Splittings and automorphisms of relatively hyperbolic groups
We study automorphisms of a relatively hyperbolic group . When is one-ended, we describe using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when is toral relatively hyperbolic, is virtually built out of mapping class groups and subgroups of fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups.
Given a malnormal quasiconvex subgroup of a hyperbolic group , we view as hyperbolic relative to and we apply the previous analysis to describe the group of automorphisms of that extend to : it is virtually a McCool group. If is infinite, then is a vertex group in a splitting of . If is torsion-free, then is of type VF, in particular finitely presented.
We also determine when is infinite, for relatively hyperbolic. The interesting case is when is infinitely-ended and has torsion. When is hyperbolic, we show that is infinite if and only if splits over a maximal virtually cyclic subgroup with infinite center. In general we show that infiniteness of comes from the existence of a splitting with infinitely many twists, or having a vertex group that is maximal parabolic with infinitely many automorphisms acting trivially on incident edge groups.
This paper studies automorphisms of hyperbolic and relatively hyperbolic groups in relation with their splittings. The first result in this direction is due to Paulin [Pau_arboreal]: if is a hyperbolic group with infinite, then has an action on an -tree with virtually cyclic (possibly finite) arc stabilizers. Rips theory then implies that splits over a virtually cyclic group.
Understanding the global structure of requires techniques which depend on the number of ends of . If is one-ended, there is an -invariant JSJ decomposition, and its study leads to Sela’s description of as a virtual extension of a direct product of mapping class groups by a virtually abelian group [Sela_structure, Lev_automorphisms]. If has infinitely many ends, one does not get such a precise description because there is no -invariant splitting. One may study by letting it act on a suitable space of splittings, the most famous being Culler-Vogtmann’s outer space for .
Before moving on to relatively hyperbolic groups, here is a basic problem about which we get new results in the context of hyperbolic, and even free, groups. Given a finitely generated subgroup of a group , consider the group consisting of outer automorphisms of which extend to automorphisms of . What can one say about this group ? For instance, is it finitely generated? finitely presented? This question was asked by D. Calegari for automorphisms of free groups, and we answer it when is malnormal.
The answer is related to splittings through the following simple remark: if is a vertex group in a splitting of , say , then any automorphism of which acts trivially (i.e. as conjugation by some element ) on each incident edge group , extends to (this is the algebraic translation of the fact that any self-homeomorphism of a closed subset which is the identity on the frontier of extends by the identity to a homeomorphism of ).
The following theorem says that this phenomenon accounts for almost all of .
Theorem 1.1 (see Corollary LABEL:cor_induit).
Let be a quasiconvex malnormal subgroup of a hyperbolic group . If is infinite, then is a vertex group in a splitting of , and the group of outer automorphisms of acting trivially on incident edge groups has finite index in .
We call the group of outer automorphisms of acting trivially on a family of subgroups a McCool group of , because of McCool’s paper about subgroups of fixing a finite set of conjugacy classes [McCool_fp]. In this language, Theorem 1.1 says that is virtually a McCool group of . It is a theme of this paper that many groups of automorphisms may be understood in terms of McCool groups, and that many results concerning the full group also apply to McCool groups (see also [GL_McCool]).
The groups considered by McCool are finitely presented [McCool_fp]. In fact, they have a finite index subgroup with a finite classifying space [CuVo_moduli]. In [GL_McCool] we extend these results to all McCool groups of torsion-free hyperbolic groups (and more generally of toral relatively hyperbolic groups). From this, one deduces that has a finite index subgroup with a finite classifying space when and are as in Theorem 1.1, with torsion-free.
Our hypotheses for Theorem 1.1, namely quasiconvexity and malnormality of ,
imply that is hyperbolic relative to (see [Bow_relhyp]).
In fact, Theorem 1.1 is just a special case of a result describing
as a virtual McCool group when is relatively hyperbolic
and is a maximal parabolic subgroup (see Theorem LABEL:caleg_general for a precise statement).
This paper also addresses the question of whether is finite or infinite. It turns out that the answer is much simpler when is torsion-free, owing to the fact that is then infinite whenever has infinitely many ends (see Lemma LABEL:freep).
Things are more complicated when torsion is allowed. For instance, characterizing virtually free groups with infinite is a non-trivial problem which was solved by Pettet [Pettet_virtually]. The following theorem gives a different characterization. We say that a subgroup of is if it is maximal for inclusion among virtually cyclic subgroups with infinite center.
