Acylindrical hyperbolicity of groups acting on trees
We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, one-relator groups, automorphism groups of polynomial algebras, -manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.
Key words and phrases:Acylindrically hyperbolic groups, groups acting on trees, one-relator groups, graph products, -manifold groups.
2010 Mathematics Subject Classification:Primary 20F67, 20F65, 20E08; secondary 20E34, 20E06, 57M05.
Recall that an isometric action of a group on a metric space is acylindrical if for every there exist such that for every two points with , there are at most elements satisfying
The notion of acylindricity goes back to Sela’s paper , where it was considered for groups acting on trees. In the context of general metric spaces, this concept is due to Bowditch . The following definition was suggested by the second author in .
A group is called acylindrically hyperbolic if there exists a generating set (possibly infinite) of such that the corresponding Cayley graph is hyperbolic, non-elementary (i.e., its Gromov boundary consists of more than points), and the action of on is acylindrical.
The above notion generalizes non-elementary word hyperbolic groups. Indeed, it is easy to see that a group is non-elementary word hyperbolic if and only if it satisfies the above definition for a finite set . For finite, the acylindricity condition is vacuous as proper and cobounded actions are always acylindrical. On the other hand, if we allow to be infinite, hyperbolicity and non-elementarity of are rather weak assumptions and acylindricity of the action turns out to be the condition that allows one to derive interesting results.
In fact, acylindrically hyperbolic groups were implicitly studied by Bestvina and Fujiwara , Bowditch , Hamenstädt , Dahmani, Guirardel, and Osin , Sisto , and many other authors before they were formally defined in . However the above-mentioned papers used different definitions stated in terms of hyperbolically embedded subgroups , weakly contracting elements , or various forms of (weakly) acylindrical actions [16, 18, 43]. Some nontrivial relations between these definitions were established in  and , and finally the equivalence of all definitions was proved in  (see Section 3 for details).
The class of acylindrically hyperbolic groups, denoted by , encompasses many examples of interest: non-(virtually cyclic) groups hyperbolic relative to proper subgroups, for , all but finitely many mapping class groups, non-(virtually cyclic) groups acting properly on proper CAT()-spaces and containing rank one elements, and so forth (see [29, 73]). On the other hand, is restricted enough to possess a non-trivial theory. Below we mention just few directions in the study of acylindrically hyperbolic groups; for a more comprehensive survey we refer the reader to .
Every acylindrically hyperbolic group contains non-degenerate hyperbolically embedded subgroups. This allows one to use methods from the paper  by Dahmani, Guirardel, and Osin to transfer a significant portion of the theory of relatively hyperbolic groups, including group theoretic Dehn filling, to the class . Despite their generality, the techniques from  are capable of answering open questions and producing new results even for well-studied classes of groups such as relatively hyperbolic groups, mapping class groups, and .
Acylindrical hyperbolicity shares many common features with “analytic negative curvature” as defined in  and . In particular, Hamenstädt  showed that is a subclass of the Monod–Shalom class (see also  and  for various generalizations). This opens doors for the Monod–Shalom measure rigidity and orbit rigidity theory . Results about quasi-cocycles on acylindrically hyperbolic groups with coefficients in the left regular representation obtained in [17, 43, 50] also seem likely to be useful in the study of the structure of group von Neumann algebras via a further generalization of methods from [24, 25].
Sisto  studied random walks on acylindrically hyperbolic groups using the language of weakly contracting elements. His main result covers many classical theorems about random elements of hyperbolic groups, mapping class groups, etc.
Yet another direction is explored by Hull , who developed a version of small cancellation theory for acylindrically hyperbolic groups and used it to construct groups with certain exotic properties.
The abundance of non-trivial results and techniques applicable to acylindrically hyperbolic groups justifies the quest for new examples, which is the main goal of our paper. We concentrate on examples arising from actions on simplicial trees. In particular, we consider fundamental groups of finite graphs of groups, one-relator groups, automorphism groups of polynomial algebras, fundamental groups of compact orientable -manifolds, and graph products. To illustrate usefulness of our main results we derive a number of corollaries about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes. However the main focus of this paper is on new examples rather than on applications.
Acknowledgements. The authors would like to thank Henry Wilton for helpful discussions of -manifolds. We are also grateful to Jack Button for his valuable comments, and to Stephane Lamy for pointing out an error in an earlier version of the paper. Finally, we would like to thank the referee for a careful reading of this article.
2. Main results
2.1. Fundamental groups of graphs of groups
We begin with a general theorem about groups acting on trees. Recall that the action of a group by automorphisms on a simplicial tree is called minimal if contains no proper -invariant subtree. As usual, by we denote the Gromov boundary of , which can be identified with the set of ends of ; no topology on is assumed.
Let be a group acting minimally on a simplicial tree . Suppose that does not fix any point of and there exist vertices of such that the pointwise stabilizer is finite. Then is either virtually cyclic or acylindrically hyperbolic.
