Largest acylindrical actions and Stability in hierarchically hyperbolic groups
Abstract
We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most groups, right-angled Artin groups, and many others. –manifold
A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions.
The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and right-angled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known.
In the appendix, it is verified that any space satisfying the a priori weaker property of being an “almost hierarchically hyperbolic space” is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.
1. Introduction
Hierarchically hyperbolic groups were recently introduced by Behrstock, Hagen, and Sisto Reference BHS17b to provide a uniform framework in which to study many important families of groups, including mapping class groups of finite type surfaces, right-angled Coxeter groups, most groups, right-angled Artin groups and many others. A hierarchically hyperbolic space (HHS) consists of: a quasigeodesic space, –manifold a set of domains, ; which index a collection of , spaces to which –hyperbolic projects; and, some additional information about these projections, including, for instance, a partial order on the domains and a unique largest element in that order, which we denote by (i.e., is comparable to and larger than every other domain in ).
Largest acylindrical actions
The study of acylindrical actions on hyperbolic spaces, as initiated in its current form by Osin Reference Osi16 building on earlier work of Sela Reference Sel97 and Bowditch Reference Bow08, has proven to be a powerful tool for studying groups with some aspects of non-positive curvature. As established in Reference BHS17b, non-virtually cyclic hierarchically hyperbolic groups admit non-elementary acylindrical actions when the space associated to the maximal element in –hyperbolic has infinite diameter, a property which holds in all the above examples except for those that are direct products.
Any given group with an acylindrical action may actually admit many acylindrical actions on many different spaces. A natural question is to try and find a “best” acylindrical action. There are different ways that one might try to optimize the acylindrical action. For instance, the notion of a universal acylindrical action, for a given group is an acylindrical action on a hyperbolic space , such that every element of which acts loxodromically in some acylindrical action on some hyperbolic space, must act loxodromically in its action on As established by Abbott, there exist finitely generated groups which admit acylindrical actions, but no universal acylindrical action .Reference Abb16; we also note that universal actions need not be unique Reference ABO19.
In Reference ABO19, Abbott, Balasubramanya, and Osin introduce a partial order on cobounded acylindrical actions which, in a certain sense, encodes how much information the action provides about the group. When there exists an element in this partial ordering which is comparable to and larger than all other elements it is called a largest action. By construction, any largest action is necessarily a universal acylindrical action and unique.
In this paper we construct a largest action for every hierarchically hyperbolic group. Special cases of this theorem recover some recent results of Reference ABO19, as well as a number of new cases. For instance, in the case of right-angled Coxeter groups (and more generally for special cubulated groups), even the existence of a universal acylindrical action was unknown. Further, outside of the relatively hyperbolic setting, our result provides a single construction that simultaneously covers these new cases as well as all previously known largest and universal acylindrical actions of finitely presented groups. The following summarizes the main results of Section 5 (where there are also further details on the background and comparison with known results).
We use this construction of a largest action to characterize stable subgroups (Theorem B) and contracting elements (Corollary 5.5) of hierarchically hyperbolic groups, and to describe random subgroups of hierarchically hyperbolic groups (Theorem E).
Stability in hierarchically hyperbolic groups
One of the key features of a Gromov hyperbolic space is that every geodesic is uniformly Morse, a property also known as (quasigeodesic) stability; that is, any uniform quasigeodesic beginning and ending on a geodesic must lie uniformly close to it. In fact, any geodesic metric space in which each geodesic is uniformly Morse is hyperbolic.
In the context of geodesic metric spaces, the presence of Morse geodesics has important structural consequences for the space; for instance, any asymptotic cone of such a space has global cut points Reference DMS10. Moreover, quasigeodesic stability in groups is quite prevalent, since any finitely generated acylindrically hyperbolic group contains Morse geodesics Reference Osi16Reference Sis16.
