Largest acylindrical actions and stability in hierarchically hyperbolic groups

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 3-manifold groups, right-angled Artin groups, and many others. 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 quasi-convexity 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.


Introduction
Hierarchically hyperbolic groups were recently introduced by Behrstock, Hagen, and Sisto [BHS17b] to provide a uniform framework in which to study many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3-manifold groups, right-angled Artin groups and many others. A hierarchically hyperbolic space consists of: a quasigeodesic space, X ; a set of domains, S, which index a collection of δ-hyperbolic spaces to which X projects; and, some additional information about these projections, including, for instance, a partial order on the domains and a unique maximal element in that order.
Largest acylindrical actions. The study of acylindrical actions on hyperbolic spaces, as initiated in its modern form by Osin [Osi16] following earlier work of [Sel97] and [Bow08], has proven to be a powerful tool for studying groups with some aspects of non-positive curvature. As established in [BHS17b], hierarchically hyperbolic groups admit non-elementary acylindrical actions when the δ-hyperbolic space associated to the maximal element in S 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 G, is an acylindrical action on a hyperbolic space X such that every element of G which acts loxodromically in some acylindrical action on some hyperbolic space, must act loxodromically in this action. As established by Abbott, there exist finitely generated groups which admit acylindrical actions, but no universal acylindrical action [Abb16]; we also note that universal actions need not be unique [ABO16].
In forthcoming work, Abbott, Balasubramanya, and Osin [ABO16], introduce a partial order on cobounded acylindrical actions; when there exist 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 action and unique.
In this paper we prove that all hierarchically hyperbolic groups have largest actions. Special cases of this theorem recover some recent results of [ABO16], as well as a number of new cases. For instance, in the case of right-angled Coxeter groups (and more generally for 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).
Theorem A (HHG have actions that are largest and universal). Every hierarchically hyperbolic group admits a largest acylindrical action. In particular, the following admit acylindrical actions which are largest and universal: (1) Hyperbolic groups and their subgroups.

3) Fundamental groups of three manifolds with no Nil or Sol in their prime decomposition. (4) Groups that act properly and cocompactly on a proper CAT(0) cube complex. This includes right angled-Artin groups and right-angled Coxeter groups.
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 (quasigeodesically) stable; 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 [DMS10]. Moreover, quasigeodesic stability in groups is quite prevalent, since any finitely generated acylindrically hyperbolic group contains Morse geodesics [Osi16,Sis16].
A subset Y Ă X is said to have D-bounded projections when diampπ U pYqq ă D for all nonmaximal U P S; when the constant doesn't matter we simply say the subset has uniformly bounded projections.
We prove a complete characterization of stability in hierarchically hyperbolic groups.
Theorem B (Equivalent conditions for subgroup stability). Any hierarchically hyperbolic group G admits a hierarchically hyperbolic group structure pG, Sq such that for any finitely generated H ă G, the following are equivalent: (1) H is stable in G; (2) H is undistorted in G and has uniformly bounded projections; (3) Any orbit map H Ñ CS is a quasi-isometric embedding, where S is Ď-maximal in S.
Theorem B generalizes some previously known results. In the case of mapping class groups: [DT15] proved equivalence of (1) and (3); equivalence of (2) and (3) follows from the distance formula; moreover, [KL08,Ham] yield that these conditions are also equivalent to convex cocompactness in the sense of [FM02]. The case of right-angled Artin groups was studied in [KMT14], 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. As a sample application of Theorem B and using work of Taylor-Tiozzo [TT16], we prove the following in Section 6.4.
Theorem C (Random subgroups are stable). Let pX , Sq be a HHS for which CS has infinite diameter, where S is the Ď-maximal element, and consider G ă AutpX , Sq which acts properly and cocompactly on X . Then any k-generated random subgroup of G stably embeds in X .
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 [MS17]. The mapping class group and rightangled Artin groups cases of this result were first established in [TT16].
On purely loxodromic subgroups. In the mapping class group setting [BBKL16] 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 [KMT14] 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 [Tra, Theorem 1.4], who proved that property is equivalent to (3). Since there often exist elements in a right-angled Coxeter group which do not act loxodromically on the contact graph, his condition is not equivalent to (1); it is the ability to change the hierarchically hyperbolic structure as we do in Theorem 3.11, 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 CS where S is the Ď-maximal domain. On the other hand, right-angled Coxeter groups often don't admit universal acylindrical actions on CS 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 [BBKL16], and the second item would generalize what is known in right-angled Artin groups as proven in [KMT14], and partial results about right-angled Coxeter groups as in [Tra]. In the case of the mapping class group, if the second item was true this would resolve a question of Farb-Mosher [FM02]. See also [ADT16, Question 1].
Question D. Let pG, Sq be a hierarchically hyperbolic group which admits a universal acylindrical action on CS, where S is Ď-maximal in S. Let H be a finitely generated subgroup of G. Are the conditions in Theorem B also equivalent to: (1) . . . H is undistorted and acts purely loxodromically on CS?
Note that in the context of Question D, an element acts loxodromically on CS 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.
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 3.11 which allows us to modify a given hierarchically hyperbolic structure pX , Sq by removing CU for some U P S and, in their place, enlarging the space CS where S is the Ď-maximal element of S. 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: Theorem E (Characterization of contracting quasigeodesics). Let G be a hierarchically hyperbolic group. For any D ą 0 and K ě 1 there exists a D 1 ą 0 depending only on D and G such that the following holds for every pK, Kq-quasigeodesic γ Ă X : the quasigeodesic γ has D-bounded projections if and only if γ is D 1 -contracting.
Since the presence of a contracting geodesic implies the group has at least quadratic divergence, an immediate consequence of Theorem E is that any hierarchically hyperbolic group has quadratic divergence whenever X projects to an infinite diameter subset of CS.
Finally, through much of this paper we impose a technical hypothesis on our hierarchically hyperbolic structures, called having clean containers. Although, in Proposition 3.5 this hypothesis is shown to hold for many groups, it does not hold in all cases. In Section 7 we introduce a technical trick which allows us to prove a weakened version of the results of Section 3 without this hypothesis. In turn, this allows us to remove this hypothesis from the remaining results in the paper, for instance allowing us in the case of groups to upgrade Theorem 5.1 to Theorem A and Theorem 4.4 to Theorem E.

Background
2.1. Hierarchically hyperbolic spaces. In this section we recall the basic definitions and properties of hierarchically hyperbolic spaces as introduced in [BHS17b,BHS15].
Definition 2.1 (Hierarchically hyperbolic space). A q-quasigeodesic space pX , d X q is said to be hierarchically hyperbolic if there exists δ ě 0, an index set S, and a set tCW | W P Su of δ-hyperbolic spaces pCU, d U q, such that the following conditions are satisfied: (1) (Projections.) There is a set tπ W : X Ñ 2 CW | W P Su of projections sending points in X to sets of diameter bounded by some ξ ě 0 in the various CW P S. Moreover, there exists K so that each π W is pK, Kq-coarsely Lipschitz.
(2) (Nesting.) S is equipped with a partial order Ď, and either S " H or S contains a unique Ď-maximal element; when V Ď W , we say V is nested in W . We require that W Ď W for all W P S. For each W P S, we denote by There is also a projection ρ W V : CW Ñ 2 CV . (3) (Orthogonality.) S has a symmetric and anti-reflexive relation called orthogonality: we write V KW when V, W are orthogonal. Also, whenever V Ď W and W KU , we require that V KU . Finally, we require that for each T P S and each U P S T for which tV P S T | V KU u ‰ H, there exists W P S T´t T u, so that whenever V KU and V Ď T , we have V Ď W ; we say W is a container associated with T P S and U P S T . Finally, if V KW , then V, W are not Ď-comparable. (4) (Transversality and consistency.) If V, W P S are not orthogonal and neither is nested in the other, then we say V, W are transverse, denoted V &W . There exists κ 0 ě 0 such that if V &W , then there are sets ρ V W Ď CW and ρ W V Ď CV each of diameter at most ξ and satisfying: for all x P X . For V, W P S satisfying V Ď W and for all x P X , we have: (5) (Finite complexity.) There exists n ě 0, the complexity of X (with respect to S), so that any set of pairwise-Ď-comparable elements has cardinality at most n. (6) (Large links.) There exist λ ě 1 and E ě maxtξ, κ 0 u such that the following holds.