If is the fundamental group of a graph of groups , one can apply Theorem 2.1 to the action of on the associated Bass-Serre tree. In this case the minimality of the action and the absence of fixed points on can be recognized from the local structure of – see Section 4.2. We mention here two particular cases and refer to Theorem 4.17 for a more general result. Below we say that a subgroup of a group is weakly malnormal if there exists such that .
Let split as a free product of groups and with an amalgamated subgroup . Suppose and is weakly malnormal in . Then is either virtually cyclic or acylindrically hyperbolic.
Note that the virtually cyclic case cannot be excluded from Corollary 2.2. Indeed it is realized if is finite and has index in both factors. However these are the only exceptions: it is easy to show that under the assumptions of the corollary all other amalgams contain non-abelian free subgroups, and so they are are acylindrically hyperbolic.
Let be an HNN-extension of a group with associated subgroups and . Suppose that and is weakly malnormal in . Then is acylindrically hyperbolic.
The assumption of Corollary 2.3 saying that is weakly malnormal in is equivalent to the existence of an element such that , because and are conjugate in .
The weak malnormality condition is essential in both corollaries. This can be illustrated in many ways, but the most convincing examples are the following.
It is worth noting that our Corollaries 2.2 and 2.3 can be used to recover some results proved by Schupp  and Sacerdote and Schupp  by using small cancellation theory for amalgamated products and HNN-extensions. In fact, our approach is slightly more general; for details we refer to Section 4.2.
Let be a one-relator group with at least generators. Then is acylindrically hyperbolic.
Corollary 2.6 is related to a conjecture of P. Neumann , claiming that if is a one-relator group then either is cyclic or is a solvable Baumslag-Solitar group, or is SQ-universal. Recall that a group is said to be SQ-universal if every countable group embeds in a quotient of . The first non-trivial example of an SQ-universal group was provided by Higman, Neumann and Neumann , who proved that the free group of rank is SQ-universal or, equivalently, every countable group embeds in a group generated by elements. It is straightforward to see that any SQ-universal group contains non-abelian free subgroups. Furthermore, since the set of all finitely generated groups is uncountable and every countable group contains countably many finitely generated subgroups, every countable SQ-universal group has uncountably many non-isomorphic quotients. Thus the property of being SQ-universal may be thought of as an indication of “algebraic largeness” of a group. For a survey of SQ-universality we refer to [4, 83].
Sacerdote and Schupp  established the SQ-universality of one-relator groups with at least generators, thus confirming Neumann’s conjecture in this case. Corollary 2.6 can be regarded as a strengthening of this result (indeed, by a theorem of Dahmani, Guirardel and Osin , every acylindrically hyperbolic group is SQ-universal). The case of one-relator groups with two generators is more complicated, as one can see by looking at the Baumslag-Solitar groups. The full proof of Neumann’s conjecture was announced by Sacerdote , but it was never published, and it seems that the conjecture is still open. In Section 4.3, we make a step towards the resolution of this conjecture by showing that every one-relator group with a sufficiently complicated Magnus-Moldavanskii hierarchy is acylindrically hyperbolic (see Proposition 4.21). More precisely, all non-acylindrically hyperbolic one-relator groups belong to the class of HNN-extensions of (finitely generated free)-by-cyclic groups. Unfortunately, our techniques do not apply to ascending HNN-extensions (cf. Examples 4.19, 4.20), therefore for groups with Magnus-Moldavanskii hierarchy of low complexity we only have partial results.
Another example having the structure described in Corollary 2.2 is , the group of automorphisms of the polynomial algebra for any field . This group is sometimes called the integral 2-dimensional Cremona groups over and is also isomorphic to the group of automorphisms of the free associative algebra [28, 60]; if it is also isomorphic to the group of automorphisms of the free -generated Poisson algebra . The algebraic structure of these groups has received a lot of attention but is still far from being well understood, see [9, 37, 94] and references therein. We prove the following.
For any field , the group is acylindrically hyperbolic.
For motivation and a survey of some related results we refer to Section 4.2. Note that is solvable and hence it is not acylindrically hyperbolic. We do not know if the analogue of Corollary 2.7 holds true for the group when , but we would rather expect some higher rank phenomena to prevent it from being acylindrically hyperbolic.
and amalgamated products (respectively, HNN-extensions) of the form (respectively, ), where is a hyperbolic group and is a quasi-convex subgroup of infinite index in . The latter examples should be compared to simple groups mentioned in Example 2.5. For details we refer to Section 4.5
2.2. -manifold groups
By default, all manifolds considered in this paper are assumed to be connected. Our next goal is to show that most orientable -manifold groups are acylindrically hyperbolic. More precisely, let denote the class of all subgroups of fundamental groups of compact orientable -manifolds. Note that by Scott’s theorem , every finitely generated group is itself the fundamental group of a compact orientable -manifold. However the groups we consider are not necessarily finitely generated.