Theorem 1.2 (see Theorems LABEL:thm_twist_hyp and LABEL:thm_MC_infini).
Let be a hyperbolic group. Then is infinite if and only splits over a subgroup ; in this case, any element of infinite order defines a Dehn twist which has infinite order in .
Moreover, one may decide algorithmically whether is finite or infinite.
The first assertion answers a question asked by D. Groves. See [Carette_automorphism] for a related result proved independently by M. Carette, and [Lev_automorphisms, DG2] for the one-ended case.
Let us now consider (in)finiteness of when is relatively hyperbolic. Suppose that is hyperbolic with respect to a finite family of finitely generated subgroups . Since automorphisms of need not respect (for instance, may be free and may consist of any finitely generated malnormal subgroup), we consider the group consisting of automorphisms mapping each to a conjugate (in an arbitrary way). Note that has finite index in the full group when the groups are small but not virtually cyclic, more generally when they are not themselves relatively hyperbolic in a nontrivial way [MiOs_fixed].
Given a splitting of , we have already pointed out that any automorphism of a vertex group acting trivially on incident edge groups extends to an automorphism of . Twists around edges of the splitting also provide automorphisms of . For instance, if , and centralizes , there is an automorphism of equal to conjugation by on and to the identity on . Note that we do not require that be virtually cyclic or that (see Subsection 2.6).
The following result says that infiniteness of comes from twists or from a McCool group of a parabolic group.
Theorem 1.3 (see Corollary LABEL:cor_outu_infini).
Let be hyperbolic relative to a family of finitely generated subgroups. Then is infinite if and only if has a splitting over virtually cyclic or parabolic subgroups, with each contained in a conjugate of a vertex group, such that one of the following holds:
the group of twists of the splitting is infinite;
some is a vertex group and there are infinitely many outer automorphisms of acting trivially on incident edge groups.
As mentioned above, one can get similar results characterizing the infiniteness of McCool groups of .
We refer to Section LABEL:outinfi, in particular Theorem LABEL:thm_marked and Corollary LABEL:cor_outu_infini,
for more detailed statements.
Let us now discuss the techniques that we use. We assume that is hyperbolic relative to , and we distinguish two cases according to the number of ends (technically, we consider relative one-endedness, but we will ignore this in the introduction).
When is one-ended, we use a canonical -invariant decomposition of , namely (see Subsection 3.3) the JSJ decomposition over elementary (i.e. parabolic or virtually cyclic) subgroups relative to parabolic subgroups (i.e. parabolic subgroups have to be contained in conjugates of vertex groups).
One may thus generalize the description of given by Sela for hyperbolic, and express in terms of mapping class groups, McCool groups of maximal parabolic subgroups, and a group of twists . For simplicity we restrict to a special case here (see Section 4 for a general statement).
Theorem 1.4 (see Corollary 4.4).
Let be toral relatively hyperbolic and one-ended. Then some finite index subgroup of fits in an exact sequence
where is finitely generated free abelian, is the group of isotopy classes of homeomorphisms of a compact surface mapping each boundary component to itself in an orientation-preserving way, and is the group of automorphisms of fixing the first generators.
More generally, McCool groups of a one-ended toral relatively hyperbolic group have a similar description (see Corollary 4.9). A more general statement (without restriction on the parabolic subgroups) is given in Theorems 4.3 and 4.6.
We also show that the modular group of , introduced by Sela [Sela_hopf, Sela_diophantine1] and usually defined by considering all suitable splittings of , may be seen on the single splitting . We refer to Section 5 for details.
To prove Theorem 1.1 when is one-ended, one applies the previous analysis, viewing as hyperbolic relative to . Note that we use a JSJ decomposition which is relative (to ), and over subgroups which are not small (any subgroup of is allowed).
Another example of the usefulness of relative JSJ decompositions is to prove the Scott conjecture about fixed subgroups of automorphisms of free groups. The proof that we give in Section LABEL:fixed, though not really new, is simplified by the use of the cyclic JSJ decomposition relative to the fixed subgroup.
We therefore work consistently in a relative context. We fix another family of finitely generated subgroups , and we define as the group of automorphisms mapping to a conjugate (in an arbitrary way) and acting trivially on (i.e. as conjugation by an element of ).