There has been much interest in developing alternative characterizations Reference DMS10Reference CS15Reference ACGH17Reference ADT17 and understanding this phenomenon in various important contexts Reference Min96Reference Beh06Reference DMS10Reference DT15Reference ADT17. This includes the theory of Morse boundaries, which encode all Morse geodesics of a group Reference CS15Reference Cor17Reference CH17Reference CD19Reference CM19. In Reference DT15, Durham and Taylor generalized the notion of stability to subspaces and subgroups.
In this paper, we obtain a complete characterization of stability in hierarchically hyperbolic groups.
Let be an HHS. We say that a subset has projections when –bounded for all non-maximal when the constant does not matter, we simply say the subset has uniformly bounded projections. ;
Theorem B generalizes some previously known results. In the case of mapping class groups: Reference Beh06 proved that 2 implies 1 for cyclic subgroups; Reference DT15 proved equivalence of 1 and 3; equivalence of 2 and 3 follows from the distance formula; moreover, Reference KL08Reference Ham yield that these conditions are also equivalent to convex cocompactness in the sense of Reference FM02. The case of right-angled Artin groups was studied in Reference KMT17, where they prove equivalence of 1 and 3.
Section 6 contains a more general version of Theorem B, as well as further applications, including Theorem 6.6, which concerns the Morse boundary of hierarchically hyperbolic groups and proves that all hierarchically hyperbolic groups have finite stable asymptotic dimension.
On purely loxodromic subgroups
In the mapping class group setting Reference BBKL20 proved that the conditions in Theorem B are also equivalent to being undistorted and purely pseudo-Anosov. Similarly, in the right-angled Artin group setting, it was proven in Reference KMT17 that 1 and 3 are each equivalent to being purely loxodromic.
Subgroups of right-angled Coxeter groups all of whose elements act loxodromically on the contact graph were studied in the recent preprint Reference Tra, Theorem 1.4, which proved that property is equivalent to 3. Since there often exist Morse elements in a right-angled Coxeter group which do not act loxodromically on the contact graph (which plays the role of in the standard HHG structure on the group), his condition is not equivalent to 1. It is the ability to change the hierarchically hyperbolic structure as we do in Theorem 3.7, discussed below, which allows us to prove our more general result which characterizes all stable subgroups, not just the ones acting loxodromically on the contact graph.
Mapping class groups and right-angled Artin groups have the property that in their standard hierarchically hyperbolic structure they admit a universal acylindrical action on where , is the hyperbolic space associated to the domain –maximal On the other hand, right-angled Coxeter groups often don’t admit universal acylindrical actions on . in their standard structure. Accordingly, we believe the following questions are interesting. The first item would generalize the situation in the mapping class group as established in Reference BBKL20, and the second item for right-angled Artin groups would generalize results proven in Reference KMT17, and for right-angled Coxeter groups would generalize results in Reference Tra. If the second item is true for the mapping class group, this would resolve a question of Farb–Mosher Reference FM02. See also Reference ADT17, Question 1.
Note that in the context of Question C, an element acts loxodromically on if and only if it has positive translation length. This holds since the action is acylindrical and thus each element either acts elliptically or loxodromically.
In an early version of this paper, we asked if the second part of Question C held for all hierarchically hyperbolic groups. In the general hierarchically hyperbolic setting, however, the undistorted hypothesis is necessary, as pointed out to us by Anthony Genevois with the following example. The necessity is shown by Brady’s example of a torsion-free hyperbolic group with a finitely presented subgroup which is not hyperbolic Reference Bra99. This subgroup is torsion-free and thus purely loxodromic. But, a subgroup of a hyperbolic group is stable if and only if it is quasiconvex. Thus, since this subgroup is not quasiconvex, we see that being purely loxodromic is strictly weaker than the conditions of Theorem B.
New hierarchically hyperbolic structures
In order to establish the above results, we provide some new structural theorems about hierarchically hyperbolic spaces.
One of the key technical innovations in this paper is provided in Section 3. There we prove Theorem 3.7 which allows us to modify a given hierarchically hyperbolic structure by removing for some and, in their place, enlarging the space For instance, this is how we construct the space on which a hierarchically hyperbolic group has its largest acylindrical action. .
