Acylindrically hyperbolic groups with exotic properties
We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group with strong fixed point properties: has property for all , and every action of on a finite dimensional contractible topological space has a fixed point. In addition, has other properties which are rather unusual for groups exhibiting “hyperbolic-like” behaviour. E.g., is not uniformly non-amenable and has finite generating sets with arbitrary large balls consisting of torsion elements.
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
A group is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space; equivalently, is not virtually cyclic and acts on a hyperbolic space acylindrically with unbounded orbits.
The class of acylindrically hyperbolic groups was introduced in  and includes many examples of interest: non-elementary hyperbolic and relatively hyperbolic groups, all but finitely many mapping class groups of punctured closed surfaces, for , finitely presented groups of deficiency at least , most -manifold groups, etc. On the other hand, many aspects of the theory of hyperbolic and relatively hyperbolic groups can be generalized in the context of acylindrical hyperbolicity (see  and references therein).
In [17, Corollary 1.6], Hull proved that any two finitely generated acylindrically hyperbolic groups have a common acylindrically hyperbolic quotient. The main goal of this paper is to prove the following strengthening of his result.
Every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient.
All corollaries below are obtained by applying Theorem 1.1 to the family of all non-elementary hyperbolic groups. In particular, all properties mentioned in Corollaries 1.2–1.4 can be realized by the same group. Constructing interesting examples as common quotients of countable families of hyperbolic and relatively hyperbolic groups is not a new idea. For instance, it was used in  to obtain groups with property (T) whose left regular representation is not uniformly isolated from the trivial representation, and in  to obtain groups with strong fixed point properties (see below). Our main contribution is that such a quotient can be made acylindrically hyperbolic.
Recall that a group is said to have property if every affine isometric action of on an -space has a fixed point. Known examples of groups having property for all include higher rank lattices  and certain Gromov’s Monsters ; none of these examples admit a non-elementary actions on hyperbolic spaces [14, 15]. On the other hand, Yu  proved that every hyperbolic group admits a proper affine action on an -space for large enough , which is a strong negation of the property . A generalization of Yu’s result to relatively hyperbolic groups has been recently obtained by Chatterji and Dahmani in . It is also known that every acylindrically hyperbolic group admits an unbounded quasi-cocycle for all (see [16, 18]), which can be seen as a violation of the “quasified” version of .
Motivated by these results, Gruber, Sisto and Tessera  asked whether there exists a group acting non-elementarily on a hyperbolic space and such that has property for all . The following corollary answers this question affirmatively.
There exists a finitely generated acylindrically hyperbolic group such that:
has property for all ;
every action of on a contractible Hausdorff topological space of finite covering dimension has a fixed point;
every simplicial action of on a finite dimensional locally finite contractible simplicial complex is trivial, i.e., it fixes the whole of pointwise.
The second and the third claims of the corollary strengthen the main results of , where the first examples of finitely generated groups that do not admit any fixed point-free actions on finite dimensional contractible Hausdorff topological spaces were constructed. These strong fixed point properties for the acylindrically hyperbolic group can be contrasted with a theorem of Rips, stating that any hyperbolic group acts properly and cocompactly on a contractible finite dimensional simplicial complex ([12, Ch. 4, Theorem 1]).
The same construction produces groups with interesting “non-uniform” behaviour. Recall that there are two potentially non-equivalent ways to define uniform non-amenability of a finitely generated group. The first one was suggested by Shalom in  and uses the Kazhdan constants of the left regular representations. More precisely, one says that the left regular representation of a finitely generated group is uniformly isolated from the trivial representation if there exists such that for every finite generating set of and every unit vector , there exists such that
where denotes the left regular representation of the group . Another possibility, considered in , is to control the Følner constants: one says that is uniformly non-amenable if there exists such that for every finite generating set of , we have , where
Here the infimum is taken over all finite subsets , and
It is not difficult to show that if the left regular representation of a finitely generated group is uniformly isolated from the trivial representation, then is uniformly non-amenable . To the best of our knowledge, it is still an open problem whether the converse is true.