In order to understand the structure of the automorphism group of a one-ended relatively hyperbolic group from its canonical JSJ decomposition, one needs to control automorphisms of rigid vertex groups. This is made possible by the following generalisation of Paulin’s theorem mentioned above:
Theorem 1.5 (see Theorem 3.9).
Let be hyperbolic relative to , with finitely generated and . Let be another family of finitely generated subgroups.
If is infinite, then splits over an elementary (virtually cyclic or parabolic) subgroup relative to .
Note that there is no quasiconvexity or malnormality assumption on groups in , but the automorphisms that we consider have to act trivially on them (see also Remark LABEL:lent).
The theorem is proved using the Bestvina-Paulin method (extended to relatively hyperbolic groups in [BeSz_endomorphisms]) to get an action on an -tree , and then applying Rips theory as developed in [BF_stable] to get a splitting. There are technical difficulties in the second step because may fail to be finitely presented (the ’s are not required to be finitely presented), and the action on may fail to be stable if the ’s are not slender; it only satisfies a weaker property which we call hypostability, and in the last section we generalize [BF_stable] to hypostable actions of relatively finitely presented groups.
Theorem 1.5 explains why McCool groups appear in Theorems 1.3 and 1.4. Indeed, given a rigid vertex group in a JSJ decomposition of a one-ended group, Theorem 1.5 implies that only finitely many outer automorphisms of extend to automorphisms of . In turn, this implies that, after passing to a finite index subgroup, automorphisms of act trivially on edge groups of the JSJ decomposition. See Subsection 4.1 for details.
When is not one-ended, one has to consider splittings over finite groups. We do not have an exact sequence as in Theorem 1.4 because there is no -invariant splitting. In order to prove Theorems 1.2 and 1.3, we use the tree of cylinders introduced in [GL4] to obtain a non-trivial splitting over finite groups which is invariant or has an infinite group of twists (Corollary LABEL:propc_induct).
The paper is organized as follows. Section 2 consists of preliminaries (JSJ decompositions, automorphisms of a tree, trees of cylinders). Section 3 contains generalities about relatively hyperbolic groups. We point out that vertex groups of a splitting over relatively quasiconvex subgroups are relatively quasiconvex, and that the canonical JSJ decomposition has finitely generated edge groups. In Section 4 we study the structure of the automorphism group of a one-ended relatively hyperbolic group. Section 5 is devoted to the modular group. In Section LABEL:sec_induced we study extendable automorphisms; Theorem 1.1 is a special case of Theorem LABEL:caleg_general. Section LABEL:outinfi is devoted to the question of whether groups like and are finite or infinite. Section LABEL:fixed contains a proof of the Scott conjecture, and a partial generalization to relatively hyperbolic groups. Theorem 1.5 is proved in Section LABEL:sec_Rips.
Acknowledgments. We thank D. Groves and D. Calegari for asking stimulating questions, and the organizers of the 2007 Geometric Group Theory program at MSRI where this research was started. We also thank the referee for helpful comments. This work was partly supported by ANR-07-BLAN-0141, ANR-2010-BLAN-116-01, ANR-06-JCJC-0099-01, ANR-11-BS01-013-01.
Unless mentioned otherwise, will always be a finitely generated group.
Given a group and a subgroup , we denote by the center of , by the centralizer of in , and by the normalizer of in . We write for .
A subgroup is malnormal if is trivial for all , almost malnormal if is finite for all .
A group is virtually cyclic if it has a cyclic subgroup of finite index; it may be finite or infinite. Its outer automorphism group is finite.
A group is slender if and all its subgroups are finitely generated. We say that is small if it contains no non-abelian free group (see [BF_bounding] for a slightly weaker definition).
Let be a family of subgroups . In most cases, will be a finite collection of finitely generated groups .
The group is finitely presented relative to if it is the quotient of by the normal closure of a finite subset, with a finitely generated free group. If is finitely presented relative to , and if , then is also finitely presented relative to .
Let be a finitely generated subgroup. If is finitely presented relative to , then is finitely presented relative to ; the converse is not true in general.
2.1 Relative automorphisms
Given and , we denote by the subgroup of consisting of automorphisms mapping each to a conjugate, and by its image in . If , we use the notations and . We also write for .
We define by restricting to automorphisms whose restriction to each agrees with conjugation by some element of (we call them marked automorphisms, or automorphisms acting trivially on ). An outer automorphism belongs to if and only if, for each , it has a representative equal to the identity on .