Let . Then exactly one of the following three conditions holds.
is acylindrically hyperbolic.
contains an infinite cyclic normal subgroup and is acylindrically hyperbolic.
is virtually polycyclic.
If is itself a fundamental group of a compact irreducible orientable -manifold, we can give a more geometric description of the non-acylindrically hyperbolic cases.
Let be a compact orientable irreducible -manifold such that is not acylindrically hyperbolic. Then either is virtually polycyclic or is Seifert fibered.
Theorem 2.8 allows to bring many results known for acylindrically hyperbolic groups to the world of -manifold groups. We discuss some examples below. Let us say that a group has type (I), (II), or (III) if the corresponding condition from Theorem 2.8 holds for .
A group is inner amenable iff it is of type (II) or (III).
Recall that a group is inner amenable if there exists a finitely additive conjugacy invariant probability measure on . Inner amenability is closely related to the Murray–von Neumann property for operator algebras. In particular, if is not inner amenable, the von Neumann algebra of does not have property . For further details and motivation we refer to  and Section 7.1.
Type is a measure equivalence invariant in . That is, if and is measure equivalent to , then and have the same type.
This result can be seen as the first step towards measure equivalence classification of -manifold groups. We note that a classification of fundamental groups of compact -manifolds up to quasi-isometry was done by Behrstock and Neumann [15, 14], while no non-trivial results about measure equivalence seem to be known.
2.3. Graph products
Let be a graph without loops and multiple edges with vertex set . Let also be a family of groups, called vertex groups, indexed by vertices of . The graph product of with respect to , denoted , is the quotient group of the free product by the relations for all , whenever and are adjacent in . Graph products generalize free and direct products of groups. Other basic examples include right angled Artin and Coxeter groups. The study of graph products and their subgroups has gained additional importance in view of the recent breakthrough results of Agol, Haglund, Wise, and their co-authors, which claim that many groups can be virtually embedded into right angled Artin groups (see [1, 41, 93] and references therein).
For any subset , the subgroup of the graph product is said to be a full subgroup; it is naturally isomorphic to to the graph product , where is the full subgraph of spanned on the vertices from . Any conjugate of a full subgroup is called parabolic. We will say that the graph is irreducible if its complement graph is connected; this equivalent to saying that the graph product does not naturally split as a direct product of two proper full subgroups.
Graph products naturally decompose as amalgamated products in many ways (see Subsection 6.2). In Section 6 we study subgroups of graph products using the actions on the associated Bass-Serre trees together with the theory of parabolic subgroups developed in . In particular, we prove the following (see Section 6.3).
Let be the graph product of non-trivial groups with respect to some finite irreducible graph with at least two vertices. Suppose that is a subgroup that is not contained in a proper parabolic subgroup of . If is not virtually cyclic then .
The corollary below covers the particular case when .
Let be the graph product of non-trivial groups with respect to some finite irreducible graph with at least two vertices. Then is either virtually cyclic or acylindrically hyperbolic.
Graph products are simplest examples of groups having “relative non-positive curvature” with respect to the vertex subgroups. From this point of view, Corollary 2.13 can be considered as another instance of a more general phenomenon: “irreducibility” of “nice” non-positively curved groups (or spaces) often implies the existence of “hyperbolic directions” in an appropriate sense; in turn, the latter condition implies acylindrical hyperbolicity. In the context of groups acting on spaces, a more formal version of this claim is sometimes called the Rank Rigidity Conjecture [8, 24] and is closely related to the classical work of Ballman, Brin, Burns, and Spatzier [6, 7, 21] as well as more recent results of Caprace and Sageev . For a more detailed discussion we refer to Section 6.4.
Theorem 2.12 is quite useful for studying properties of subgroups of graph products. For example, it is used in  to prove that is residually finite for any virtually compact special (in the sense of Haglund and Wise ) group . Theorem 2.12 also implies that every subgroup of a graph product is either acylindrically hyperbolic, or “reducible”, or “elementary” in an appropriate sense (see Theorem 6.25). A particular application to right angled Artin groups gives rise to the following algebraic alternative:
Let be a non-cyclic subgroup of a finitely generated right angled Artin group. Then exactly one of the following holds:
is acylindrically hyperbolic;
contains two non-trivial normal subgroups such that .
We also mention one application of graph products to groups with hyperbolically embedded subgroups. Recall that if a group is hyperbolic relative to a subgroup , then many “nice” properties of (e.g., solvability of algorithmic problems, finiteness of the asymptotic dimension, various analytic properties, etc.) are inherited by [35, 75, 76, 32]. As shown in , many properties of acylindrically hyperbolic groups resemble those of relatively hyperbolic groups, and hyperbolically embedded subgroups (see Subsection 3.1 for the definition) serve as analogues of peripheral subgroups in the relatively hyperbolic case. Thus one may wonder if the structure of an acylindrically hyperbolic group is determined, to some extent, by the structure of its hyperbolically embedded subgroups. The following result provides a strong negative answer; it is a simplified version of Theorem 7.7.