Another important tool is Theorem 4.4 which provides a simple characterization of contracting geodesics in a hierarchically hyperbolic space.
The following is a restatement of that result in the case of groups (see Theorem 4.4 for the precise statement):
Since the presence of a contracting geodesic implies the group has at least quadratic divergence, an immediate consequence of Theorem D is that any hierarchically hyperbolic group has quadratic divergence whenever projects to an infinite diameter subset of .
As a sample application of Theorem D and using work of Taylor–Tiozzo Reference TT16, we prove the following in Section 6.4 as Theorem 6.8.
We note that one immediate consequence of this result is a new proof of a theorem of Maher–Sisto: any random subgroup of a hierarchically hyperbolic group which is not the direct product of two infinite groups is stable Reference MS19. The mapping class group and right-angled Artin group cases of this result were first established in Reference TT16.
Finally, at the end of the paper we discuss a technical condition on hierarchically hyperbolic structures, called having clean containers. While in Proposition 7.2 this hypothesis is shown to hold for many groups, it does not hold in all cases. This condition was used in earlier versions of this paper in which it was assumed for the proof of Theorem 3.7, and then the general result was bootstrapped from there. In light of Theorem A.1 in the Appendix, this property is no longer required for this paper. We keep the contents of this section in the paper nonetheless, since they have found independent interest and already been used elsewhere, e.g., Reference BRReference HS16Reference Rus20, as well as in several papers in progress.
2. Background
We begin by recalling some preliminary notions about metric spaces, maps between them, and group actions. Given metric spaces we use , to denote the distance functions in respectively. A map , is if there exists a constant –Lipschitz such that for every , it is ; Lipschitz if –coarsely The map is a embedding if there exist constants –quasi-isometric and such that for all ,
If, in addition, is contained in the of –neighborhood then , is a . For any interval –quasi-isometry the image of an isometric embedding , is a geodesic and the image of a embedding –quasi-isometric is a . –quasigeodesic
If any two points in can be connected by a then we say –quasigeodesic, is a space. If –quasigeodesic we may simply say that , is a space. A subspace –quasigeodesic is if there exists a constant –quasi-convex such that any geodesic in connecting points in is contained in the of –neighborhood For all of the above notions, if the particular constants . are not important, we may drop them and simply say, for example, that a map is a quasi-isometry.
Throughout this paper, we will assume that all group actions are by isometries. The action of a group on a metric space which we denote by , is proper if for every bounded subset , the set , is finite. The action is cobounded (respectively, cocompact) if there exists a bounded (respectively, compact) subset such that If a group . acts on metric spaces and we say a map , is if for every –equivariant and every we have A quasi-action of . on associates to each a quasi-isometry of with uniform quasi-isometry constants, such that is within uniformly bounded distance of .
2.1. Hierarchically hyperbolic spaces
In this section we recall the basic definitions and properties of hierarchically hyperbolic spaces as introduced in Reference BHS17bReference BHS19.
An important consequence of being a hierarchically hyperbolic space is the following distance formula, which relates distances in to distances in the hyperbolic spaces for The notation . means include in the sum if and only if .
We now recall an important construction of subspaces in a hierarchically hyperbolic space called standard product regions introduced in Reference BHS17b, Section 13 and studied further in Reference BHS19. First we define a consistent tuple, which will be used to define the two factors in the product space.
We often abuse notation slightly and use the notation and , to refer to the image in of the associated set. In Reference BHS19, Construction 5.10 it is proven that these standard product regions have the property that they are “hierarchically quasiconvex subsets” of We leave out the definition of hierarchical quasiconvexity, because its only use here is that product regions have “gate maps,” as given by the following in .Reference BHS19, Lemma 5.5:
We also need the notion of a hierarchy path, whose existence was proven in Reference BHS19, Theorem 4.4 (although we use the word path, since they are quasi-geodesics, typically we consider them as discrete sequences of points):
We call a domain relevant to a pair of points, if the projections to that domain are larger than some fixed (although possibly unspecified) constant depending only on the hierarchically hyperbolic structure. We say a domain is relevant for a particular quasi-geodesic if it is relevant for the endpoints of that quasi-geodesic.