The group is denoted by PMCG in [Lev_automorphisms], by in [DG2], and is called a (generalized) McCool group in [GL_McCool].
Note that is trivial, and that has finite index in if is a finite collection of finite groups (more generally, of groups with finite).
If is another family of subgroups, we define . We allow or to be empty, in which case we do not write it.
The groups defined above do not change if we replace each or by a conjugate, or if we add conjugates of the ’s to or conjugates of the ’s to .
A splitting of a group is an isomorphism between and the fundamental group of a graph of groups . Equivalently, using Bass-Serre theory, we view a splitting of as an action of on a simplicial tree , with . This tree is well-defined up to equivariant isomorphism, and two splittings are considered equal if there is an equivariant isomorphism between the corresponding Bass-Serre trees.
The group acts on the set of splittings of (by changing the isomorphism between and , or precomposing the action on ).
Trees will always be simplicial trees with an action of without inversion. We usually assume that the splitting is minimal (there is no proper -invariant subtree). Since is assumed to be finitely generated, this implies that is a finite graph.
A splitting is trivial if fixes a point in (minimality then implies that is a point).
A splitting is relative to if every is conjugate to a subgroup of a vertex group, or equivalently if is elliptic (i.e. fixes a point) in . If , we also say that the splitting is relative to .
The group splits over a subgroup (relative to ) if there is a non-trivial minimal splitting (relative to ) such that is an edge group. The group is one-ended relative to if it does not split over a finite group relative to .
If is a graph of groups, we denote by its set of vertices, and by its set of oriented edges. The origin of an edge is denoted by . A vertex or an edge carries a group or , and there is an inclusion
2.3 Trees and deformation spaces [For_deformation, Gl2]
In this subsection we consider trees rather than graphs of groups. We denote by or the stabilizer of a vertex or an edge.
We often restrict edge stabilizers of by requiring that they belong to a family of subgroups of which is stable under taking subgroups and under conjugation. We then say that is an -tree. For instance, may be the set of finite subgroups, of cyclic subgroups, of abelian subgroups, of elementary subgroups of a relatively hyperbolic group (see Section 3). We then speak of cyclic, abelian, elementary trees (or splittings).
Besides restricting edge stabilizers, we also often restrict to trees relative to : every is elliptic in . We then say that is an -tree.
A tree is a collapse of if it is obtained from by collapsing each edge in a certain -invariant collection to a point; conversely, we say that refines . In terms of graphs of groups, one passes from to by collapsing edges; for each vertex , the vertex group is the fundamental group of the graph of groups occuring as the preimage of in .
Given two trees and , we say that dominates if there is a -equivariant map , or equivalently if every subgroup which is elliptic in is also elliptic in . In particular, dominates any collapse .
Two trees belong to the same deformation space if they dominate each other. In other words, a deformation space is the set of all trees having a given family of subgroups as their elliptic subgroups. We denote by the deformation space containing a tree , and by the group of automorphisms leaving invariant. The set of -trees contained in a deformation space is called a deformation space over (and usually denoted by also).
A tree is reduced if whenever an edge has its endpoints in different -orbits. Equivalently, no tree obtained from by collapsing the orbit of an edge belongs to the same deformation space as . If is not reduced, one may collapse edges so as to obtain a reduced tree in the same deformation space.
Any two reduced trees in a deformation space over finite groups may be joined by slide moves (see [For_deformation, GL2] for definitions). In particular, they have the same set of edge and vertex stabilizers.
2.4 Induced structures
Definition 2.1 (Incident edge groups ).
Given a vertex of a graph of groups , we denote by the collection of all subgroups of , for an edge with origin . We call the set of incident edge groups. We also use the notation .
Similarly, if is a vertex of a (minimal) tree, there are finitely many -orbits of incident edges, and is the family of stabilizers of some representatives of these orbits. This is a finite collection of subgroups of , each well-defined up to conjugacy.
Any splitting of relative to extends (non-uniquely) to a splitting of , whose edges are those of together with those of ; an edge of incident to is attached to a vertex of whose group contains (up to conjugacy). We call this refining at using . One recovers from by collapsing edges of .
Consider subgroups such that, if and , then (this holds in particular when is a vertex stabilizer of a tree , and is a subgroup which fixes no edge of ).
