion map

.

We have the following lemmas from [12] which are useful as we proceed further.

Lemma 5.2.

Given , there exists , such that if are vertices of a -hyperbolic metric space , with , and then lies in a -neighbourhood of any geodesic joining and .

Lemma 5.3.

Let be a -hyperbolic metric space and be a geodesic segment in . Let be the nearest point projection map. Then for every , where depends only on .

Lemma 5.4.

(Nearest point projection and quasi-isometry almost commute)

Let and be -hyperbolic metric spaces. Let be a geodesic joining , in and let . Let be a -quasi-isometry. Let be a geodesic joining and . Then for some which depends on , and .

We recall few tools required for the proof from [13].

5.1. Hyperbolic ladder

By Lemma 3.6, the maps induced by are also qi-embeddings. Thus, is a -quasiconvex subset of .

Let , with from Lemma 5.2. Let be the constant from Lemma 5.2.

Definition 5.5.

Let and be the initial and terminal vertices of the edge in . We define a map such that for if there exists with , then .

We recall the construction of the ladder. We have a tree of metric spaces . Fix the vertex as the base point. Let be a vertex of .

Let be a geodesic such that the initial and final point lie outside the horosphere-like sets. Consider the set of all edges incident on except for the edge lying in the geodesic joining to in . Among them, choose the collection of all edges such that . From each such set , choose , such that is maximal. Further, choose a subcollection of edges , with , such that .

For each , let be the edge joining to . Let be the electric geodesic in joining and .

Then,

Here, is the electric geodesic joining and in .

The convex hull of is a subtree of .

Now, suppose we have constructed . Let and let , where is a geodesic in .

Then

The ladder

Convex hull of is a subtree of and we denote it by .

Remark 5.6.

One can construct a hyperbolic ladder in the simialr way for a geodesic ray in , as well.

5.1.1. Retraction map

Let be a geodesic metric space hyperbolic relative to . It is easy to see that the inclusion of in is an isometric embedding with in the electric metric. Hence, these spaces are quasi-isometric.

Let (identified with ).

Let be an electric geodesic in and let be its electro-ambient quasigeodesic in . Let be a nearest point projection of onto .

Definition 5.7.

Electric projection is the map given by:

If is a cone point of a horosphere like set , choose some and define

Lemma 5.8.

([13], Lemma 1.16) Let be hyperbolic relative to . There exists a constant depending upon , , such that for any and and a geodesic in , then .

This implies that the electric projection is coarsely well-defined.

Definition 5.9.

For each , let be an electric projection of onto , for each vertex .

The retraction map is defined by :

If , we choose such that .

Then,

Theorem 5.10.

([13], Theorem 2.2) If is hyperbolic, then is uniformly quasiconvex (independent of ).

5.2. Vertical quasigeodesic rays

Let be an electric geodesic in with its endpoints outside all horospheres. Let be its electro-ambient quasigeodesic. We have the ladder .

For any electric geodesic , we denote the union of subsegments of lying outside the horosphere-like sets by . This gives us .

For any , there exists such that . Let be the geodesic in with and .

A vertical quasigeodesic ray starting at is a map satisfying the following for a constant :

for all . Here, is the original metric on .

Clearly, and .

5.3. Proof of the theorem

Let , be vertices connected by an edge . We know that the induced maps and are qi-embeddings and they induce the embeddings and . This gives a partially defined map from to with the domain restricted to . However, we denote the map simply by i.e  .

Definition 5.11.

Let and . Then we say is a flow of and that can be flowed into .

If and is the sequence of consecutive vertices in the geodesic in then we say can be flowed into if there exists such that and for . And is called a flow of .

Note : Let and let be the CT map. Then by saying is a conical limit point of we mean that it is a conical limit point for the subgroup acting on .

Lemma 5.12.

Suppose can be flowed to and let be the flow. Then and map to the same limit point in under the respective CT maps.

Proof.

It is enough to prove the case when and are adjacent vertices. Let be the edge in joining to .

By the definition of flow, .

There exists such that , for .

Let be a sequence in with as .

Then, for , is a sequence in with as and .

This implies that .

So, under the CT map, both and map to the same element of . ∎

Conversely, we have the following proposition:

Proposition 5.13.

Let . For both , suppose map to the limit point under the CT map such that is a conical limit point for both and . Then there exists such that and can be flowed to .

Proof.

Assume cannot be flowed into and cannot be flowed into . Then there exists such that can be flowed till in the direction of and can be flowed till in the direction of .

Figure 1. Case when .

Case 1: Suppose .

Here, take to be .

Case 2: Suppose .

We will show that this is not possible. We prove by contradiction.

Using Lemma 5.12, without loss of generality, assume