It is often convenient to work with equivariant hierarchically hyperbolic structures, we now recall the relevant structures for doing so. For details see Reference BHS19.
2.2. Acylindrical actions
We recall the basic definitions related to acylindrical actions; the canonical references are Reference Bow08 and Reference Osi16. We also discuss a partial order on these actions which was recently introduced in Reference ABO19.
Recall that given a group acting on a hyperbolic metric space an element , is loxodromic if the map defined by is a quasi-isometric embedding for some (equivalently any) However, an element of . may be loxodromic for some actions and not for others. Consider, for example, the free group on two generators acting on its Cayley graph and acting on the Bass-Serre tree associated to the splitting In the former action, every non-trivial element is loxodromic, while in the latter action, no powers of . and are loxodromic.
Notice that if every acylindrical action of a group on a hyperbolic space has bounded orbits, then does not contain any generalized loxodromic elements, and the action of on a point (which is acylindrical) is a universal acylindrical action.
The following notions are discussed in detail in Reference ABO19. We give a brief overview here. Fix a group Given a (possibly infinite) generating set . of let , denote the word metric with respect to and let , be the Cayley graph of with respect to the generating set Given two generating sets . and we say , is dominated by and write if
Note that when the action , provides more information about the group than and so, in some sense, is a “larger” action. The two generating sets , and are equivalent if and when this happens we write ; .
Let be the set of equivalence classes of generating sets of such that is hyperbolic and the action is acylindrical. We denote the equivalence class of by The preorder . induces an order on which we also denote , .
Given a cobounded acylindrical action of on a hyperbolic space a Milnor–Svarc argument gives a (possibly infinite) generating set , of such that there is a quasi-isometry between –equivariant and By a slight abuse of language, we will say that a particular cobounded acylindrical action . on a hyperbolic space is largest, when, more precisely, it is the equivalence class of the generating set associated to this action through the above correspondence, that is the largest element in , .
2.3. Stability
Stability is strong coarse convexity property which generalizes quasiconvexity in hyperbolic spaces and convex cocompactness in mapping class groups Reference DT15. In the general context of metric spaces, it is essentially the familiar Morse property generalized to subspaces, so we begin there.
We can now define a notion of stable embedding of one metric space in another which is equivalent to the one introduced by Durham and Taylor Reference DT15:
The following generalizes the notion of a Morse quasigeodesic to subgroups:
Stability of a subset is preserved under quasi-isometries. Note that stable subgroups are undistorted in their ambient groups and, moreover, they are quasiconvex with respect to any choice of finite generating set for the ambient group.
3. Altering the hierarchically hyperbolic structure
The goal of this section is to prove that any hierarchically hyperbolic space satisfying a technical assumption—the bounded domain dichotomy—admits a hierarchically hyperbolic structure with unbounded products, i.e., every non-trivial product region in the ambient space has unbounded factors; see Theorem 3.7 below.
In particular, this establishes that all hierarchically hyperbolic groups admit a hierarchically hyperbolic group structure with unbounded products. It is for this reason that our complete characterization of the contracting property in spaces with unbounded products in Section 4 yields a characterization of the contracting property for all hierarchically hyperbolic groups, as stated in Theorem D.
3.1. Unbounded products
Fix a hierarchically hyperbolic space .
Let and let be the set of domains such that there exists and satisfying: , and , .
Recall that a set of domains is closed under nesting if whenever and then , .
Recall that for every hierarchically hyperbolic group the set of domains , contains finitely many and each –orbits induces an isometry for each (see Definition 2.15). It thus follows that every hierarchically hyperbolic group has the bounded domain dichotomy. (Also, note that this property implies the space is “asymphoric” as defined in Reference BHS17c.)
3.2. Almost hierarchically hyperbolic spaces
In this section we introduce a tool for verifying a space is hierarchically hyperbolic.