If is conjugate to in , it is conjugate to in .
If leaves invariant and maps to , then . ∎
This lemma is trivial, but very useful. Given a vertex stabilizer of a tree , it allows us to define a family as follows (like , it is a finite set of subgroups of , each well-defined up to conjugacy).
Definition 2.3 (Induced structure ).
Let be a collection of subgroups of , and let be a vertex stabilizer in a tree relative to . For each such that fixes a point in the orbit of , but fixes no edge of , let be a conjugate of . When defined, is unique up to conjugacy in by Lemma 2.2. We define as this collection of subgroups ; we define similarly if is a vertex group of .
Given and , one of the following always holds: fixes a vertex of not in the orbit of , or some conjugate of fixes and an edge incident to , or is conjugate to a group in .
In this definition, depends not only on and , but also on the incident edge groups of . In practice, we will not work with alone, but with . This is the case for instance in the following lemma.
If is a splitting of relative to , refining at using yields a splitting of which is relative to .
The refinement is possible because is relative to . Each is elliptic in by Remark 2.4. ∎
2.5 JSJ decompositions [GL3a]
Fix as in Subsection 2.3, and a (possibly empty) family of subgroups. All trees considered here are -trees.
A subgroup is universally elliptic if it is elliptic (fixes a point) in every tree. A tree is universally elliptic if its edge stabilizers are. A tree is a JSJ tree over relative to if it is universally elliptic and dominates every universally elliptic tree (see Section 4 of [GL3a]). JSJ trees exist if is finitely presented relative to . The set of JSJ trees, if non-empty, is a deformation space called the JSJ deformation space over relative to . When is the set of cyclic (abelian, elementary…) subgroups, we refer to the cyclic (abelian, elementary…) JSJ.
When is the set of finite subgroups, and , the JSJ deformation space is the Stallings-Dunwoody deformation space, characterized by the property that its trees have vertex stabilizers with at most one end (see Section 6 of [GL3a]). Because it is a deformation space over finite groups, all its reduced trees have the same edge and vertex stabilizers (see Subsection 2.3). This space exists if and only if is accessible, in particular when is finitely presented [Dun_accessibility]. If is torsion-free, the Stallings-Dunwoody deformation space is the Grushko deformation space; edge stabilizers are trivial, vertex stabilizers are freely indecomposable and non-cyclic.
More generally, the JSJ deformation space over finite subgroups relative to will be called the Stallings-Dunwoody deformation space relative to . We will also consider JSJ spaces over finite subgroups of cardinality bounded by some ; these exist whenever is finitely generated by Linnell’s accessibility [Linnell].
If is a tree (in particular, if it is a JSJ tree), a vertex stabilizer of not belonging to (or itself) is rigid if it is universally elliptic. Otherwise, (or ) is flexible. In many situations, flexible vertex stabilizers of JSJ trees are quadratically hanging subgroups (see Section 7 of [GL3a]).
Definition 2.7 (QH vertex).
A vertex stabilizer (or ) is quadratically hanging, or QH, (relative to ) if there is a normal subgroup (called the fiber of ) such that is isomorphic to the fundamental group of a hyperbolic -orbifold (usually with boundary); moreover, if is an incident edge stabilizer, or is the intersection of with a conjugate of a group in , then the image of in is finite or contained in a boundary subgroup (a subgroup conjugate to the fundamental group of a boundary component).
Definition 2.8 (Full boundary subgroups).
Let be a QH vertex stabilizer. For each boundary component of , we select a representative for the conjugacy class of its fundamental group in , and we consider its full preimage in . This defines a finite family of subgroups of .
If is QH with finite fiber, every infinite incident edge stabilizer is virtually cyclic and (up to conjugacy in ) contained with finite index in a group of . If is one-ended relative to , then every incident edge stabilizer is infinite, so is contained in a group of .
If is a flexible QH vertex stabilizer with finite fiber, and is universally elliptic, then the image of in is finite or contained in a boundary subgroup (see Proposition 7.6 of [GL3a]; this requires a technical assumption on , which holds in all cases considered in the present paper). In particular, is virtually cyclic.
2.6 The automorphism group of a tree [Lev_automorphisms]
Let be a tree with a minimal action of . We assume that is not a line with acting by translations.
We denote by the group of automorphisms leaving invariant: there exists an isomorphism which is -equivariant in the sense that for and .