Let W P S and let x, x 1 P X . Let N " λd W pπ W pxq, π W px 1 qq`λ. Then there exists There exists a constant α with the following property. Let tV j u be a family of pairwise orthogonal elements of S, and let p j P π V j pX q Ď CV j . Then there exists x P X so that: ‚ d V j px, p j q ď α for all j, (9) (Uniqueness.) For each κ ě 0, there exists θ u " θ u pκq such that if x, y P X and dpx, yq ě θ u , then there exists V P S such that d V px, yq ě κ.
Notation 2.2. Note that below we will often abuse notation by simply writing pX , Sq or S to refer to the entire package of an hierarchically hyperbolic structure, including all the associated spaces, projections, and relations given by the above definition.
Notation 2.3. When writing distances in CU for some U P S, we often simplify the notation slightly by suppressing the projection map π U , i.e., given x, y P X and p P CU we write d U px, yq for d U pπ U pxq, π U pyqq and d U px, pq for d U pπ U pxq, pq. Note that when we measure distance between a pair of sets (typically both of bounded diameter) we are taking the minimum distance between the two sets. Given A Ă X and U P S we let π U pAq denote Y aPA π U paq.
It is often convenient to work with equivariant hierarchically hyperbolic structures, we now recall the relevant structures for doing so. For details see [BHS15].
Definition 2.4 (Hierarchically hyperbolic groups). Let pX , Sq be a hierarchically hyperbolic space. The automorphism group of pX , Sq is denoted AutpX , Sq and is defined as follows. An element of AutpX , Sq consists of a map g : X Ñ X , together with a bijection g ♦ : S Ñ S and, for each U P S, an isometry g˚pU q : CU Ñ Cpg ♦ pU qq so that the following diagrams coarsely commute whenever the maps in question are defined (i.e., when U, V are not orthogonal): A finitely generated group G is said to be a hierarchically hyperbolic group (HHG) if there is a hierarchically hyperbolic space pX , Sq and an action G Ñ AutpX , Sq so that the induced uniform quasi-action of G on X is metrically proper, cobounded, and S contains finitely many G-orbits. Note that when G is a hyperbolic group then with respect to any word metric it inherits a hierarchically hyperbolic structure.
An important consequence of being a hierarchically hyperbolic space is the following distance formula, which relates distances in X to distances in the hyperbolic spaces CU for U P S. The notation t txu u s means include x in the sum if and only if x ą s.
Theorem 2.5 (Distance formula for HHS; [BHS15]). Let pX , Sq be a hierarchically hyperbolic space. Then there exists s 0 such that for all s ě s 0 , there exist C, K so that for all x, y P X , We now recall an important construction of subspaces in a hierarchically hyperbolic space called standard product regions introduced in [BHS17b, Section 13] and studied further in [BHS15]. First we define the two factors in the product space.
Definition 2.6 (Nested partial tuple (F U )). Recall S U " tV P S | V Ď U u. Fix κ ě κ 0 and let F U be the set of κ-consistent tuples (i.e., tuples satisfying the conditions of Definition 2.1.(4)) in ś V PS U 2 CV . Definition 2.7 (Orthogonal partial tuple (E U ) ). Let S K U " tV P S | V KU u Y tAu, where A is a Ď-minimal element W such that V Ď W for all V KU . Fix κ ě κ 0 , let E U be the set of κ-consistent tuples in ś Definition 2.8 (Product regions in X ). Given X and U P S, there are coarsely well-defined maps φ Ď , φ K : F U , E U Ñ X which extend to a coarsely well-defined map φ U : F UˆEU Ñ X . Indeed, for each p a, bq P F UˆEU , and each V P S, the coordinate pφ U p a, bqq V is defined as We refer to F UˆEU as a product region, which we denote P U .
We often abuse notation slightly and use the notation E U , F U , and P U to refer to the image in X of the associated set. In [BHS15,Lemma 5.9] it is proven that these standard product regions have the property that they are "hierarchically quasiconvex subsets" of X .
Here we leave out the definition of hierarchically quasiconvexity, because its only use here is that product regions have "gate maps," as given by the following in [BHS15, Lemma 5.4]: Lemma 2.9 (Existence of coarse gates; [BHS15]). If Y Ď X is k-hierarchically quasiconvex and non-empty, then there exists a gate map for Y, i.e., for each x P X there exists y P Y such that for all V P S, the set π V pyq (uniformly) coarsely coincides with the projection of π V pxq to the kp0q-quasiconvex set π V pYq.
Remark 2.10 (Surjectivity of projections). As one can always change the hierarchical structure so that the projection maps are coarsely surjective [BHS15, Remark 1.3], we will assume that S is such a structure. That is, for each U P S, if π U is not surjective, then we identify CU with π U pX q.
We also need the notion of a hierarchy path, whose existence was proven in [BHS15, Theorem 5.4]: for each W P S, π W˝γ is an unparametrized pR, Rq-quasigeodesic. An unbounded hierarchy path r0, 8q Ñ X is a hierarchy ray.
2.2. Acylindrical actions. We recall the basic definitions related to acylindrical actions; the canonical references are [Bow08] and [Osi16]. We also discuss a partial order on these actions which was recently introduced in [ABO16].
Definition 2.12 (Acylindrical). The action of a group G on a metric space X is acylindrical if for any ε ą 0 there exist R, N ě 0 such that for all x, y P X with dpx, yq ě R, |tg P G | dpx, gxq ď ε and dpy, gyq ď εu| ď N.
Recall that given a group G acting on a hyperbolic metric space X, an element g P G is loxodromic if the map Z Ñ X defined by n Þ Ñ g n x is a quasi-isometric embedding for some (equivalently any) x P X. However, an element of G 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 F 2 » xxy˚xyy. In the former action, every non-trivial element is loxodromic, while in the latter action, no powers of x and y are loxodromic.
Definition 2.13 (Generalized loxodromic). An element of a group G is called generalized loxodromic if it is loxodromic for some acylindrical action of G on a hyperbolic space.
Remark 2.14. By [Osi16, Theorem 1.1], every acylindrical action of a group on a hyperbolic space either has bounded orbits or contains a loxodromic element. By [Osi16, Theorem 1.2.(L4)] and Sisto [Sis16, Theorem 1], every generalized loxodromic element is Morse. Therefore, if a group H does not contain any Morse elements, it does not contain any generalized loxodromics, and thus H must have bounded orbits in every acylindrical action on a hyperbolic space. This is the case when, for example, H is a non-trivial direct product.

Definition 2.15 (Universal acylindrical action). An acylindrical action of a group on a hyperbolic space is a universal acylindrical action if every generalized loxodromic element is loxodromic.
Notice that if every acylindrical action of a group G on a hyperbolic space has bounded orbits, then G does not contain any generalized loxodromic elements, and the action of G on a point (which is acylindrical) is a universal acylindrical action.
The following notions are discussed in detail in [ABO16]. We give a brief overview here. Fix a group G. Given a (possibly infinite) generating set X of G, let |¨| X denote the word metric with respect to X. Given two generating sets X and Y , we say X is dominated by Y and write X ĺ Y if sup yPY |y| X ă 8.
Note that when X ĺ Y , then the action G ñ ΓpG, Y q provides more information about the group than G ñ ΓpG, Xq, and so, in some sense, is a "larger" action. The two generating sets X and Y are equivalent if X ĺ Y and Y ĺ X; when this happens we write X " Y . Let AHpGq be the set of equivalence classes of generating sets X of G such that ΓpG, Xq is hyperbolic and the action G ñ ΓpG, Xq is acylindrical, where ΓpG, Xq is the Cayley graph of Γ with respect to the generating set X. We denote the equivalence class of X by rXs. The preorder ĺ induces an order on AHpGq, which we also denote ĺ.