The following is a weaker version of the orthogonality axiom:
(Bounded pairwise orthogonality) has a symmetric and anti-reflexive relation called orthogonality: we write when are orthogonal. Also, whenever and we require that , Moreover, if . then , are not Finally, the cardinality of any collection of pairwise orthogonal domains is uniformly bounded by –comparable. .
By Reference BHS19, Lemma 2.1, the orthogonality axiom (Definition 2.1, (3)) for an hierarchically hyperbolic structure implies axiom However, the converse does not hold; that is, the last condition of . does not directly imply the container statement in (3), and thus this is an a priori strictly weaker assumption. However, as is proven in the appendix in Theorem A.1, this weakened version of the axiom is enough to produce a hierarchically hyperbolic structure.
We now introduce the notion of an almost hierarchically hyperbolic space:
In the appendix, Berlyne and Russell prove Theorem A.1, establishing that if a space is almost hierarchically hyperbolic, then the associated structure can be modified to obtain a hierarchically hyperbolic structure on the original space. This result is used in our proof of Theorem 3.7.
3.3. A new hierarchically hyperbolic structure
In this section we describe a new hierarchically hyperbolic structure on hierarchically hyperbolic spaces with the bounded domain dichotomy. We first describe the hyperbolic spaces that will be part of the new structure.
Let be a hierarchically hyperbolic space with the domain dichotomy. Recall that we define –bounded to be the set of such that there exists with for which there exists a satisfying For each . define , similarly.
The next result uses the above spaces to obtain a hierarchically hyperbolic structure with particularly nice properties from a given hierarchically hyperbolic structure.
4. Characterization of contracting geodesics
For this section, fix a hierarchically hyperbolic space with the bounded domain dichotomy; denote the element –maximal .
In this section, we will focus our attention to the case of Definition 4.2 where is a quasigeodesic. In Section 6 we will consider results about arbitary subsets with the contracting property.
We note that sometimes authors refer to any quasigeodesic satisfying 3 as contracting. Nonetheless, for applications one also needs to assume the coarse idempotence and coarse Lipschitz properties given by (1) and (2), so for convenience we combine them all in one property.
A useful well-known fact is stability of contracting quasigeodesics. Two different proofs of the following occur as special cases of the results Reference MM99, Lemma 6.1 and Reference Beh06, Theorem 6.5; this explicit statement is also in Reference DT15, Section 4.
The following result and argument both generalize and simplify the analogous result for mapping class groups in Reference Beh06.
5. Universal and largest acylindrical actions
The goal of this section is to show that for every hierarchically hyperbolic group the poset has a largest element. Recall that the action associated to such an element is necessarily a universal acylindrical action.
We prove the following stronger result which, in addition to providing new largest and universal acylindrical actions for cubulated groups, gives a single construction that recovers all previously known largest and universal acylindrical actions of finitely presented groups that are not relatively hyperbolic.
The following is Theorem A of the introduction:
Before giving the proof, we record the following result which gives a sufficient condition for an action to be largest. This result follows directly from the proof of Reference ABO19, Theorem 4.13; we give a sketch of the argument here. Recall that an action is elliptic if has bounded orbits.
This remark directly implies that the action on is a universal acylindrical action. (The universality of the action can also be proven using the classification of elements of described in Reference DHS17.)
Another immediate consequence of the above remark is the following, which for hierarchically hyperbolic groups strengthens a result obtained by combining Reference Osi16, Theorem 1.4.(L4) and Reference Sis16, Theorem 1, which together prove that a generalized loxodromic element in an acylindrically hyperbolic group is quasi-geodesically stable.
The next result provides information about the partial ordering of acylindrical actions. Of the groups listed below, the largest and universal acylindrical action of the class of special CAT(0) cubical groups is new; the other cases were recently established to be largest in Reference ABO19.
We give an explicit description of these actions for each hierarchically hyperbolic group in the corollary, in the sense that we describe the set of domains which are removed from the standard hierarchical structure of the group and whose associated hyperbolic space is infinite diameter. Recall that the space is constructed from by coning off all elements of which consists of those components of whose associated product regions have both factors with infinite diameter. Coning off all of yields a space which is is quasi-isometric to the space obtained by just coning off .