Following [Lev_automorphisms], we describe the image of in in terms of the graph of groups . Our assumptions on imply that is minimal, and is not a mapping torus (as defined in [Lev_automorphisms]).
The group acts on the finite graph , and we define as the finite index subgroup acting trivially. We use the notations , and (see Section 2.1).
Action on vertex groups.
If is a vertex of , there is a natural map defined as follows. Let . When acts on by inner automorphisms, in particular when fixes a unique point in (in this case ), one defines simply by choosing any representative of leaving invariant and considering its restriction to . In general, one has to choose more carefully: one fixes a vertex of mapping to such that the stabilizer of is , one chooses so that fixes , and is represented by the restriction .
If is an edge of , one may define similarly.
Let be the product map. As pointed out in Subsection 2.3 of [Lev_automorphisms], the image contains and is contained in . More precisely:
Let be two families of subgroups of . Let be a tree relative to . Then
for every , and
If is finite for all edges (resp. for all edges incident to ), all inclusions (resp. all inclusions in (1)) have images of finite index.
Recall that was defined in Definition 2.3.
The inclusion is proved in [Lev_automorphisms] by extending any “by the identity” to get , with , acting as a conjugation on each edge group and on each for . The left hand side inclusions in the lemma follow from Remark 2.4.
The inclusion follows from the fact that, given an edge of containing the lift of used to define , any has a representative such that fixes ; this representative induces an automorphism of leaving invariant. To prove the right hand side inclusions, apply Lemma 2.2 with , recalling that groups in or fix a unique point in .
If is finite for all incident edge groups, has finite index in (see Proposition 2.3 in [Lev_automorphisms]). Intersecting with , we get that has finite index in . This conludes the proof. ∎
We can also consider automorphisms which do not leave invariant, but only leave some vertex stabilizer invariant. Assuming that equals its normalizer, there is a natural map and
As in Subsection 2.5 of [Lev_automorphisms], we now consider the kernel of the product map . It consists of automorphisms in having, for each , a representative in whose restriction to is the identity. If is relative to , the group is contained in .
To study , we need to introduce the group of twists associated to or equivalently to (we write or if there is a risk of confusion).
Let be a separating edge of with origin and endpoint . Then with and . Given , one defines the twist by around near as the (image in of the) automorphism of equal to the identity on and to conjugation by on . There is a similar definition in the case of an HNN extension : given , the twist by is the identity on and sends to .
The group is the subgroup of generated by all twists. It is a quotient of and is contained in . The following facts follow directly from Section 2 of [Lev_automorphisms].
If every is finite, then has finite index in .
Assume that every non-oriented edge of has an endpoint such that acts on by inner automorphisms (this holds in particular if is abelian, or if is malnormal in , or if is infinite and almost malnormal in ). Then . ∎
The kernel of the epimorphism from to is the image of a natural map
where is the set of non-oriented edges of (see Proposition 3.1 of [Lev_automorphisms]). The image of an element of is called a vertex relation at , the image of an element of is an edge relation.
For instance, if is a non-trivial amalgam , then is the image of the map sending to the class of the automorphism acting on as conjugation by and on as conjugation by . The kernel of is generated by the elements with and with (vertex relations), together with the elements with (edge relations).
Let be an edge of with origin . If and are finite, but is infinite, then the image of in is infinite. In particular, is infinite.
Note that is infinite if is infinite and is finite.
It is pointed out in [Lev_GBS, Lemma 3.2] that the image of in maps onto the quotient . Since and are commuting finite subgroups, this quotient is infinite. ∎
Let be a graph of groups with fundamental group , and a connected subgraph. We view as a graph of groups, with fundamental group and associated group of twists .
If is infinite, then so is .
Let be the set of oriented edges of . The projection from to obtained by keeping only the factors with is compatible with the vertex and edge relations, so induces an epimorphism from to . ∎
If is a graph of groups with infinite, there is an edge such that the graph of groups obtained by collapsing every edge except has infinite.