Definition 2.16 (Largest). We say an equivalence class of generating sets is largest if it is the largest element in AHpGq under this ordering.
Given a cobounded acylindrical action of G on a hyperbolic space S, a Milnor-Schwartz argument gives a (possibly infinite) generating set Y of G such that there is a G-equivariant quasi-isometry between G ñ S and G ñ ΓpG, Y q. By a slight abuse of language, we will say that a particular cobounded acylindrical action G ñ S 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, rY s, that is the largest element in AHpGq.
Remark 2.17. Notice that by definition, every largest acylindrical action is a universal acylindrical action.
2.3. Stability. Stability is strong coarse convexity property which generalizes quasiconvexity in hyperbolic spaces and convex cocompactness in mapping class groups [DT15]. In the general context of metric spaces, it is essentially the familiar Morse property generalized to subspaces, so we begin there.
We call N the stability gauge for γ and say γ is N -stable if we want to record the constants.
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 [DT15]: Definition 2.19 (Stable embedding). We say a quasi-isometric embedding f : X Ñ Y between quasigeodesic metric spaces is a stable embedding if there exists a stability gauge N such that for any quasigeodesic constants K, C and any pK, Cq-quasigeodesic γ Ă X, then f pγq is an N -stable quasigeodesic in Y .
The following generalizes the notion of a Morse quasigeodesic to subgroups: Definition 2.20 (Subgroup stability). Let H be subgroup of a finitely generated group G. We say H is a stable subgroup of G if some (equivalently, any) orbit map of H into some (any) Cayley graph (with respect to a finite generating set) of G is a stable embedding.
If for some h P G, H " xhy is stable, then we call h stable. Such elements are often called Morse elements.
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.

Altering the hierarchically hyperbolic structure
The goal of this section is to prove that any hierarchically hyperbolic space satisfying two technical assumptions-the bounded domain dichotomy and the clean container propertyadmits a hierarchically hyperbolic structure with unbounded products, i.e., every non-trivial product region in the ambient space has unbounded factors; see Theorem 3.11 below.
In particular, this establishes that most of the standard examples of hierarchically hyperbolic groups admit a hierarchically hyperbolic group structure with unbounded products. This is a key ingredient in our complete characterization of the contracting property in such spaces, which we establish in Section 4.
Let M ą 0 and let S M Ă S be the set of domains U P S such that there exists V P S and W P S K V satisfying: We note that any hierarchically hyperbolic group has the bounded domain dichotomy. (Also, note that this property implies the space is "asymphoric" as defined in [BHS17c].) Definition 3.3 (Unbounded products). We say that a hierarchically hyperbolic space pX , Sq has unbounded products if it has the bounded domain dichotomy and the property that if U P S has diampF U q " 8, then diampE U q " 8.
3.2. Clean containers. The clean container property is a technical assumption related to the orthogonality axiom.
Definition 3.4 (Clean containers). A hierarchically hyperbolic space pX , Sq has clean containers if for each T P S and each U P S T with tV P S T | V K U u ‰ H, the associated container provided by the orthogonality axiom is orthogonal to U .
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 [BHS15,Sections 8 & 9] and [BHS17a, Section 6] for details on the structure in the new spaces.
Proposition 3.5. The following spaces admit hierarchically hyperbolic structures with clean containers.
Proof. Hierarchically hyperbolic structures for these spaces were constructed in [BHS17b] and [BHS15].
(1) The statement is immediate for hyperbolic groups, as they all admit hierarchically hyperbolic structure with no orthogonality, and thus the containers axiom is vacuous.
(2) For mapping class groups, in the standard structure, a container for domains orthogonal to a given subsurface U is the complementary subsurface, which is orthogonal to U . (4) Given a geometric 3-manifold M of the above form, π 1 pM q is quasi-isometric to a (possibly degenerate) product of hyperbolic spaces, and so has clean containers by Lemma 3.6. Given an irreducible non-geometric graph manifold M , the hierarchically hyperbolic structure comes from considering π 1 pM q as a tree of hierarchically hyperbolic spaces with clean containers and hence has clean containers by Lemma 3.8. Finally, the general case of a non-geometric 3-manifold M follows immediately from Lemma 3.7 and the fact that π 1 pM q is hyperbolic relative to its maximal graph manifold subgroups. l Lemma 3.6. The product of two hierarchically hyperbolic spaces which both have clean containers has clean containers.
Proof. Let pX 0 , S 0 q and pX 1 , S 1 q be hierarchically hyperbolic spaces with clean containers.
In the hierarchically hyperbolic structure pX 0ˆX1 , Sq given by [BHS15,Theorem 8.25] there are two types of containers, those that come from one of the original structures and those that do not. Containers of the first type are clean, as both original structures have clean containers.
The second type of domain consists of new domains obtained as follows. Given a domain U P S i , a new domain V U is defined with the property that it contains under nesting any domain in S i which is orthogonal to U and also any domain in S i`1 . Thus, by construction V U is a container for everything orthogonal to U . As V U K U , the result follows. l Lemma 3.7. If G is hyperbolic relative to a collection of hierarchically hyperbolic spaces which all have clean containers, then G is a hierarchically hyperbolic space with clean containers.
Proof. That G is a hierarchically hyperbolic space follows from [BHS15, Theorem 9.1]. In the hierarchically hyperbolic structure on G, no new orthogonality relations are introduced, and thus all containers are containers in the hierarchically hyperbolic structure of one of the peripheral subgroups. As each of these structures have clean containers, it follows that G does, as well. l Lemma 3.8. Let T be a tree of hierarchically hyperbolic spaces such that XpT q is hyperbolic. If for each v P T , the hierarchically hyperbolic space pX v , S v q has clean containers, then so does XpT q.
Proof. This follows immediately from the proof of [BHS15, Lemma 8.10] and the fact that edge-hieromorphisms are full and preserve orthogonality. In the notation from that result, we note that, if S v has clean containers for each v P T , then the domain A v P S v described in the proof also has the property that A v K U v . Therefore, as edge-hieromorphisms are full and preserve orthogonality, rA v s K rU s. l The following uses the notion of hyperbolically embedded subgroups introduced in [DGO17].
Lemma 3.9. Let pG, Sq be a hierarchically hyperbolic group with clean containers, and let H ãÑ hh pG, Sq. Then there exists a finite set F Ă H´t1u such that for all N Ÿ H with F X N " H and H{N is hyperbolic, then the group G{N , obtained by quotienting by the normal closure, is a hierarchically hyperbolic group with clean containers.
Proof. Recall that in the hierarchically hyperbolic structure pG{N , S N q obtained in [BHS17a, Theorem 6.2] (and in the notation used there), two domains U, To prove the container axiom, we consider domains T, U, V P S such that T P T, U P U and V P V for all V P V, and such that any pair is a linked pair. Then the orthogonality axiom for pG, Sq provides a domain W such that W Ě V and W Ď T . As pG, Sq has clean containers, we also have that W K U . This implies that ρ U S and ρ W S are coarsely equal by [DHS15, Lemma 1.5], and so tU, W u is a linked pair. Therefore, W K U. l 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 and clean containers. We first describe the hyperbolic spaces that will be part of the new structure. Let pX , Sq be a hierarchically hyperbolic space with the M -bounded domain dichotomy. Let S M Ă S be the set of U P S such that there exists a V P S and W P S K V satisfying diampCV q ą M and diampCW q ą M . For each U P S, define S M U Ă S U similarly. Remark 3.10 (Factored spaces). As defined in [BHS17a], given pX , Sq and T Ă S the factored space p F T is the space obtained from X by coning-off all F V for V P T. Sometimes we abuse language slightly and refer to this as the factored space obtained from X by collapsing T. In particular, when S is the Ď-maximal element of S, then CS is identified with the space p F SztSu , which is obtained from X by coning-off F U for all U P SztSu.