Any finitely generated group can be embedded into a finitely generated acylindrically hyperbolic group such that every proper hyperbolically embedded subgroup of is finitely generated and virtually free.
2.4. Geometric vs. analytic negative curvature
Recall that a countable group belongs to the Monod–Shalom class if . An a priori different class was defined by Thom : a countable group is in if is non-amenable and there exists an unbounded quasi-cocycle . Groups from and exhibit behavior typical to non-elementary hyperbolic groups. Moreover, most known examples from these classes come from the world of groups acting on hyperbolic spaces. Motivated by these observations, Monod  and Thom  suggested to consider and as analytic analogues of the class of “negatively curved” groups. It is worth noting that no non-trivial relation between and is known, although they are likely to coincide (see Remark 2.7 in )
Monod [66, Problem J] asked whether can be characterized by a geometric condition. In this paper we discuss the relation between , , and . First we recall that . This was originally proved by Hamenstädt  in slightly different terms and also follows from the results of Hull-Osin  and Bestvina–Bromberg–Fujiwara . Moreover, acylindrically hyperbolic groups constitute the majority of examples from or . Indeed the only examples of non-acylindrically hyperbolic groups from known up to now are those constructed by Osin  and Lück and Osin . These are finitely generated infinite torsion groups with positive first -betti numbers. Note that implies while torsion groups are never acylindrically hyperbolic (see Example 3.8 (b)).
Groups from  and  are not finitely presented. On the other hand, there is an evidence that for finitely presented groups, existence of non-trivial quasi-cocycles may be used to construct non-elementary actions on hyperbolic spaces . This motivates the following question:
Question (Geometric vs. Analytic Negative Curvature).
Are the conditions , , and equivalent for a finitely presented group ?
We do not really believe in a positive answer to this question in full generality. However, in Section 7.2 we obtain the following as immediate applications of other results from the paper.
The conditions , , and are equivalent for any group from the following classes:
Subgroups of fundamental groups of compact orientable -manifolds.
Subgroups of graph products of amenable groups. In particular, this class includes subgroups of right angled Artin groups.
3.1. Equivalent definitions of acylindrically hyperbolic groups
Given a group acting on a hyperbolic space , an element is called elliptic if some (equivalently, any) orbit of is bounded, and loxodromic (or hyperbolic) if the map defined by is a quasi-isometry for some (equivalently, any) .
In papers devoted to groups acting on general hyperbolic spaces, the terms “loxodromic” and “hyperbolic” are used as synonyms. Recent papers on relatively hyperbolic and acylindrically hyperbolic groups (including papers by the second author) tend to use the term “loxodromic” more often. In this paper we switch to “hyperbolic” since we mostly deal with actions on trees for which this term is well established.
Every hyperbolic element has exactly limit points on the Gromov’s boundary . Hyperbolic elements are called independent if the sets and are disjoint. The next theorem classifies acylindrical group actions on hyperbolic spaces.
Theorem 3.2 ([73, Theorem 1.1]).
Let be a group acting acylindrically on a hyperbolic space. Then satisfies exactly one of the following three conditions.
has bounded orbits.
is virtually cyclic and contains a hyperbolic element.
contains infinitely many independent hyperbolic elements.
Recall that the action of a group on a hyperbolic space is called elementary if the limit set of in the Gromov boundary consists of at most points and non-elementary otherwise. According to Theorem 3.2, an acylindrical action of is non-elementary if and only if the orbits are unbounded and is not virtually cyclic.
The next theorem provides equivalent characterizations of acylindrically hyperbolic groups. In practice, (AH) is the most useful condition for proving that a certain group is acylindrically hyperbolic, while (AH), (AH), and (AH) are more useful for proving theorems about acylindrically hyperbolic groups.
Theorem 3.3 ([73, Theorem 1.2]).
For any group , the following conditions are equivalent.
The group is acylindrically hyperbolic in the sense of Definition 1.1.
admits a non-elementary acylindrical action on a hyperbolic space.
is not virtually cyclic and admits an action on a hyperbolic space such that at least one element of is hyperbolic and satisfies the WPD condition.
contains a proper infinite hyperbolically embedded subgroup.
Note that every group has an obvious acylindrical action on a hyperbolic space, namely the trivial action on a point. Thus considering elementary acylindrically hyperbolic groups does not make much sense. For this reason we include non-elementarity in the definition. The terminology used in conditions (AH) and (AH) will also be used in our paper, so we explain it below.
Given a group acting on a metric space , a subset , and , we define the pointwise -stabilizer of in as the set of all that move every point of by at most . That is,
Thus the pointwise -stabilizer is the same as the usual pointwise stabilizer, which will be denoted . Note that, in general, pointwise -stabilizers are not subgroups.