There is an edge such that has infinite image in . Twists of around are also twists of . ∎
2.7 Trees of cylinders [Gl4]
We recall some basic properties of the tree of cylinders (see Section 4 of [GL4] for details). Besides (and possibly ), we have to fix a conjugacy-invariant subfamily and an admissible equivalence relation on . Rather than giving a general definition, we describe the examples that will be used in this paper (with consisting of all finite, elementary, or abelian subgroups respectively):
consists of all finite subgroups of a fixed order , and is equality.
is relatively hyperbolic (see Section 3), consists of all infinite elementary subgroups (parabolic or loxodromic), and is co-elementarity: if and only if is elementary.
is a torsion-free CSA group, consists of all infinite abelian subgroups, and is commutation: if and only if is abelian (recall that is CSA if centralizers of non-trivial elements are abelian and malnormal).
Let be a tree with edge stabilizers in . We declare two (non-oriented) edges to be equivalent if . The union of all edges in an equivalence class is a subtree , called a cylinder of . Two distinct cylinders meet in at most one point. The tree of cylinders of is the bipartite tree such that is the set of vertices of which belong to at least two cylinders, is the set of cylinders of , and there is an edge between and in if and only if . In other words, one obtains from by replacing each cylinder by the cone on its boundary (defined as the set of vertices of belonging to some other cylinder). Note that may be trivial even if is not.
The tree is dominated by (in particular, it is relative to if is). It only depends on the deformation space containing (we sometimes say that it is the tree of cylinders of ). In particular, is invariant under any automorphism of leaving invariant.
The stabilizer of a vertex is the stabilizer of , viewed as a vertex of . The stabilizer of a vertex is the stabilizer of the equivalence class containing stabilizers of edges in , for the action of on by conjugation (see Subsection 5.1 of [GL4]). It is the normalizer of a finite subgroup in case 1, a maximal elementary (resp. abelian) subgroup in case 2 (resp. 3). Note that if and is its equivalence class.
The stabilizer of an edge of is ; it is elliptic in . In cases 2 and 3, belongs to . But, in case 1, it may happen that edge stabilizers of are not in , so we also consider the collapsed tree of cylinders obtained from by collapsing each edge whose stabilizer does not belong to (see Subsection 5.2 of [GL4]). It is an -tree if is, and .
3 Relatively hyperbolic groups
In this section we assume that is hyperbolic relative to a finite family of finitely generated subgroups; we say that is the parabolic structure.
The group is finitely generated. It is not necessarily finitely presented, but it is finitely presented relative to [Osin_relatively], so JSJ decompositions relative to exist. In particular, there is a deformation space of relative Stallings-Dunwoody decompositions (see Subsection 2.5).
We refer to [Hruska_quasiconvexity] for equivalent definitions of relative hyperbolicity. In particular, acts properly discontinuously on a proper geodesic -hyperbolic space (which may be taken to be a graph [GroMan_dehn]), the action is cocompact in the complement of a -invariant union of disjoint horoballs, and the ’s are representatives of conjugacy classes of stabilizers of horoballs. Any horoball has a unique point at infinity , and the stabilizer of (for the action of on ) coincides with the stabilizer of .
For each constant , one can change the system of horoballs so that any two distinct horoballs are at distance at least . Indeed, for each horoball with stabilizer defined by a horofunction , the function is another (well-defined) horofunction which is -equivariant; then is a new -invariant horoball such that for large enough. Doing this for a chosen horoball in each orbit, and extending by equivariance, one gets a system of horoballs at distance at least from each other.
A subgroup of is parabolic if it is contained in a conjugate of some , loxodromic if it is infinite, virtually cyclic, and not parabolic, elementary if it is parabolic or virtually cyclic (finite or loxodromic). Any small subgroup is elementary. The group itself is elementary if it is virtually cyclic or equal to a . We say that is an elementary subgroup of if it is elementary and contained in .
One may remove any virtually cyclic subgroup from , without destroying relative hyperbolicity (see e.g. [DrSa_tree-graded, Cor 1.14]). Conversely, one may add to a finite subgroup or a maximal loxodromic subgroup (see e.g. [Osin_elementary]). These operations do not change the set of elementary (or relatively quasiconvex, as defined below) subgroups, and it is sometimes convenient (as in [Hruska_quasiconvexity]) to assume that every is infinite. Any infinite is a maximal elementary subgroup.
The following lemma is folklore, but we have not found it in the literature.
Given a relatively hyperbolic group , there exists a number such that any elementary subgroup of cardinality is contained in a unique maximal elementary subgroup . There are finitely many conjugacy classes of non-parabolic finite subgroups.