We often consider the case of a fixed pX , Sq and U P S and then applying this construction to the hierarchy hyperbolic structure pF U , S U q. For this application, note that π U pX q is quasi-isometric to p F S U ztU u , by [BHS17a, Corollary 2.9], and thus so is CU , by Remark 2.10.
Consider the factored space p F S M . By Lemma 3.1, S M is closed under nesting and hence p X S M is a hierarchically hyperbolic space. Moreover, since this hierarchically hyperbolic space has the property that no pair of orthogonal domains both have diameter larger than M , by [BHS17c, Corollary 2.16] it is hyperbolic for some constant depending only on pX , Sq and M .
For each U P S M , we similarly define T U to be the factored space obtained from F U by collapsing S M U . This setup again satisfies the assumptions in Lemma 3.1 and [BHS17c, Corollary 2.16], so we obtain that T U is δ-hyperbolic for some δ which we can take to be uniform by finite complexity of pX , Sq.
The next result uses the above spaces to obtain a hierarchically hyperbolic structure with particularly nice properties from a given hierarchically hyperbolic structure.
Theorem 3.11. Every hierarchically hyperbolic space with the bounded domain dichotomy and clean containers admits a hierarchically hyperbolic structure with unbounded products.
Proof. Let pX , Sq be a hierarchically hyperbolic space. Let T Ă S which we define to include S as well as the set of domains U with both F U and E U unbounded.
We begin to define our new hierarchically hyperbolic structure on X by taking T as the index set. We let T S " p X S M be the hyperbolic space associated to the top level domain S, and let T U be the hyperbolic spaces associated to each U P T, as defined in the discussion before the theorem.
Notice that for each U P T, the space T U is identical to the space CU if and only if for every W P S for which W Ď U satisfies W P T. Now, suppose T U ‰ CU (and hence that there exists at least one W P S with W Ď U satisfying W R T). As noted above, we can take F U , T U , and CU to have the same underlying space with different metrics. Even though the identity map on F U no longer induces an isometry ψ U : T U Ñ CU , this map is still 1-Lipschitz. In fact, we now show that ψ U is actually bi-Lipschitz for all U P TztSu. By definition of T, any W P S for which W R T either F W or E W is bounded. Since U ‰ S, E U is unbounded, and it follows that E W is unbounded. Since W R T, it follows that F W must be bounded; moreover, pX , Sq has the M -bounded domain dichotomy, so F W is uniformly bounded. Thus, by the distance formula in F U , for any pair of points in in F U , their distance in T U can be at most a uniformly bounded multiple of their distance in CU , and so ψ U is a bi-Lipschitz map, with constant M .
To avoid confusion, if U P T, we use the notation d U for distance in T U and the notation d CU for distance in CU .
We take the associated projections π U to be the composition of the nearest point projection X Ñ F U and the factor map F U Ñ T U . If U, V P T, we take the relative projections ρ V U to be the preimage under ψ U of the corresponding relative projections in pX , Sq whenever U ‰ S. In the case that U " S, we take ρ V S to be the image of F V under the factor map X Ñ T S . We now check the axioms for pX , Tq. Projections: Since the metrics on our new spaces T U for U P T are not the same, we need to check that these new projections are still coarsely Lipschitz. This, however, is clear, as π U is the composition of a nearest point projection and a factor map, both of which are coarsely Lipschitz.
Nesting: The partial order and projections are given by construction. The diameter bound in the case of nesting projections is immediate from the bound from pX , Sq and the fact that the maps ψ U are bi-Lipschitz for all U P T.
Orthogonality: This is essentially inherited from the hierarchically hyperbolic structure pX , Sq. The first and last statements are immediate. To prove the second statement, let T P T and let U P T T be such that tV P T T | V K U u ‰ H. Then as T Ď S, by the orthogonality axiom for pX , Sq there is a domain W P S such that W Ĺ T and whenever V Ď T and V K U , we have V Ď W . We will show that W P T.
Indeed, F W is unbounded as there exists some V P T with V Ď W . Furthermore, as the hierarchically hyperbolic structure pX , Sq has clean containers, W K U . As U P T, it follows that E W is unbounded, as well.
Transversality and consistency: For U, V P T Ď S with U &V , the relative projections ρ U V , ρ V U are well-defined and satisfy the required bounds using the constant M¨ξ, where ξ is the original constant from the hierarchically hyperbolic structure on pX , Sq.
We now check the consistency conditions. For all U P T distances in T U can only increase from distances in CU by factor of M , so the consistency inequalities clearly hold by this fact and the consistency axiom from pX , Sq. Moreover, we may take the constant to be M¨κ 0 , where κ 0 is the original constant from pX , Sq.
Partial realization: If tV j u is a family of pairwise orthogonal domains of T, then tV j u is a family of pairwise orthogonal domains of S. By the partial realization axiom for pX , Sq, there is a constant α and a point x P X such that the conclusion holds for all W P S. By increasing the constant to M¨α, we also have a bound on the appropriate distances in T W , and the axiom holds.
Finite complexity: This clearly holds by construction. Large link axiom: Let λ and E be the constants from the large link axiom for pX , Sq, let W P T, and let x, x 1 P X . Consider the set tT i u Ă S W´t W u provided by the large link axiom for pX , Sq. Since T i Ď W , it follows that E T i is unbounded for each i. Let T P T W´t W u. If d T px, x 1 q ą E¨M , it follows that F T is unbounded. Furthermore, d CT px, x 1 q ą E, whence T Ď T i for some i by the large link axiom for pX , Sq. Therefore F T i is unbounded, and so T i P T. The result follows.
Bounded geodesic image: For all domains in TztSu, we have increased distances in the corresponding hyperbolic spaces by no more than M . Hence this axiom holds with the original constants multiplied by M .
Uniqueness: Let κ ą 0. We can take θ 1 u ą maxtθ u pκq, M u, where θ u pκq is the original constant from the uniqueness axiom for pX , Sq. Then if x, y P X with dpx, yq ą θ 1 u , then uniqueness for pX , Sq implies there exists U P S with d CU px, yq ą M . Either U P T or diampCU q " 8 and E U is bounded. We are done in the first case. In the second case, by construction T U is uniformly quasi-isometrically embedded in T S , and hence d S px, yq is at least a uniform constant depending only on M and the quasi-isometry constants. l Corollary 3.12. If pG, Sq is a hierarchically hyperbolic group with clean containers, then pG, Tq is a hierarchically hyperbolic group with unbounded products.
Proof. Recall that every hierarchically hyperbolic group has the bounded domain dichotomy. Thus by Theorem 3.11, pG, Tq is a hierarchically hyperbolic space with unbounded products. It remains only to show that is a hierarchically hyperbolic group structure. The action of G on itself is clearly metrically proper and cobounded, so it only remains to show that T contains finitely many G-orbits. If U P S but U R T, then either F U or E U must be bounded. Then for each g P G, the same will be true for F gU or E gU , which shows that gU R T. Thus G¨U Ć T. Since S has finitely many G-orbits, the result follows. l

Characterization of contracting geodesics
For this section, fix a hierarchically hyperbolic space pX , Sq with the bounded domain dichotomy; denote the Ď-maximal element S P S. Definition 4.1 (Bounded projections). Let Y Ă X and D ą 0. We say that Y has Dbounded projections if for every U P SztSu, we have d U pYq ă D.
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 [MM99, Lemma 6.1] and [Beh06, Lemma 6.2]; this explicit statement can also be found in [DT15, Section 4].
Lemma 4.3. If γ is a D-contracting pK, Kq-quasigeodesic in a metric space X, then γ is D 1 -stable for some D 1 depending only on D and K.
The following result and argument both generalize and simplify the analogous result for mapping class groups in [Beh06].