In these terms, the action of on is acylindrical if and only if for every there exists such that for all with , the sizes of pointwise -stabilizers of are uniformly bounded by a constant which only depends on .
An element of a group acting isometrically on a metric space satisfies the weak proper discontinuity condition (or is a WPD element) if for some (or, equivalently, for all ) and every , exists such that
This definition is essentially due to Bestvina and Fujiwara .
The notion of a hyperbolically embedded subgroup was introduced in ; it generalizes the notion of a peripheral subgroup of a relatively hyperbolic group. More precisely, let be a group, , . We assume that and denote by the Cayley graph of with respect to the generating set (even though and might intersect as subsets of , for the purposes of constructing we consider them to be disjoint) and by the Cayley graph of with respect to the generating set . Clearly is a complete subgraph of .
Given two elements , we define to be the length of a shortest path in that connects to and does not contain edges of . If no such path exists we set . Clearly is an extended metric on . We say that is hyperbolically embedded in with respect to (and write ) if the following conditions hold:
and is hyperbolic.
is a locally finite space, i.e., every ball (of finite radius) is finite.
We also say that is hyperbolically embedded in (and write ) if for some .
Let us consider three basic examples.
For any group , we have . Indeed take . Then the Cayley graph has diameter and whenever . Further, if is a finite subgroup of a group , then . Indeed for . These cases are referred to as degenerate.
Let , , where is a generator of . Then is quasi-isometric to a line and hence it is hyperbolic. However for every . If is infinite, then , and moreover .
Let , , where is a generator of . In this case is quasi-isometric to a tree and unless . Thus .
It is known that a group is hyperbolic relative to a subgroup if and only if for some finite set .
3.2. Properties of acylindrically hyperbolic groups
We begin with some elementary algebraic properties of acylindrically hyperbolic groups. The theorem below summarizes some results from [73, Section 7] and [29, Theorems 2.24, 2.27, 2.33]. Recall that a subgroup is -normal in if for every , we have . Clearly every infinite normal subgroup is -normal. Recall also that a group is SQ-universal if every countable group embeds in a quotient of .
For any acylindrically hyperbolic group the following conditions hold.
The amenable radical of is finite.
If , then for or .
Every -normal subgroup of is acylindrically hyperbolic.
If for some subgroups of . Then is acylindrically hyperbolic for at least one . In particular, is not boundedly generated.
is SQ-universal. In particular, contains non-abelian free subgroups.
contains uncountably many normal subgroups.
It is useful to keep this theorem in mind when searching for examples of acylindrically hyperbolic groups as it allows one to show that certain groups are not acylindrically hyperbolic.
No amenable group is acylindrically hyperbolic.
No group without non-abelian free subgroups is acylindrically hyperbolic. In particular, no torsion group or a group satisfying a non-trivial identity is acylindrically hyperbolic.
The Baumslag-Solitar groups
are not acylindrically hyperbolic unless or , because the cyclic subgroup is -normal in in this case.
is not acylindrically hyperbolic for , since it is boundedly generated. Another argument is based on the Margulis Theorem, which states that for every normal subgroup of is either finite or of finite index and hence has only countably many normal subgroups. For a generalization, see [73, Example 7.5].
Let be an acylindrically hyperbolic group. Suppose that is a finite index subgroup of , or a quotient of modulo a finite normal subgroup, or an extension of with finite kernel. Then is also acylindrically hyperbolic.
Firstly, if and is a finite index subgroup of , then is -normal in and thus by Theorem 3.7.
Further, let be a group, a finite normal subgroup and let be the natural epimorphism. Then for any generating set of , generates and the natural map is a -equivariant quasi-isometry. Thus if is non-elementary hyperbolic then so is . Also observe that the action of on factors through the canonical action of . Hence if the former is acylindrical then so is the latter. Thus if then .
Conversely, if is some generating set of then is a generating set of and the Cayley graphs and are -equivariantly quasi-isometric. Since , if the action of on is acylindrical, then so is the natural action of on . This allows to argue as above to conclude that implies . ∎
The next two results will be used in Section 7 of our paper. The first one was proved by Dahmani, Guirardel and the second author in [29, Theorems 2.24 and 2.35].
Theorem 3.10 ([29, Theorem 2.32]).
For any acylindrically hyperbolic group , there exists a maximal normal finite subgroup and the following conditions are equivalent.
has infinite conjugacy classes (equivalently, the von Neumann algebra of is a factor).
is not inner amenable. In particular, does not have property of Murray and von Neumann.
The reduced -algebra of is simple with unique trace.
The next theorem was first proved in  under a certain assumption equivalent to acylindrical hyperbolicity (see ); it also follows from the main result of , where the language of hyperbolically embedded subgroups was used. For details about quasi-cocycles and bounded cohomology we refer to [43, 50, 66, 90].
Theorem 3.11 (Hamenstädt ).
Finally we mention a theorem from . Recall that the conjugacy growth function of a finitely generated group for every measures the number of conjugacy classes intersecting the ball of radius centered at in the word metric. For details we refer to .