We may assume that every is infinite. Let be elementary. The existence of is well-known if is infinite (see for instance [Osin_relatively]), so assume is finite. We also assume that the distance between any two distinct horoballs in is bigger than . Given , let be the set of points of which are moved less than by . It follows from Lemma 3.3 p. 460 of [BH_metric] (existence of quasi-centres) that is nonempty.
Arguing as in the proof of Lemma 1.15 page 407 and Lemma 3.3 page 428 in [BH_metric], one sees that any geodesic joining two points of , or a point of to a fixed point of in , is contained in .
If meets , properness of the action of on implies that there are only finitely many possibilities for up to conjugacy, so we can choose to ensure . This implies that is contained in a unique horoball of . This horoball is -invariant since horoballs are -apart, so fixes the point at infinity of and is contained in the maximal parabolic subgroup . In particular, is parabolic.
There remains to prove uniqueness of . It suffices to check that cannot fix any point in . If it did, a geodesic joining to a point of would be contained in and meet . This contradicts our choice of . ∎
Since maximal elementary subgroups are equal to their normalizer, we get:
Corollary 3.2 ([DrSa_groups, Lemma 4.20]).
Maximal elementary subgroups are uniformly almost malnormal: if has cardinality , then . ∎
Definition 3.3 (Hruska [Hruska_quasiconvexity]).
Let and be as above, and . A subspace is relatively -quasiconvex if, given , any geodesic has the property that lies in the -neighbourhood of . The space is relatively quasiconvex if it is relatively -quasiconvex for some . A subgroup is relatively quasiconvex if some (equivalently, every) -orbit is relatively quasiconvex in .
Let be hyperbolic relative to , with finitely generated. If acts on a simplicial tree relative to with relatively quasiconvex edge stabilizers, then vertex stabilizers are relatively quasiconvex.
The proposition applies in particular if edge stabilizers are elementary, since elementary subgroups are relatively quasiconvex. It was proved by Bowditch [Bo_cut, Proposition 1.2] and Kapovich [Kapovich_quasiconvexity, Lemma 3.5] for hyperbolic. We generalize Bowditch’s argument.
As usual, we assume that acts on minimally and without inversion. Since is finitely generated, the graph is finite. We also assume that is a connected graph, and edges of have length 1.
Since is elliptic in , and stabilizers of points of are finite, hence elliptic in , there exists an equivariant map sending vertices to vertices, mapping each edge linearly to an edge path, and constant on each horoball of .
For each edge of , let be the midpoint of , and . Let be a vertex of , and let be the set of edges of with origin . Let be the preimage under of the closed ball of radius around in . Note that for all and . Also note that by minimality of and connectedness of .
If for and , then fixes . Since acts cocompactly on , it follows that acts cocompactly on and acts cocompactly on . In particular, is the -orbit of a finite set. Relative quasiconvexity of implies that is relatively quasiconvex. Since is a finite graph, there exists a common constant such that all subsets are relatively -quasiconvex.
We now fix a vertex , and we show that is relatively quasiconvex. Choose . Since acts cocompactly on , the Hausdorff distance between and the -orbit of is finite, so it suffices to prove that is relatively quasiconvex. Let be a geodesic of joining two points of , and let be a maximal subgeodesic contained in . Considering the image of in , we see that both endpoints of belong to the same , for some . Thus is -close to , hence to . This shows that is relatively -quasiconvex. ∎
A relatively quasiconvex subgroup is relatively hyperbolic in a natural way ([Hruska_quasiconvexity, Theorem 9.1]). In particular:
If is an infinite vertex stabilizer of a tree with finite edge stabilizers, it is hyperbolic relative to the family of Definition 2.3.
This follows from Theorem 9.1 of [Hruska_quasiconvexity], adding finite groups belonging to to the parabolic structure if needed. ∎
3.3 The canonical JSJ decomposition
In this section we recall the description of the canonical relative JSJ decomposition. The content of the word canonical is that the JSJ tree (not just the JSJ deformation space) is invariant under automorphisms.
Let be hyperbolic relative to , and denote by the family of elementary subgroups. In this subsection we fix another (possibly empty) family of finitely generated subgroups and we assume that is one-ended relative to .
We consider the canonical -invariant JSJ tree defined (under the name ) in Theorem 13.1 of [GL3b] (see also Theorem 7.5 of [GL4]). It is the tree of cylinders (see Subsection 2.7) of the JSJ deformation space over elementary subgroups relative to