Theorem 4.4. Let pX , Sq be a hierarchically hyperbolic space. For any D ą 0 and K ě 1 there exists a D 1 ą 0 depending only on D and pX , Sq such that the following holds for every pK, Kq-quasigeodesic γ Ă X . If γ has D-bounded projections, then γ is D 1 -contracting. Moreover, if pX , Sq has the bounded domain dichotomy and clean containers, then X admits a hierarchically hyperbolic structure pX , Tq with unbounded products where, additionally, we have that if γ is D-contracting, then γ has D 1 -bounded projections.
Remark 4.5. In Section 7, we introduce a technical trick that allows us to remove the assumption that pX , Sq has clean containers. Thereby Theorem E follows from Theorem 4.4.
Proof. First suppose that γ has D-bounded projections. It follows immediately from the definition that γ is a hierarchically quasiconvex subset of X . Hierarchical quasiconvexity is the hypothesis necessary to apply [BHS17a, Lemma 5.4], which then yields existence of a coarsely Lipschitz gate map g : X Ñ γ, i.e., for each x P X , the image gpxq P γ has the property that for all U P S the set π U pgpxqq is a uniformly bounded distance from the projection of π U pxq to π U pγq.
We will use g as the map to prove γ is contracting. Gate maps satisfy condition (1) of Definition 4.2 by definition and condition (2) since they are coarsely Lipschitz. Hence it remains to prove that condition (3) of Lemma 4.3 holds.
Fix a point x P X with d X px, γq ě B 0 and let y P X be any point with d X px, yq ă B 1 d X px, γq for constants B 0 and B 1 as determined below.
Since g is a gate map and γ has D-bounded projections, for all U P S´tSu we have d U pgpxq, gpyqq ă D. Thus, by taking a threshold L for the distance formula (Theorem 2.5) larger than D, we have d X pgpxq, gpyqq -pK,Cq d S pgpxq, gpyqq, for uniform constants K, C. Thus it suffices to prove that d S pgpxq, gpyqq is bounded by some uniform constant B 2 . We also choose L to be larger than the constants in Definition 2.1.(4). By Definition 2.1.(1), the maps π U are Lipschitz with a uniform constant. Taking B 0 sufficiently large, it follows that there exists U P S so that d U px, gpxqq ą L. By choosing B 1 to be sufficiently small, and applying the distance formula to the pairs px, yq and px, gpxqq, the fact that the projections π U are Lipschitz implies that the sum of the terms in the distance formula associated to px, gpxqq is much greater than the sum of those associated to px, yq. Having chosen B 1 ă 1 2 , we have ř d U px, gpxqq ą 2 ř d U px, yq ą ř pd U px, yq`Lq. Thus, there exists W P S for which d W px, gpxqq ą d W px, yq`L.
If W " S, then having d S px, gpxqq ą d S px, yq`L (where we enlarge L if necessary) would already show that the CS-geodesic between x and y was disjoint from π S pγq and then hyperbolicity of CS would yield a uniform bound on the d S pgpxq, gpyqq.
Otherwise, we may assume W ‰ S. By the triangle inequality, we have d W py, gpxqq ą L. Further, since, as noted above, the CW projections between gpxq and gpyq are uniformly bounded, by choosing B 0 large enough and B 1 small enough, we also have d W py, gpyqq ą L.
By the bounded geodesic image axiom (Definition 2.1.(7)), any geodesic in CS either has bounded projection to CU or satisfies π S pγq X N E pρ U S q ‰ H for any U P S´tSu. For any geodesic from π S pxq to π S pgpxqq (or from π S pyq to π S pgpyq), the above argument implies that the first condition doesn't hold for W . Thus, in both cases, we know that any such geodesic must pass uniformly close to ρ W S . Hence the hyperbolicity of CS implies γ is contracting, and the first implication holds.
We prove the second implication by contradiction. By Theorem 3.11, we obtain a new structure pX , Tq which has unbounded products. For every U P TztSu we have that both F U and E U are unbounded, hence every U P TztSu yields a non-trivial product region P U " E UˆFU which is uniformly quasi-isometrically embedded in X .
Suppose γ is contracting but doesn't have D-bounded projections. Then we obtain a sequence tU i u P TztSu with diampπ CU i pγqq Ñ 8. Thus there is a sequence of pairs of points x i , y i P γ, so that d U i px i , y i q -K i , with K i Ñ 8. For each i, let q i be a R-hierarchy path between x i , y i . By [BHS15,Proposition 8.24], there exists ν ą 0 depending only on R and pX , Sq, such that diam U i pq i X N ν pP U i qq -K i . Since γ is contracting, it is uniformly stable by Lemma 4.3. Since γ is uniformly stable and the q i are uniform quasigeodesics, it follows that each q i is contained in a uniform neighborhood of γ. Hence arbitrarily long segments of γ are uniformly close to the product regions P U i . This contradicts the assumption that γ is contracting and completes the proof. l

Universal and largest acylindrical actions
The goal of this section is to show that for every hierarchically hyperbolic group pG, Sq with clean containers AHpGq 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.
Theorem 5.1. Every hierarchically hyperbolic group with clean containers admits a largest acylindrical action.
Remark 5.2. In Section 7, we introduce a technical trick that allows us to remove the assumption that pX , Sq has clean containers. Whence Theorem 5.1 implies Theorem A.
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 Theorem 6.3 in [ABO16]; we give a sketch of the argument here. Recall that an action H ñ S is elliptic if H has bounded orbits.

Proposition 5.3 ([ABO16]
). Let G be a group, tH 1 , . . . , H n u a finite collection of subgroups of G, and F be a finite subset of G such that F Y p Ť n i"1 H i q generates G. Assume that: (1) ΓpG, F Y p Ť n i"1 H i qq is hyperbolic and acylindrical. (2) Each H i is elliptic in every acylindrical action of G on a hyperbolic space. Then rF Y p Ť n i"1 H i qs is the largest element in AHpGq. Proof. First notice that by assumption (1), ΓpG, F Y p Ť n i"1 H i q is an element of AHpGq. Let G ñ S be a cobounded acylindrical action of G on a hyperbolic space, S, and fix a basepoint s P S. Then there exists a bounded subspace B Ă S such that S Ď Ť gPG g¨B. By assumption (2), the orbit H i¨s is bounded for all i " 1, . . . , n. Since |F | ă 8, we know diampF¨sq ă 8 and thus K " maxtdiampBq, diampH 1¨s q, . . . , diampH n¨s q, diampF¨squ is finite. Let C " ts 1 P S | dps 1 , sq ď 3Ku, and let Z " tg P G | g¨C X C ‰ Hu.
The standard Milnor-Schwartz Lemma argument shows that Z is an infinite generating set of G and there exists a G-equivariant quasi-isometry S Ñ ΓpG, Zq. It is clear that Z contains F , as well as H i for all i " 1, . . . , n and thus rZs ĺ rF Y p Ť n i"1 H i s. The result follows. l Proof of Theorem 5.1. Let pG, Sq be a hierarchically hyperbolic group with finite generating set F . By Corollary 3.12, there is a hierarchically hyperbolic group structure pG, Tq with unbounded products. Recall that S is the Ď-maximal element of T with associated hyperbolic space T S . The action on T S is acylindrical by [BHS17b,Theorem K]. Moreover, the action of G on T S is cobounded, so let B be a fundamental domain for G ñ T S and U " tU P T | π S pF U q Ă B and U is Ď-maximal in TztSuu.
Notice that U will contain exactly one representative from each G-orbit of domains, and so must be a finite set. Indeed, for a hierarchically hyperbolic group, this follows from the fact that the action of G on T is cofinite.