Theorem 3.12 (Hull-Osin ).
Every acylindrically hyperbolic group has exponential conjugacy growth.
4. Fundamental groups of graphs of groups
4.1. Hyperbolic WPD elements in groups acting on trees
Given an oriented edge of a graph , and will denote the initial vertex and the terminal vertex of respectively, and will denote the inverse edge to (i.e., equipped with the opposite orientation).
In this paper, a tree means a simplicial tree, unless specified otherwise. Let be a tree. The natural metric on , induced by identification of edges with , is denoted by . We will think of as a simplicial tree and a metric space simultaneously; in particular we will talk about vertices and edges as well as points of . Given two points , will denote the unique geodesic segment connecting and .
Throughout this subsection let denote a group acting on by isometries. It is well known that any element either fixes a point of (which is either a vertex or the midpoint of an edge) or there exists a unique minimal -subtree of , which is a bi-infinite geodesic path, called the axis of and denoted by , on which acts by translation . In the former case is said to be elliptic, and in the latter case it is said to be hyperbolic. For a hyperbolic element , we denote by the limit points of the axis of in .
The translation length , of an element , is defined as the minimum of the distances , where runs over the set of all vertices of . Since is a simplicial tree, is a non-negative integer. The translation length can be used to classify the elements of : if and only if is elliptic. And if is hyperbolic, then the set set of all points of such that is exactly the axis of .
Consider an edge of with some fixed orientation. We will say that an element translates if and the vertices and lie outside of the geodesic segment .
Note that, in this terminology, if acts as a hyperbolic isometry and is an edge on the axis of then is translated by exactly one of the elements or (the above definition implies that in the sense of [31, I.4.10]). Conversely, if translates some edge of then is hyperbolic and the axis of contains the segment (see [31, I.4.11]).
Our main goal in this section is to establish a criterion for the existence of hyperbolic WPD elements in . We begin with a technical result about pointwise -stabilizers defined in Section 3.1.
Fix any . Let be any two points in with and let be any two vertices on the geodesic segment such that . Then is contained in the union of at most left cosets of .
Let , , be the set of all translations of by elements of such that and . Obviously (note that edge inversions are possible). For each , choose any element such that .
Let be an element of . That is,
Let (respectively, ) be the point on closest to (respectively, ). Thus is a concatenation of and and, similarly, is a concatenation of and . Using (1) we obtain
Since , (2) implies that and hence .
Let be a group acting on a simplicial tree and let be a hyperbolic element. Suppose that for some vertices , the pointwise stabilizer is finite (the possibility is allowed). Then satisfies the WPD condition. In particular, either is virtually cyclic or .
Without loss of generality we can assume that for some (otherwise, exchange and ). We will show that in this case satisfies Definition 3.5 for . Take any . Evidently we can increase to assume that . Let be such that and let and . Then and . Applying Lemma 4.2, we obtain that is finite. Since , is conjugate to in , and hence it is also finite. Thus the definition of WPD is satisfied for and .
The action of on itself by translations shows that Corollary 4.3 does not extend to actions on -trees.
Recall that the action of on is minimal if does not contain any proper -invariant subtree. Let be the natural compactification of . Take any point . The limit set , of in the boundary , is the intersection of the closure of in with . Since acts on isometrically, does not depend on the choice of the point .
Let be a tree with at least vertices. Suppose that a group acts minimally on and does not fix any point of . Then for any two vertices of , there exists a hyperbolic element such that contains and .
Since has at least vertices, there exists a vertex of degree at least . Let be the minimal subtree of containing the orbit . Obviously is -invariant and no vertex of has degree in . Since the action of is minimal, . Thus does not contain vertices of degree . This implies that any two vertices of are contained in a bi-infinite geodesic . In particular, is unbounded.
If the action of is elliptic (i.e., orbits are bounded), then for any vertex of , the minimal subtree containing the orbit is bounded. Obviously is -invariant. However this contradicts minimality of the action since is unbounded. Thus the action of cannot be elliptic. Since does not fix any point of , the action of on cannot be parabolic or quasi-parabolic in Gromov’s terminology either (see [40, Section 8.2]). Hence the set , where is the set of all hyperbolic elements of , is dense in [40, Corollary 8.2.G] (for a detailed proof in a more general situation see ). Moreover, since the action of on is minimal, for any vertex , coincides with the convex span of the orbit . Thus any point of lies on a geodesic segment connecting two vertices from . It follows that , and so the set is dense in . This implies that for any segment of , there is a hyperbolic element with , as claimed. ∎
Note that the lemma does not hold for the tree consisting of a single edge. Indeed acts on by inversion. The action is minimal and obviously does not fix any point of as the latter is empty. Similar counterexamples can be constructed by using any group that surjects onto . However if we assume that acts of without inversions, then the condition in the lemma can be relaxed to .