- (1)
Hyperbolic groups have a canonical simplest hierarchically hyperbolic group structure given by taking where , is the Cayley graph of the group with respect to a finite generating set. For this structure, and the action on the Cayley graph is clearly largest. ,
- (2)
For mapping class groups, the natural hierarchically hyperbolic group structure is the set of homotopy classes of non-trivial non-peripheral (possibly disconnected) subsurfaces of the surface; the maximal element is the surface itself, and the hyperbolic space is the curve complex of For this structure . (Note that to form . one must remove the nest-maximal collections of disjoint subsurfaces; the hyperbolic space associated to each of these, except has finite diameter). Additionally, we emphasize that although the new hyperbolic space , is not it is quasi-isometric to , the action on which is known to be universal. Universality of this action was shown by Osin in ,Reference Osi16, and follows from results of Masur-Minsky and Bowditch Reference Bow08Reference MM99.
- (3)
If is a compact with no Nil or Sol in its prime decomposition and –manifold then , is exactly the set of vertex groups in the prime decomposition that are fundamental groups of hyperbolic 3–manifolds (each of which has exactly one domain in its hierarchically hyperbolic structure).
- (4)
If is a group that acts properly and cocompactly on a special CAT(0) cube complex then by ,Reference BHS17b, Proposition B, has a factor system. This factor system gives a hierarchically hyperbolic group structure in which –equivariant is the closure under projection of the set of hyperplanes along with a maximal element where , is the contact graph as defined in Reference Hag14. In this structure, is the set of indices whose stabilizer in contains a power of a rank one element.
In the particular case of right-angled Artin groups, no power of a rank one element will stabilize a hyperplane, so In this case, the contact graph . is quasi-isometric to the extension graph defined by Reference KK14. That the action on the extension graph is a universal acylindrical action follows from the work of Reference KK14 and the centralizer theorem for right-angled Artin groups. This action is also shown to be largest in Reference ABO19.
We give a concrete example of the situation in the case of a right-angled Coxeter group.
6. Characterizing stability
In this section, we will give several characterizations of stability which hold in any hierarchically hyperbolic group. In fact, we will characterize stable embeddings of geodesic metric spaces into hierarchically hyperbolic spaces with unbounded products. One consequence of this will be a description of points in the Morse boundary of a proper geodesic hierarchically hyperbolic space with unbounded products as the subset of the hierarchically hyperbolic boundary consisting of points with bounded projections.
6.1. Stability
While it is well-known that contracting implies stability Reference Beh06Reference DMS10Reference MM99, the converse is not true in general. Nonetheless, in several important classes of spaces the converse holds, including in hyperbolic spaces, CAT(0) spaces, the mapping class group, and Teichmüller space Reference Sul14Reference Beh06Reference DT15Reference Min96. We record the following corollary of Theorem 4.4 which gives a relationship between stability and contracting subsets that holds in a fairly general context.
The following provides a general characterization of stability in HHSs, a special case of which is Theorem B.
6.2. The Morse boundary
In the rest of this section, we turn to studying the Morse boundary and use this to give a bound on the stable asymptotic dimension of a hierarchically hyperbolic space. We begin by describing two notions of boundary.
In Reference DHS17, Durham, Hagen, and Sisto introduced a boundary for any hierarchically hyperbolic space. We collect the relevant properties we need in the following theorem:
Building on ideas in Reference CS15, Cordes introduced the Morse boundary of a proper geodesic metric space Reference Cor17, which was then refined further by Cordes–Hume in Reference CH17. The Morse boundary is a stratified boundary which encodes the asymptotic classes of Morse geodesic rays based at a common point. Importantly, it is a quasi-isometry invariant and generalizes the Gromov boundary of a hyperbolic space Reference Cor17.
We briefly discuss the construction of the Morse boundary and refer the reader to Reference Cor17Reference CH17 for details.