Let H i ď G be the stabilizer of F U i for each U i P U . By a standard Milnor-Schwartz argument (see [ABO16] for details) there is a G-equivariant quasi-isometry between ΓpG, F Y p Ť n i"1 H i qq and T S , where n " |U |. Therefore condition (1) of Proposition 5.3 is satisfied. By definition, each H i sits inside a non-trivial direct product in G, the product region P U i associated to each U i P U . It follows that H i must be elliptic in every acylindrical action of G on a hyperbolic space (see Remark 2.14), satisfying condition (2). Therefore, by Proposition 5.3, the action is largest. l Remark 5.4. The proof of Theorem 5.1 can be extended to treat a number of groups which are hierarchically hyperbolic spaces, but not hierarchically hyperbolic groups. For example, it was shown in [BHS15, Theorem 10.1] that every fundamental group of a 3-manifold with no Nil or Sol in its prime decomposition admits a hierarchically hyperbolic space structure, but as explained in [BHS15, Theorem 10.2] it is likely that these don't all admit hierarchically hyperbolic group structures. Nonetheless, the proof of the above theorem works in this case by replacing the use of the fact that the action of G on T is cofinite, with the fact that for π 1 pM q, U is precisely the set of Ď-maximal domains in the hierarchically hyperbolic structure on each of the Seifert-fibered components of the prime decomposition of M , and so is finite.
Remark 5.5. There is an instructive direct proof of the universality of the above action using the characterization of contracting elements in Section 4, which we now give. Let g P G be an infinite order element and consider the geodesic xgy in ΓpG, F q.
If xgy is contracting in ΓpG, F q, then by Theorem 4.4 all proper projections are bounded, and thus by the distance formula, g is loxodromic for the action on T S .
If xgy is not contracting in ΓpG, F q, then there exists some U P T such that π U pxgyq is unbounded. Thus for any increasing sequence of constants pK i q with K i Ñ 8, there are sequences of pairs of points x i , y i P xgy such that dpx i , y i q Ñ 8 as i Ñ 8 and d U px i , y i q ě K i . For each i, let γ i be an R-hierarchy path between x i and y i . By definition, γ i is a uniform quasigeodesic. Then by [BHS15,Proposition 8.24], there exists ν ą 0 depending only on R and pX , Tq such that diam U pγ i X N ν pP U qq ě K i . If g is a generalized loxodromic, then xgy is stable, by [Sis16], and so the subgeodesic rx i , y i s stays within a uniform bounded distance of γ i . Thus arbitrarily long subgeodesics of xgy stay within a uniformly bounded distance of a product region, P U . This contradicts xgy being Morse, and therefore g is not a generalized loxodromic element.
This remark directly implies that the action on T S is a universal acylindrical action. (The universality of the action can also be proven using the classification of elements of AutpSq described in [DHS15].) Another immediate consequence of the above remark is the following, which for hierarchically hyperbolic groups strengthens a result obtained by combining Osin [Osi16, Theorem 1.2.(L4)] and Sisto [Sis16, Theorem 1], which together prove that a generalized loxodromic element in an acylindrically hyperbolic group is quasi-geodesically stable.
Corollary 5.6. Let pG, Sq be a hierarchically hyperbolic group. An element g P G is generalized loxodromic if and only if xgy is contracting in Γ.
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 CAT(0) cubical groups is new; the other cases were recently established to be largest in [ABO16]. Proof. With the exception of p3q the above are all hierarchically hyperbolic groups [BHS17b, BHS15,HS16] and therefore have the bounded domain dichotomy. If G is the fundamental group of a three-manifold with no Nil or Sol in its prime decomposition, then while G is not always a hierarchically hyperbolic group, it has a hierarchically hyperbolic structure pX , Sq such that X is the Cayley graph of G and G ă AutpSq. Additionally, all of the associated hyperbolic spaces are infinite, and therefore pX , Sq has the bounded domain dichotomy. By Theorem 3.5, the above groups each have clean containers, so the result follows. l We give an explicit description of these actions for each hierarchically hyperbolic group in the corollary, in the sense that we describe the set W of domains which are removed from the standard hierarchical structure of the group. Recall that the space T S is constructed from X by coning off all elements of T " SzW.
(1) Hyperbolic groups (and their subgroups) have a canonical simplest hierarchically hyperbolic group structure given by taking S " tSu, where CS is the Cayley graph of the group with respect to a finite generating set. For this structure, W " H, and the action on the Cayley graph is clearly a universal acylindrical action.
(2) For mapping class groups, the natural hierarchically hyperbolic group structure is S is the set of stabilizers of simple closed curves on the surface and a maximal element S, where CS is the curve complex. For this structure, W " H, and the action on the curve complex is universal. Universality of this action was shown by Osin in [Osi16], and follows from results of Masur-Minsky and Bowditch [Bow08,MM99]. (3) If M is a 3-manifold with no Nil or Sol in its prime decomposition and G " π 1 M , then W 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 G is a group that acts properly and cocompactly on a CAT(0) cube complex X, then by [HS16], X has a G-equivariant factor system. This factor system gives a hierarchically hyperbolic group structure in which S is the closure under projection of the set of hyperplanes along with a maximal element S, where CS is the contact graph as defined in [Hag14]. In this structure, W is the set of indices whose stabilizer in G 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 W " H. In this case, the contact graph CS is quasi-isometric to the extension graph defined by [KK14]. That the action on the extension graph is a universal acylindrical action follows from the work of [KK14] and the centralizer theorem for right-angled Artin groups. This action is also shown to be largest in [ABO16].
We give a concrete example of the situation in the case of a right-angled Coxeter group.
Example 5.8. Let G be the right-angled Coxeter group whose defining graph is a pentagon. Then G " xa, b, c, d, e | ra, bs, rb, cs, rc, ds, rd, es, ra, esy, and the Cayley graph of G is the tiling of the hyperbolic plane by pentagons. We consider the dual square complex to this tiling. To form the contact graph CS, we start with the square complex and cone off each hyperplane carrier, which is equivalent to coning off the hyperplane stabilizers in the Cayley graph. The result is a quasi-tree. Thus a fundamental domain for the hierarchically hyperbolic group structure of G is tU a , U b , U c , U d , U e , Su where U v is associated to the stabilizer of the hyperplane labeled by v and S is associated to the contact graph described above.
Consider the hyperplane J b that is labeled by b. Then the stabilizer of J b is the link of the vertex b, which contains the infinite order element ac. As G is a hyperbolic group, all infinite order elements are generalized loxodromic, but ac is not loxodromic for the action on the contact graph since its axis lies in a hyperplane stabilizer that has been coned-off. Thus the action on the contact graph is not universal.
Let U b P S be the element associated to StabpJ b q. Then StabpJ b q " xa, b, c | ra, bs, rb, csy » D 8ˆZ {2Z » F U bˆE U b is a product region, and the maximal orthogonal component E U b is bounded. Thus U b P W, as is U v , for each vertex v of the defining graph. The contact graph associated to pF U b , S U b q is a line, and the element ab is loxodromic for the action on this space.
Note that once W has been removed from S, the resulting hierarchically hyperbolic structure is pG, tSuq, the canonical hierarchically hyperbolic structure for a hyperbolic group, in which CS " ΓpG, ta, b, c, d, euq.

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 [Beh06,DMS10,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 [Sul14,Beh06,DT15,Min96]. We record the following relation between stability and contracting subsets which holds in a fairly general context: Proof. Lemma 4.3 shows that contracting implies stable (the assumption on unbounded products is not necessary for this implication). For the other direction, the fact that X has unbounded products implies that Y has bounded projections, since otherwise one could find large segments of quasigeodesics contained inside direct products, contradicting stability. The result now follows from Theorem 4.4. l The following provides a general characterization of stability: Corollary 6.2. Let i : Y Ñ X be map from a metric space into a hierarchically hyperbolic space pX , Sq with unbounded products. The following are equivalent: (1) i is a stable embedding; (2) ipYq is undistorted and has uniformly bounded projections; (3) π S˝i : Y Ñ CS is a quasi-isometric embedding.
Proof. Items (1) and (2) are equivalent via Corollary 6.1. Equivalence of (2) and (3) follows from the distance formula and the assumption that i is a quasi-isometric embedding. l 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 [DHS15], Durham, Hagen, and Sisto introduced a boundary for any hierarchically hyperbolic space. We collect the relevant properties we need in the following theorem: Theorem 6.3 (Theorem 3.4 and Proposition 5.8 in [DHS15]). If pX , Sq is a proper hierarchically hyperbolic space, then there exists a topological space BX such that BX Y X " X compactifies X , and the action of AutpX , Sq on X extends continuously to an action on X .