The next result is obtained in the course of proving part (3) of Proposition 6 in . Note that although the authors assume that every hyperbolic element satisfies WPD there, the proof only uses the existence of a single hyperbolic WPD element.
Lemma 4.7 (Bestvina-Fujiwara).
Let be a group acting on a hyperbolic space and containing a hyperbolic WPD element. If is not virtually cyclic, then it contains two hyperbolic elements such that .
We are now ready to prove the main result of this subsection.
Let be a group acting minimally on a simplicial tree . If is not virtually cyclic then the following conditions are equivalent:
contains a hyperbolic WPD element (for the given action of on ).
does not fix any point of and there exist vertices of such that is finite.
Suppose that is non-elementary and contains a hyperbolic WPD element . Let be the axis of and let be any vertex. Let be the constant from the definition of the WPD condition for corresponding to . Set , then is finite by the WPD condition. Since the only points of fixed by a hyperbolic element are , Lemma 4.7 implies that does not fix any point of .
Conversely, suppose does not fix any point of and there exist vertices of such that is finite. If were finite then would have finite index in implying that , which would contradict the assumption that is not virtually cyclic. Hence the tree must be infinite, so, by Lemma 4.5, there exists a hyperbolic element such that passes through and . Therefore, according to Corollary 4.3, satisfies the WPD condition. ∎
4.2. Fundamental groups of graphs of groups
Let be a graph of groups, where is a connected graph and is a collection consisting of
edge groups and vertex groups associated to edges and vertices of ;
maps and , which for every edge of , define the embeddings of the edge group into the vertex groups and , respectively.
Let also denote the fundamental group of . As we know, this fundamental group naturally acts on the associated Bass-Serre tree . In this section we will investigate various conditions ensuring that this action satisfies the assumptions of Theorem 2.1.
First we will characterize the situation when fixes a point of . The proof of the following statement is essentially contained in the proof of Theorem 4.1 from :
Suppose that is a group acting isometrically on a simplicial tree . If fixes a point of then all elliptic elements in form a normal subgroup such that the quotient is either trivial or infinite cyclic.
Let be the point fixed by and let be an elliptic element. Then fixes some vertex of , and hence it must fix (pointwise) the unique geodesic ray from to . If is another elliptic element, then fixes the geodesic ray for some vertex of . The intersection of the rays and is again a geodesic ray for some vertex . It follows that is fixed by both and , hence the product is elliptic. Evidently an inverse and a conjugate of an elliptic element is again elliptic, therefore the set of elliptic elements forms a normal subgroup .
If has no hyperbolic elements, then and the lemma is proved. So, assume that has at least one hyperbolic element. Then there is a hyperbolic element which has minimal translation length . Recall that the only fixed points of on are . After replacing with , if necessary, we can assume that . Take any other hyperbolic element . We want to show that . As above, we can suppose that . Hence the intersection of and contains an infinite geodesic ray for some vertex of , and positive powers of and translate this ray into itself.
By the construction of , , hence there exist such that and , . Set and observe that translates the ray into itself and . So, if then is a hyperbolic element with , contradicting the choice of . Therefore , i.e., fixes , and so it is elliptic. Thus , which implies that . Since , we see that is infinite cyclic, and the lemma is proved. ∎
Suppose that is an edge between vertices and in a graph of groups . We will say that is good if the natural images of in and are proper subgroups. Any edge that is not good will be called bad.
If the graph of groups has at least one good edge then the elliptic elements of do not form a subgroup. In particular, does not fix any point of , where is the corresponding Bass-Serre tree.
Let be any edge of which maps onto a good edge of . Fixing some orientation of and abusing the notation, we let , and denote the -stabilizers of , and respectively. Since the image of in is good, there is an element and an element . Clearly and are elliptic, and to prove the statement it suffices to show that the element is hyperbolic. Let , then but as . On the other hand, the edge starts at and is also distinct from (see Figure 1).
We will say that an oriented edge of is reducible if and at least one of the maps , is onto. A graph of groups is reduced if it does not contain reducible edges.
Thus a reducible edge is a bad edge which is not a loop. It is easy to see that any reducible edge can be contracted without affecting the fundamental group. Therefore every finite graph of groups can be contracted to a reduced graph of groups with the same fundamental group (see  for details, where reducible edges are called directed).
Recall that an HNN-extension of a group with associated subgroups and is called ascending if or . Ascending HNN-extension gives a basic example of a reduced graph of groups that has a single vertex and a single bad loop at this vertex. The following proposition characterizes the case when the fundamental group of a reduced graph of groups fixes a point on the boundary of the corresponding Bass-Serre tree:
Suppose that is a reduced graph of groups with at least one edge. Let and let be the corresponding Bass-Serre tree. Then the following are equivalent:
fixes a point of the boundary ;
has only one vertex and one (unoriented) bad edge with . In other words, is an ascending HNN-extension of .
The fact that (ii) implies (i) is well-known and we leave it as an exercise for the reader. Now, suppose that (i) holds. Then has no good edges by Lemma 4.11. Since has no reducible edges by the assumptions and is connected, one concludes that has only one vertex and all the edges of are loops at .
The quotient of by the normal closure , of in , is the free group, whose rank is equal to the number of (unoriented) loops at . Since is the only vertex of , every elliptic element is conjugate to an element of in . So, by Lemma 4.9, is the set of all elliptic elements, and the quotient is cyclic. Thus can have at most one loop at , which is a bad edge (as defined above), i.e., (ii) holds. ∎
For a general graph of groups , the situation when the canonical action of on the corresponding Bass-Serre tree is minimal was characterized by Bass in . The next lemma follows from [10, Prop. 7.12]:
Let be a reduced graph of groups of finite diameter. Then the action of on the corresponding Bass-Serre tree is minimal.
A combination of Proposition 4.13 with Lemma 4.14 and Theorem 2.1 gives a criterion to show that the fundamental group of a finite reduced graph of groups is acylindrically hyperbolic (see Theorem 4.17 below). Before formulating it we need to make a couple of observations.
Suppose that is a group acting on a simplicial tree . If the tree has at least two vertices, the existence of vertices with is equivalent to the existence of two edges of such that . (Note that we allow the possibilities and/or .)
Indeed, in one direction this is evident. For the other direction, suppose that and are edges of with . Without loss of generality we can assume that the segment contains both of these edges. Then , i.e., one can take and .
Suppose that is the fundamental group of a graph of groups and is the corresponding Bass-Serre tree. In algebraic terms, the existence of vertices of such that is finite means that for some and some vertices , of . Similarly, the existence of two edges of such that corresponds to for some and some edges , of .
Let be the fundamental group of a finite reduced graph of groups with at least one edge such that the condition (ii) from Proposition 4.13 does not hold. Suppose that there are edges of (not necessarily distinct) and an element such that . Then is either virtually cyclic or acylindrically hyperbolic.
Let be the Bass-Serre tree associated to . By Lemma 4.14, the action of on is minimal, and, by Proposition 4.13, does not fix any point of . Since is reduced and has at least one edge, it is easy to see that the tree must be infinite (any loop gives rise to a hyperbolic element; if there are no loops, then all the edges must be good, hence a hyperbolic element can be produced as in the proof Lemma 4.11). By Remarks 4.16 and 4.15, the assumptions imply that contains two vertices such that is finite. Therefore, we can apply Theorem 2.1 to conclude that is either virtually cyclic or acylindrically hyperbolic. ∎
Corollary 2.2 from Section 2 is an immediate consequence of Theorem 4.17 applied in the situation when consists of two vertices and a good edge connecting them. Corollary 2.3 corresponds to the case when has one vertex and one (good) loop at this vertex. Thus its claim also follows from Theorem 4.17 modulo the observation that a non-ascending HNN-extension of any group contains non-abelian free subgroups (see, for example, [11, Thm. 6.1]), and so it cannot be virtually cyclic.
It is easy to see that the condition is necessary in Corollary 2.2. As for the requirement in Corollary 2.3, the situation is more complicated and counterexamples are not so obvious. We note the following.
Let be an ascending HNN-extension of a group . If is acylindrically hyperbolic, then so is .
Let be the stable letter of the HNN-extension, . Then the subgroup is normal in and . Consequently we have . Thus by part (d) of Theorem 3.7 must be acylindrically hyperbolic. ∎
Proposition 4.18 allows to construct examples of ascending HNN-extensions with weakly malnormal base which are not acylindrically hyperbolic.
Using methods from [72, Section 4.2] it is not hard to construct a proper malnormal subgroup of a free Burnside group of rank and large odd exponent such that . Let be the corresponding ascending HNN-extension of . Then and are weakly malnormal in , but is not acylindrically hyperbolic by a combination of Proposition 4.18 and Example 3.8 (b).
If is acylindrically hyperbolic, then its ascending HNN-extensions can be acylindrically hyperbolic as well. For example, many ascending HNN-extensions of free groups are hyperbolic. Nevertheless the example below shows that acylindrical hyperbolicity (even relative hyperbolicity) of together with weak malnormality is still not sufficient to derive acylindrical hyperbolicity of .
In the proof of [76, Prop. 18], the second author constructed a boundedly generated finitely presented group (denoted by in ) which is universal, i.e., contains an isomorphic copy of every recursively presented group. Let us denote this group by . Let and let be a generator of the subgroup . Then is acylindrically hyperbolic (in fact, it is hyperbolic relative to ). Since is finitely presented, it is isomorphic to a subgroup . Let be the corresponding ascending HNN-extension of with the stable letter . Obviously and hence and are weakly malnormal in . Arguing as in the proof of Proposition 4.18, we obtain . Since is boundedly generated, so is . Hence is not acylindrically hyperbolic by part (d) of Theorem 3.7.