Consider a a proper geodesic metric space with a basepoint Given a stability gauge . define a subset , to be the collection of points such that and can be connected by an geodesic in –stable Each such . is for some –hyperbolic depending on and Reference CH17, Proposition 3.2; here, we use the Gromov product definition of hyperbolicity, as need not be connected. Moreover, any stable subset of embeds in for some Reference CH17, Theorem A.V.
The set of stability gauges admits a partial order: if and only if for all constants In particular, if . then , .
Since each is Gromov hyperbolic, each admits a Gromov boundary Take the direct limit with respect to this partial order to obtain a topological space . called the Morse boundary of .
We fix a hierarchically hyperbolic structure with unbounded products. ,
The boundary contains for each by construction. The next lemma shows that the boundary points with bounded projections are contained in , as a subset of , where , is the element. In general, the set of boundary points with bounded projections may be a very small subset of –maximal For instance, in the boundary of the Teichmüller metric, these points are a proper subset of the uniquely ergodic ending laminations and have measure zero with respect to any hitting measure of a random walk on the mapping class group. .
6.3. Bounds on stable asymptotic dimension
The asymptotic dimension of a metric space is a coarse notion of topological dimension which is invariant under quasi-isometry. Introduced by Cordes–Hume Reference CH17, the stable asymptotic dimension of a metric space is the maximal asymptotic dimension of a stable subspace of .
The stable asymptotic dimension of a metric space is always bounded above by its asymptotic dimension. Behrstock, Hagen, and Sisto Reference BHS17a proved that all proper hierarchically hyperbolic spaces have finite asymptotic dimension (and thus have finite stable asymptotic dimension, as well). The bounds on asymptotic dimension obtained in Reference BHS17a are functions of the asymptotic dimension of the top level curve graph.
In the following theorem, we prove that a hierarchically hyperbolic space has finite stable asymptotic dimension under the assumption that where , is the hyperbolic space associated to the domain –maximal in .
Recall that asymptotic dimension is monotonic under taking subsets. Thus, if is assumed to be proper, so that then , (and therefore its stable subsets) have finite asymptotic dimension by Reference BHS17a. Here, using some geometry of stable subsets we obtain a sharper bound on than .
The following corollary is immediate:
6.4. Random subgroups
Let be any countable group and a probability measure on whose support generates a non-elementary semigroup. A random subgroup of –generated, denoted is defined to be the subgroup generated by the step of independent random walks on where , For other recent results on the geometry of random subgroups of acylindrically hyperbolic groups, see .Reference MS19.
Following Taylor-Tiozzo Reference TT16, we say a random subgroup –generated of has a property if
In particular, one consequence is a new proof of the following result of Maher–Sisto. This result follows from the above, together with Rank Rigidity for HHG (Reference DHS17, Theorem 9.14) which implies that a hierarchically hyperbolic group which is not a direct product of two infinite groups has infinite diameter.
7. Clean containers
The clean container property is a condition related to the orthogonality axiom. In Proposition 7.2 this property is shown to hold for many, but not all, hierarchically hyperbolic groups. Unlike earlier versions of this paper, this condition is no longer needed to prove the main theorems of the earlier sections. However, we keep the content in this paper since this property has found independent interest and is used elsewhere.
We first describe some interesting examples with clean containers. Then we show that this property is preserved under some combination theorems for hierarchically hyperbolic spaces. We refer the reader to Reference BHS19, Sections 8 & 9 and Reference BHS17a, Section 6 for details on the structure in the new spaces.
The following example relies on the combination theorem Reference BHS19, Theorem 8.6. We provide this as another example of hierarchically hyperbolic spaces with clean containers, but since we don’t rely on this elsewhere in the paper, we refer to that reference for the relevant definitions. Nonetheless, we include a full proof for the expert, since it is short. (We note that after this paper was circulated, Berlai and Robbio proved a combination theorem under weaker conditions than Reference BHS19, Theorem 8.6 and, in the process, also proved that if all the vertex spaces have clean containers, then so does the combined space, see Reference BR, Theorem A.)