Moreover, if Y is a hierarchically quasiconvex subspace of X , then, with respect to the induced hierarchically hyperbolic structure on Y, the limit set of ΛY of Y in BX is homeomorphic to BY and the inclusion map i : Y Ñ X extends continuously an embedding Bi : BY Ñ BX .
Building on ideas in [CS15], Cordes introduced the Morse boundary of a proper geodesic metric space [Cor15], which was then refined further by Cordes-Hume in [CH16]. 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 [Cor15].
We briefly discuss the construction of the Morse boundary and refer the reader to [Cor15,CH16] for details.
Consider a a proper geodesic metric space X with a basepoint e P X. Given a stability gauge N : R 2 ě0 Ñ R ě0 , define a subset X pN q e Ă X to be the collection of points y P X such that e and y can be connected by an N -stable geodesic in X. Each such X pN q e is δ Nhyperbolic for some δ N ą 0 depending on N and X [CH16, Proposition 3.2]; here, we use the Gromov product definition of hyperbolicity, as X pN q e need not be connected. Moreover, any stable subset of X embeds in X pN q e for some N [CH16, Theorem A.V]. The set of stability gauges admits a partial order: N 1 ă N 2 if and only if N 1 pK, Cq ă N 2 pK, Cq for all constants K, C. In particular, if N 1 ă N 2 , then X Since each X pN q e is Gromov hyperbolic, each admits a Gromov boundary BX pN q e . Take the direct limit with respect to this partial order to obtain a topological space B s X called the Morse boundary of X.
We fix pX , Sq, a hierarchically hyperbolic structure with unbounded products.
Definition 6.4. We say λ P BX has bounded projections if for any e P X , there exists D ą 0 such that any R-hierarchy path re, λs has D-bounded projections. Let B c X denote the set of points λ P BX with bounded projections.
The boundary BX contains BCU for each U P S, by construction. The next lemma shows that the boundary points with bounded projections are contained in BCS, as a subset of BX , where S is the Ď-maximal element. In general, the set of cobounded boundary points may be a very small subset of BCS. 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.
Lemma 6.5. The inclusion B c X Ă BCS holds for any pX , Sq with unbounded products where S is the Ď-maximal element of S. Moreover, if X is also proper, then for any D ą 0 there exists D 1 ą 0 depending only on D and pX , Sq such that if px n q Ă X is a sequence with x n Ñ λ P BX such that re, x n s has D-bounded projections for some e P X and each n, then re, λs has D 1 -bounded projections.
Proof. Let λ P B c X . If re, λs is an R-hierarchy path, then re, λs has an infinite diameter projection to some CU , see, e.g., [DHS15,Lemma 3.3]. As λ has bounded projections, we must have U " S. Since π S pre, λsq Ă CS is a quasigeodesic ray, the first statement follows. Now suppose that X is also proper. For each n, let γ n " re, x n s be any R-hierarchy path between e and x n in X . The Arzela-Ascoli theorem implies that after passing to a subsequence, γ n converges uniformly on compact sets to some R 1 -hierarchy path γ with R 1 depending only on R and pX , Sq. Hence γ has D 1 -bounded projections for some D 1 depending only on D and pX , Sq. Moreover, since x n Ñ λ in CS, it follows that π S pγq is asymptotic to λ in CS.
If re, λs is any other R 1 -hierarchy path, it follows from uniform hyperbolicity of the CU and the definition of hierarchy paths that d Haus U pγ, re, λsq is uniformly bounded for all U P S. Since γ has D 1 -bounded projections, the distance formula implies that re, λs has D 2 -bounded projections for some D 2 depending only on D and pX , Sq, as required. l 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 [CH16], the stable asymptotic dimension of a metric space X is the maximal asymptotic dimension a stable subspace of X.
The stable asymptotic dimension of a metric space X is always bounded above by its asymptotic dimension. Behrstock, Hagen, and Sisto, [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 [BHS17a] are functions of the asymptotic dimension of the top level curve graph.
In the following theorem, we prove that a hierarchically hyperbolic space pX , Sq has finite stable asymptotic dimension under the assumption that asdimpCSq ă 8. Recall that asymptotic dimension is monotonic under taking subsets. Thus, if X is assumed to be proper, so that asdimpCSq ă 8, then X (and therefore the stable subsets) have finite asymptotic dimension by [BHS17a]. Here, using some geometry of stable subsets we obtain a sharper bound on asdim s pX q than asdimpX q.
Theorem 6.6. Let pX , Sq be a hierarchically hyperbolic space with unbounded products such that CS has finite asymptotic dimension. Then asdim s pX q ď asdimpCSq. Moreover, if X is also proper and geodesic, then there exists a continuous bijection p i : B s X Ñ B c X .
Proof. By [CH16, Lemma 3.6], for any stability gauge N there exists N 1 such that X pN q e is N 1 -stable. Hence, there exists D 1 ą 0 depending only on N 1 and pX , Sq such that X pN q e is D 1 -cobounded. By Corollary 6.2, it follows that the projection π S : X pN q e Ñ CS is a quasiisometric embedding with constants depending only on D 1 and pX , Sq. Since every stable subset of X embeds into some X pN q e [CH16, Theorem A.V], the first conclusion then follows from the definition of stable asymptotic dimension. Now suppose that X is proper.
Since each X pN q e is stable in X , these sets have bounded projections by Corollary 6.2; from this it follows that X pN q e is hierarchically quasiconvex for each N . Hence by [DHS15, Proposition 5.8], the canonical embedding i pN q : X pN q e ãÑ X extends to an embedding p i pN q : BX pN q e ãÑ BX . By Corollary 6.2 and Lemma 6.5, we have p i pN q´B X pN q e¯Ă B c X Ă BCS. Let p i : B s X Ñ B c X be the direct limit of the p i pN q . Since it is injective on each stratum, p i is injective.
To prove surjectivity, let λ P B c X . Let e P X and fix a hierarchy path re, λs. Since λ P B c X , re, λs has D-bounded projections for some D ą 0. Let x n P re, λs be such that x n Ñ λ in X . If re, x n s is a sequence of geodesics between e and x n , then, by properness, the Arzela-Ascoli theorem, and passing to a subsequence if necessary, there exists a geodesic ray γ : r0, 8q Ñ X with γp0q " e such that re, x n s converges on compact sets to γ. Since each re, x n s has D-bounded projections, it follows that γ has D 1 -bounded projections for some D 1 depending only on D and pX , Sq. Moreover, by hyperbolicity of CS and the construction of γ we have that d Haus CS pπ S pγq, re, λsq is uniformly bounded and thus, by the distance formula, so is d Haus X pγ, re, λsq. Since rpx n qs " rγs by construction, it follows that p ipγq " λ, as required. Continuity of p i pN q for each N follows from [DHS15, Proposition 5.8], as above. This and the definition of the direct limit topology implies continuity of p i. l The following corollary is immediate: Corollary 6.7. If G is a hierarchically hyperbolic group with clean containers, then G has finite stable asymptotic dimension.
6.4. Random subgroups. Let G be any countable group and µ a probability measure on G whose support generates a non-elementary semigroup. A k-generated random subgroup of G, denoted Γpnq is defined to be the subgroup xw 1 n , w 2 n , . . . , w k n y Ă G generated by the n th step of k independent random walks on G, where k P N. For other recent results on the geometry of random subgroups of acylindrically hyperbolic groups, see [MS17].
Following Taylor-Tiozzo [TT16], we say a k-generated random of G has a property P if PrΓpnq has P s Ñ 1 as n Ñ 8.
Theorem 6.8. Let pX , Sq be a HHS for which the Ď-maximal element, S, has CS infinite diameter, and consider G ă AutpX , Sq which acts properly and cocompactly on X . Then any k-generated random subgroup of G stably embeds in X .