It was proved in  that the left regular representations of certain Baumslag-Solitar groups are not uniformly isolated from the trivial representations. These groups were also shown to be not uniformly non-amenable in . On the other hand, it follows from a result of Koubi  that every non-elementary hyperbolic group is uniformly non-amenable. This is also true for non-elementary relatively hyperbolic groups , mapping class groups of closed surfaces of genus and for . Nevertheless, we have the following.
There exists a finitely generated acylindrically hyperbolic group which is not uniformly non-amenable. In particular, its left regular representation is not uniformly isolated from the trivial representation.
The first step in the proofs of uniform non-amenability of the groups considered in [5, 22, 35] consists of showing that there exists a constant such that for any finite generating set of , one can find an element , of word length , which is loxodromic with respect to a certain action of on a hyperbolic space. In particular, such an element has infinite order. Our last corollary shows that even this property may fail to be true in a general acylindrically hyperbolic group.
There exists an acylindrically hyperbolic group with the following property. For every and every sufficiently large , there is a finite generating set of such that all elements of length have order at most .
It is worth noting that we do not know whether every finitely generated acylindrically hyperbolic group has uniform exponential growth. For the definition of uniform exponential growth and a survey of known results we refer to .
The paper is organized as follows. In the next section we collect necessary definitions and results about hyperbolically embedded subgroups. Section 3 is devoted to the small cancellation component of our proof. Its main result, Proposition 3.3, strengthens the work of  and seems to be of independent interest. To apply Proposition 3.3 in our settings we show that every acylindrically hyperbolic group contains infinite proper hyperbolically embedded subgroups with universal associated hyperbolicity constant. This is the key novelty of our paper, which is discussed in Section 4. The proofs of Theorem 1.1 and Corollaries 1.2–1.4 are contained in Section 5.
In this section we recall the definition of a hyperbolically embedded collection of subgroups. This notion was introduced in  and plays a crucial role in our paper.
In what follows, we will use Cayley graphs of groups with respect to generating alphabets which are not necessarily subsets of the group. More precisely, by a generating alphabet of a group we mean an abstract set together with a possibly non-injective map such that is generated by in the usual sense. Given a word , where and , we say that it represents an element if in .
By the Cayley graph of with respect to a generating alphabet , denoted , we mean a graph with the vertex set and the set of edges defined as follows. For every and every , there is an oriented edge in labelled by . If is not injective, may have multiple edges. By (respectively, ) we denote the standard edge-path metric on (respectively, the word length on with respect to the generating set ).
By abuse of notation, we often identify words in the alphabet and the elements of represented by them. For two words and , we write to denote the letter-by-letter equality between them, and if these words represent the same element in the group . For a word , denotes its length. If is a simplicial path in the Cayley graph , and will denote the initial and the terminal vertices of respectively, and will denote the length of this path. The label of , denoted by , is the word obtained by reading off the labels of its oriented edges from to .
Suppose that we have a group , a collection of subgroups of , and a subset such that and the union of all together generate . In this case we say that is a relative generating set of with respect to . We think of and subgroups as abstract sets and consider the disjoint unions
Obviously is a generating alphabet of , where the map is induced by the natural inclusions and . Note that may not be injective.
Henceforth we always assume that all generating sets and relative generating sets are symmetric. That is, if , then . In particular, every element of can be represented by a (positive) word in . Given a word in , we denote by the word , where are letters of : whenever and whenever .
In these settings, we consider the Cayley graphs and , , and naturally think of the latter as a subgraphs of the former. For each , we introduce an extended metric as follows.
For every , let denote the length of a shortest path in that connects the elements and contains no edges of . If no such a path exists, we set .
Clearly satisfies the triangle inequality, where the addition is extended to in the natural way.
A collection of subgroups of is hyperbolically embedded in with respect to a subset , denoted , if the following conditions hold.
The group is generated by the alphabet defined in (2) and the Cayley graph is hyperbolic.
For every and every , the ball contains finitely many elements.
We say that is hyperbolically embedded in and write if for some .
A group is acylindrically hyperbolic if and only if it contains a proper infinite hyperbolically embedded subgroup.
3. Small cancellation quotients of groups with hyperbolically embedded subgroups
The proof of Theorem 1.1 makes use of small cancellation theory over acylindrically hyperbolic groups. The idea of generalizing classical small cancellation to groups acting on hyperbolic spaces is due to Gromov . For hyperbolic groups it was elaborated by Olshanskii in . This approach was generalized to relatively hyperbolic groups by the second author in , and further extended to acylindrically hyperbolic groups by Hull . We begin by recalling necessary definitions.
Let be a group generated by an alphabet , and let be a symmetrized set of words over ; that is, we assume that for every , contains all cyclic permutations of .
Recall that a path in a metric space is said to be -quasi-geodesic for some , , if
for any subpath of . Further, a word in is -quasi-geodesic, if some (equivalently, any) path labelled by in is -quasi-geodesic.
The set satisfies the small cancellation condition (with respect to ) for some , , , , , if
for any ;
any word is -quasi-geodesic;
suppose that for two words we have , , for some words in such that
We will need the following elementary observation.
Let and be generating alphabets of a group with the corresponding maps and . Let be a surjective map such that . Suppose that is a symmetrized set of words in the alphabet satisfying the small cancellation condition (with respect to ) for some , , , , . Let be the set of words in obtained from words in by replacing each letter with . Then satisfies the same small cancellation condition (with respect to ).
We extend the map to free monoids in the obvious way. Using surjectivity of it is straightforward to check that preserves the property of a word to be -quasi-geodesic as well as property (c) in Definition 3.1 and the claim follows. ∎
Let be a group, , two collections of subgroups of , subset of . Suppose that
Then there exists and finite subsets , , such that the following holds.
Let , be the alphabets defined by (2) and let be any set of words in of the form
satisfying the following conditions for all :
there exist and in such that and , for all ;
if a letter occurs in for some , then it occurs only in and only once; in addition, does not occur in any word from .
Then the restriction of the natural homomorphism to the set is injective and
We first note that the subsets can be chosen so that for every and every , every cyclic permutation of the word satisfies the conditions (W1)–(W4) from [17, Section 5]. Indeed (W1) is obvious, (W2) can be ensured by taking to be the (finite) set of all elements satisfying in the notation of [17, Section 5], (W3) is guaranteed by (b) and the structure of (the first alternative in the conclusion of (W3) always holds), and (W4) also follows from (b).
From now on, we assume that the subsets are chosen as explained above. Thus Lemmas 5.1 and 5.2 from [17, Section 5] hold in our settings. Together with our assumption (c), they allow us to repeat the proof of [17, Proposition 5.3] and obtain the following.
Claim. For every there exists a constant such that the set of all cyclic shifts of all words satisfies the small cancellation condition.
Although [17, Proposition 5.3] deals with the case , its proof works almost verbatim in the general case; the only change we need to make is to replace the phrase “since only appears once in ” in the beginning of the third paragraph of the proof with the reference to the condition (c). Note also that [17, Proposition 5.3] proves the condition, which is stronger than (see [17, Definition 4.3]); however, we do not need this stronger condition in our paper.
Applying Lemma 3.2 to the generating alphabets defined by (2) and , of , with the obvious maps , , and , we obtain that the set of words in , obtained from words in by replacing each letter with , satisfies the same small cancellation condition.
Note that by definition (cf. [6, Remark 4.26]). Since no letter from occurs in words , , the set of words is strongly bounded with respect to the hyperbolically embedded subgroups and the relative generating set of in the terminology of [17, Section 3, p. 1089]. Let , , be the constants provided by [17, Lemma 4.4] for the group , the hyperbolically embedded collection , the relative generating set , , and .
By choosing sufficiently large we can ensure that and . Observe that the condition becomes stronger as decreases and increases. Thus taking large enough, we can apply [17, Lemma 4.4] to the set of words and conclude that the natural homomorphism is injective on the set and ∎
4. Constructing uniformly hyperbolically embedded subgroups
We say that a metric space is -hyperbolic for some constant if it satisfies the Gromov -point condition: for all one has
This condition is well-known to be equivalent to other definitions of hyperbolicity (see [12, Ch. 2, §§ 2.4, 2.3, 2.21]). We say that a metric space is hyperbolic if it is -hyperbolic for some .
Let be a graph and let be a natural number. The -expansion of is a graph obtained from by adding an edge between every two vertices that are at most apart in .
Let be a connected -hyperbolic graph, let and let be the -expansion of . Then is -hyperbolic.
Let and denote the standard path metrics on and respectively. Let be any two vertices of . The definition of implies that are also vertices of and
Now consider arbitrary points in , and choose vertices of such that . Then, in view of (5), we have
Recalling that is -hyperbolic and using (5) again, we get
Hence the graph satisfies Gromov -point condition with the constant . ∎
Recall that a subset of a geodesic metric space is -quasiconvex, for some , if for any two points any geodesic path joining and in lies within the closed -neighborhood of in .
There exists a constant such that for any the following holds. Let be a connected -hyperbolic graph and let be an -quasiconvex subset of vertices of , for some . Then for each positive integer , the -expansion of is -hyperbolic and is -quasiconvex in .
Suppose that , . Then , the -expansion of equipped with the standard path metric , is -hyperbolic by Lemma 4.2. Let denote the geometric realization of the graph where all the edges are assumed to be isometric to the interval . Thus the metric on , can be obtained by rescaling the standard path metric on : for all one has . Obviously is -hyperbolic (as ) and is -quasiconvex in (as ).
The inequalities (5) clearly imply that for all vertices of we have
i.e., the natural identification of the vertex set of with the vertex set of is a -quasi-isometry.
Thus for every geodesic path connecting two vertices in , its vertex set forms the image of a (discrete) -quasi-segment in , using the terminology of [12, Ch. 5, § 1.2]. Therefore, by [12, Ch. 5, § 1.6], this vertex set lies within an -neighborhood of any geodesic path in which has the same endpoints as , where is some global constant.
Now, let be a geodesic path connecting two vertices in , and let be any point of . Then there is a vertex of such that . Let be a geodesic between and in . By the above discussion, lies in the -neighborhood of a vertex in . On the other hand, the -quasiconvexity of in implies that there is a vertex such that , hence . Recalling (6), we obtain , hence .
Thus we can conclude that is -quasiconvex in , where is a global constant (independent of , , etc.), and the lemma is proved. ∎
The following observation will be useful.
Suppose that is a group, and is a collection of subgroups in . Let be a subset such that and . Then if and only if .
Let . Obviously the set generates if and only if the set generates , as . Moreover, the assumptions and imply that the natural inclusion of the Cayley graph in is -bi-Lipschitz. Hence hence the Cayley graph is hyperbolic if and only if is hyperbolic (cf. [12, Ch. 5, § 2.12]).
For each and let denote the metric, given by Definition 2.2, coming from the Cayley graph . Note that for all . Therefore the metric is locally finite on if and only if is locally finite on . Thus if and only if . ∎
For every , there exists such that the following holds. Let be a graph obtained from a connected -hyperbolic graph by adding edges. Suppose that for any two vertices of , connected by an edge in , and any geodesic in going from to , the diameter of in is at most . Then is -hyperbolic.
Suppose that a group acts on a hyperbolic metric space . This is said to be non-elementary if for some the orbit has more than two points on the boundary (see [13, § 8.2.D]). An element is loxodromic if for any point the map , defined by , is a quasi-isometry. Clearly the order of any loxodromic element is infinite.
Recall that every acylindrically hyperbolic group contains a maximal finite normal subgroup called the finite radical of and denoted (see [6, Theorem 2.24]).
There exists a constant with the following property. Let be an acylindrically hyperbolic group with trivial finite radical. Then there exist elements of infinite order and a generating set of satisfying the following conditions:
The Cayley graph is -hyperbolic.
By [31, Theorem 1.2] there exists a generating set of such that the Cayley graph is -hyperbolic, for some , and the natural action of on it is acylindrical and non-elementary. Since the finite radical of is assumed to be trivial, is a ‘suitable’ subgroup of itself, in the terminology of . Therefore we can use [17, Corollary 5.7] to find loxodromic elements such that .
Now, by [2, Lemma 2.5] the cyclic subgroups , are -quasiconvex in , for some . Choose any with , and let
Observe that the Cayley graph is the -expansion of the Cayley graph , so, according to Lemma 4.3, is -hyperbolic and the subgroups , are -quasiconvex in it (where is the global constant from that lemma). Moreover, Lemma 4.4 ensures that , so it remains to show that is -hyperbolic for some global constant .
Let and . We will now check that the assumptions of Lemma 4.5 are satisfied, where and . Let be two adjacent vertices of , let be any geodesic joining with in , and let be arbitrary points on .
If is a single edge then . Otherwise, must be labelled by a power of or . Let us consider the case when is labelled by , for some , as the other case is similar. Then the -quasiconvexity of in implies that there exist such that and . Moreover, since , the vertices and are adjacent in . Therefore
Thus the universal constant is an upper bound for the diameter of in . Now we can apply Lemma 4.5 to find a universal constant such that the Cayley graph is -hyperbolic, as required. ∎
Let be a collection of groups. Suppose that for every we have a collection of subgroups and a generating set of , relative to , such that
, for all , and
the hyperbolicity constants of the Cayley graphs , where , , are uniformly bounded.
Then the collection is hyperbolically embedded in the free product with respect to .
Suppose that the Cayley graphs are -hyperbolic, for some fixed , and all . After increasing , we can assume that and geodesic triangles in each of these Cayley graphs are -slim, i.e, each side of such a triangle is contained in the -neighborhood of the two other sides (see [12, Ch. 2, § 3.21]). Let us first show that all geodesic triangles in are also -slim.
Let us first establish some convenient terminology. Given a path in , its label is a concatenation of subwords, each of which is written over the alphabet , for some . Given any , a -component of is a subpath of labelled by a non-empty word over the alphabet , which is not contained in a larger subpath of with this property.
Let us now make the following observation, which is an immediate consequence of the uniqueness of normal forms of elements in free products.
If and are two simple paths in with and then , , where for each there is such that and are -components of and respectively, and , .
Suppose that is a simplicial geodesic triangle in , and is any side of it. Let and denote the two remaining sides of . Without loss of generality we can re-orient so that , and . Let be any point of . Then belongs to a -component of , for some . Let be the first vertex of which also belongs to (see Figure 1), and let , be the subpaths of and respectively, such that , and . Note that the paths , and are simple because they are geodesic in , and the choice of implies that the path is also simple. Since and , we can use Remark 4.8 to find a -component , of , such that and .
Now, by construction, is either a subpath of or or it is a concatenation of a subpath of ending at with a subpath of starting at . Thus is geodesic and is the union of at most two geodesics in the copy of in based at . The -slimness of geodesic triangles in implies that is at most away from a point of in . But is contained in the union of and , so the geodesic triangle is -slim. Since the latter is true for an arbitrary simplicial geodesic triangle in , we can conclude that this Cayley graph is hyperbolic by [12, Ch. 2, § 3.21].
Now, choose any and and consider the subgroup , of . Suppose that , , is the edge joining and and labelled by in , and is any path from to in this graph, of length at most , which avoids the edges from the copy of . Let be a simple path with the same endpoints as , whose set of edges is a subset of the set of edges of (we can get from by removing all maximal subpaths of which start and end at the same vertex). Then, according to Remark 4.8, must consist of a single -component (since this is true for and , ). Thus the path lies inside the Cayley graph . Clearly the length of does not exceed and it still avoids the edges of . Recalling that , we see that there are only finitely many possibilities for the element (for a given ). Since this is true for any and arbitrary , we can conclude that , as claimed. ∎
5. Proofs of the main results
We begin with the proof of Theorem 1.1. Recall that for an acylindrically hyperbolic group , denotes its finite radical, i.e., is the unique maximal finite normal subgroup of . It is shown in [25, Lemma 3.9] (see also [26, Lemma 1]) that the quotient group is also acylindrically hyperbolic and has trivial finite radical.
Proof of Theorem 1.1.
Let , , be a countable set of countable acylindrically hyperbolic groups. Passing to if necessary, we can assume that the finite radicals of all groups are trivial.
By Lemma 4.6, there exist infinite cyclic subgroups and generating sets such that , and the hyperbolicity constant of the Cayley graph is bounded from above by a universal constant for all .
Let and . Note that generates and, by Lemma 4.7, we have . Combining [2, Corollary 5.3] with [2, Corollary 3.12] we can find an infinite virtually cyclic subgroup such that . Let and let . If , we define to uniformize our notation. For every , we consider a word of the form
where , for all . Note that the intersection of distinct subgroups from a hyperbolically embedded collection is always finite by [6, Proposition 4.33] and therefore for all . By Proposition 3.3, applied to the group , the set , and the collections and , we can chose and finite subsets , , so that the conclusion of the proposition holds as long as the elements are chosen to satisfy conditions (b) and (c) from the proposition. Since and are infinite cyclic, , such a choice is clearly possible.
Let be the quotient group defined by (4). By Proposition 3.3, the image of in is an infinite proper hyperbolically embedded subgroup. Therefore is acylindrically hyperbolic by Theorem 2.4. On the other hand, for every , the relations , guarantee that the restriction of the natural homomorphism to is surjective, i.e., is a quotient group of . ∎
Let us explain why the countability assumptions in Theorem 1.1 cannot be dropped. Observe that, given a simple group , any non-trivial quotient of the free product must contain a copy of . So, if is an uncountable simple group (e.g., ), then the acylindrically hyperbolic group cannot have a common quotient with the free group of rank , as is not a subgroup of any countable group. On the other hand, in  Camm showed that there exists a continuum of pairwise non-isomorphic -generated infinite simple groups , . Note that the groups are acylindrically hyperbolic, , and any non-trivial common quotient of the family will be generated by elements and will contain a copy of , for each . But the latter is impossible since a countable group can have at most countably many finitely generated subgroups. Thus the only common quotient of is the trivial group.
Let be any common quotient of all non-elementary hyperbolic groups. Then the following hold:
has property for all ;
every action of on a finite dimensional contractible topological space has a fixed point;
every simplicial action of on a finite dimensional locally finite contractible simplicial complex is trivial;
is not uniformly non-amenable;
for every sufficiently large and all , there exists a finite generating set of such that every element of length has order at most .
(a) For every , there exists a non-elementary hyperbolic group such that for all every isometric action of on an -space fixes a point. In fact, a random hyperbolic group, in a suitable density model, satisfies this property for every given almost surely [11, 27] (note that, by [7, Theorem 1.3] and [8, Corollary D], property for follows from property ). Since passes to quotients, the result follows.
(b) Let denote the class of contractible Hausdorff topological spaces of covering dimension , . By [3, Theorem 1.7] for each there exists a non-elementary hyperbolic group such that any action of by homeomorphisms on any space has a global fixed point. Since is a quotient of , for every , does not admit a fixed point-free action on any space from