The following uses the notion of hierarchically hyperbolically embedded subgroups introduced in Reference BHS17a; see also Reference DGO17 for the related notion of hyperbolically embedded subgroups.
Appendix A. Almost HHSs are HHSs By Daniel Berlyne and Jacob Russell
The main result in this appendix is that every almost HHS structure can be promoted to an HHS structure. Recall that, as introduced in Section 3.2, an almost HHS is a space which satisfies all the axioms of an HHS except for the orthogonality axiom, which is instead replaced by a weaker axiom without a container requirement. In Theorem A.1, we show that an almost HHS structure can be made into an actual HHS structure by adding appropriately chosen “dummy domains” to serve as the containers. This result provides a useful method for producing an HHS structure while only needing to verify the weaker axioms of an almost HHS. This method is used in the main text in the proof of Theorem 3.7, where it is shown that every hierarchically hyperbolic space with the bounded domain dichotomy admits an HHS structure with unbounded products.
To prove Theorem A.1, we will need to collect three additional tools about almost HHSs. Each of these tools was proved in the setting of hierarchically hyperbolic spaces, but they continue to hold in the almost HHS setting. Indeed, the only use of the containers in their proofs is Reference BHS19, Lemma 2.1, which proves that the cardinality of any collection of pairwise orthogonal domains is uniformly bounded by the complexity of the HHS.
The first tool says the relative projections of orthogonal domains coarsely coincide. Note, and are both defined when or and or .
The second tool we will need is the realization theorem for almost HHSs. The realization theorem characterizes which tuples in the product are coarsely the image of a point in Essentially, it says if a tuple . satisfies the consistency inequalities of an almost HHS (see Definition 2.6), then there exists a point such that is uniformly close to for each .
The last result we need is that the relative projections of an almost HHS also satisfy the inequalities in the consistency axiom.
We are now ready to prove that every almost HHS is an HHS (Theorem A.1). If is an almost HHS, then the only HHS axiom that is not satisfied is the container requirement of the orthogonality axiom. The most obvious way to address this is to add an extra element to every time we need a container. That is, if with and there exists some with then we add a domain , to serve as the container for in i.e., every , nested into and orthogonal to will be nested into However, this approach is perilous as once a domain . is nested into we may now need a container for , in To avoid this, we add domains ! where is a pairwise orthogonal set of domains nested into that is, ; contains all domains that are nested into and orthogonal to all This allows for all the needed containers to be added at once, avoiding an iterative process. .
We say is an almost HHG if there exists an almost HHS such that and satisfy the definition of a hierarchically hyperbolic group where ‘HHS’ is replaced with ‘almost HHS’. The above proof shows that if is an almost HHG, then the structure from Theorem A.1 is an HHG structure for .
The following corollary gives criteria for the HHS structure from Theorem A.1 to have unbounded products. This is the version of Theorem A.1 that is applied in Theorem 3.7 to prove that every hierarchically hyperbolic space with the bounded domain dichotomy admits an HHS structure with unbounded products.
Let be an almost HHS with the bounded domain dichotomy. If for every non– domain –maximal there exist , so that , and , then the HHS structure , obtained by applying Theorem A.1 to has unbounded products.
Assume for every non– domain –maximal there exist , so that , and Let . be the HHS structure obtained from using Theorem A.1. If and is not then the above property implies that –maximal, and are both infinite diameter. Thus, we need only verify unbounded products for elements of Using the notation of Theorem .A.1, let and assume Now, . for all and by construction of , there exists , so that and Since . this implies , Therefore . is an HHS with unbounded products.
■Acknowledgments
The authors thank Mark Hagen and Alessandro Sisto for lively conversations about hierarchical hyperbolicity. The second author thanks Chris Leininger for an interesting conversation which led to the formulation of Question C. The authors thank Daniel Berlyne, Ivan Levcovitz, Jacob Russell, Davide Spriano, and the anonymous referee for helpful feedback. The authors thank Anthony Genevois for resolving a question asked in an early version of this article.