Proof. By [BHS17b, Theorem K], G acts acylindrically on CS. Let Γpnq be generated by k-random independent walks as above. Now, [TT16, Theorem 1.2] implies that Γpnq a.a.s. quasi-isometrically embeds into CS, and hence any orbit of Γpnq in X has bounded projections. By Theorem 4.4, having bounded projections implies contracting; thus any orbit of Γpnq in X is a.a.s. contracting, which gives the conclusion. l 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 (i.e, [DHS15, Theorem 9.14]) which implies that a hierarchically hyperbolic group which is not a direct product of two infinite groups has CS infinite diameter. Corollary 6.9 (Maher-Sisto; [MS17]). If G is a hierarchically hyperbolic group which is not the direct product of two infinite groups, then any k-generated random subgroup of G is stable.

Almost hierarchically hyperbolic spaces
While many hierarchically hyperbolic groups of interest have clean containers (see Proposition 3.5), groups exist that do not. In this section, we explain a trick which allows us to generalize the results of the previous sections to remove the clean containers hypothesis.
In the case of a hierarchically hyperbolic group pG, Sq without clean containers, the construction in Theorem 3.11 yields a structure which is not a hierarchically hyperbolic space. However, the only aspect of the definition of a hierarchically hyperbolic space that may fail to hold is part of the orthogonality axiom, Definition 2.1.(3). Indeed, given T P T and some U P T T for which tV P T T | V K U u ‰ H, the container W P S T´t T u provided by the structure pX , Sq may not be an element of T. In particular, it is possible that E W is bounded.
We introduce the notion of an almost hierarchically hyperbolic space, and use it to show that such spaces satisfy many of the same properties as hierarchically hyperbolic spaces.
The following is a weaker version of the orthogonality axiom: p3 1 q (Bounded pairwise orthogonality) T has a symmetric and anti-reflexive relation called orthogonality: we write V K W when V, W are orthogonal. Also, whenever V Ď W and W K U , we require that V K W . Moreover, if V K W , then V, W are not Ď-comparable. Finally, the cardinality of any collection of pairwise orthogonal domains is uniformly bounded by ξ. By [BHS15, Lemma 2.1], the orthogonality axiom (Definition 2.1, (3)) for an hierarchically hyperbolic structure implies axiom p3 1 q. However, the converse does not hold; that is, the last condition of p3 1 q does not imply the container statement in (3), and thus this is a strictly weaker assumption. However, it suffices for the applications in this paper.
Definition 7.1 (Almost HHS). If pX , Sq satisfies all axioms of a hierarchically hyperbolic space except (3) and additionally satisfies axiom p3 1 q, then pX , Sq is an almost hierarchically hyperbolic space.
Theorem 7.2. Given a hierarchically hyperbolic space pX , Sq with the bounded domain dichotomy, there exists an almost hierarchically hyperbolic structure pX , Tq on X with unbounded products satisfying the following properties: (1) (Distance formula) There exists s 0 such that for all x ě x 0 there exist constants K, C such that for all x, y P X , d X px, yq -pK,Cq ÿ W PT t td W px, yquu s .
(2) (Acylindricity) Let G ď AutpX , Tq act properly and cocompactly on X and let S be the unique Ď-maximal element of T. Then G acts acylindrically on the hyperbolic space T S associated with S. (3) (Realization) For each κ ě 1 there exist θ e , θ u ě 0 such that the following holds.
Let b P ś W PT 2 T W be κ-consistent; for each W , let b W denote the T W -coordinate of b. Then there exists x P X such that d W pb w , π W pxqq ď θ e for all W P T. Moreover, x is coarsely unique in the sense that the set of all x which satisfy d W pb W , π W pxqq ď θ e in each W P T has diameter at most θ u . (4) (Hierarchy paths) There exists D 0 such that any x, y P X are joined by a D 0hierarchy path. (5) (Gate maps) If Y Ď X is k-hierarchically quasiconvex and non-empty, then there exists a gate map for Y, i.e., for each x P X there exists y P Y such that for all W P T, the set π V pyq (uniformly) coarsely coincides with the projection of π V pxq to the kp0q-quasiconvex set π V pYq. (6) (Coarse median structure) The space X is coarse median of rank at most the complexity of pX , Tq.
Remark 7.3. In fact, many of the above results hold for all almost hierarchically hyperbolic spaces and do not require the existence of an original hierarchically hyperbolic structure. However, to avoid complicated notation and proofs, we restrict ourselves to this situation.
Proof. The new structure pX , Tq is as described in section 3.4.2. All of the axioms of Definition 2.1 except (3) hold as in the proof of Theorem 3.14. Finally, we now show axiom p3 1 q is satisfied by this new structure. Indeed, the first three conditions are clear, since T Ď S. For the last condition, any collection of pairwise orthogonal domains in T is also a collections of pairwise orthogonal domains in S, and thus by [BHS15, Lemma 2.2] has uniformly bounded size. Therefore pX , Tq is an almost hierarchically hyperbolic space. That it has unbounded products is clear from the construction. We now prove the properties listed above.
(1) The proof of the distance formula from [BHS15, Theorem 4.5] goes through almost verbatim. The only use of the container part of the orthogonality axiom throughout that entire paper is in that proof of [BHS15, Lemma 2.2] which proves that the cardinality of any collection of pairwise orthogonal domains is uniformly bounded by ξ. As we have adopted the conclusion of that result as part of p3 1 q, the result follows.
(2) The original proof of acylindricity in [BHS17b,Theorem 14.3] does not use the orthogonality axiom, only the distance formula, and therefore goes through as written.
(3) The proof of realization is more involved, and so we provide it in detail.
First, let R " tU P S | F U is boundedu, and let S 1 " SzR. As R is closed under nesting, by [BHS17a, Proposition 2.4] pX , S 1 q is a hierarchically hyperbolic space. Notice that T Ď S 1 .
We follow the original proof of realization in [BHS15, Theorem 3.1]. Let tV j u be a family of pairwise orthogonal elements of T, all of level at most l. By the last clause of the new orthogonality axiom (Definition 7.1 (3 1 )), we have |tV j u| ď ξ. Thus, there exists some l 1 such that tV j u is a family of pairwise orthogonal elements of S 1 , all of level at most l 1 . Then Claim 1 of the proof of [BHS15, Theorem 3.1] provides a constant θ e " θ e pl 1 , κq ą 100Eκα and a collection tU i u of pairwise orthogonal elements of S 1 so that: (a) Each U i is nested into some V j , (b) For each V j there exists some U i nested into it, and (c) Any E-partial realization point x for tU i u satisfies d W pb W , xq ď θ e for each W P S 1 for which there exists j with W Ď V j . As U i P S 1 for all i, F U i is unbounded. By (1), it follows that E U i is unbounded for all i, as well, and therefore U i P T for all i. After possibly increasing the constant, condition (c) still holds for W P T. Therefore, Claim 1 holds for pX , Tq, as well. Now, applying Claim 1 when l " l S , where S P T is the unique Ď-maximal element, along with the partial realization axiom, completes the proof of existence. If x, y both have the desired property, then d V px, yq ď 2θ e`κ for all V P T, whence the uniqueness axiom ensures that dpx, yq ď θ u , for an appropriate θ u . (4) The hierarchy paths in the new structure pX , Tq are the same as those in pX , Sq. As T Ă S, all the required properties hold. (5) The definition of a hierarchically quasiconvex subset passes through to the almost hierarchically hyperbolic setting without issue. The proof of the existence of gate maps to hierarchically quasiconvex subsets [BHS15, Lemma 5.4] uses only the existence of hierarchy paths, realization, and consistent centers [BHS15, Lemma 2.6].
The first two are shown to hold for almost hierarchically hyperbolic spaces above, and the proof of consistent centers does not use the orthogonality axiom. Therefore, the proof of the existence of gate maps goes through as written. (6) The proof of [BHS15,Theorem 7.3] relies only on consistent centers and realization, both of which hold in the present setting, as discussed above.
l As all hierarchically hyperbolic groups satisfy the bounded domain dichotomy, the following is immediate: