Boundaries and automorphisms of hierarchically hyperbolic spaces

Abstract

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract. Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space X , we build a bordification of X compatible with its hierarchically hyperbolic structure. If X is proper, e.g. a hierarchically hyperbolic group such as the mapping class group, we get a compactification of X ; we also prove that our construction generalizes the Gromov boundary of a hyperbolic space. In our first main set of applications, we introduce a notion of geometrical finiteness for hierarchically hyperbolic subgroups of hierarchically hyperbolic groups in terms of boundary embeddings. As primary examples of geometrical finiteness, we prove that the natural inclusions of finitely generated Veech groups and the Leininger-Reid combination subgroups extend to continuous embeddings of their Gromov boundaries into the boundary of the mapping class group, both of which fail to happen with the Thurston compactification of Teichmüller space. Our second main set of applications are dynamical and structural, built upon our classification of automorphisms of hierarchically hyperbolic spaces and analysis of how the various types of automorphisms act on the boundary. We prove a generalization of the Handel-Mosher "omnibus subgroup theorem" for mapping class groups to all hierarchically hyperbolic groups, obtain a new proof of the Caprace-Sageev rank-rigidity theorem for many CAT(0) cube complexes, and identify the boundary of a hierarchically hyperbolic group as its Poisson boundary; these results rely on a theorem detecting irreducible axial elements of a group acting on a hierarchically hyperbolic space (which generalize pseudoAnosov elements of the mapping class group and rank-one isometries of a cube complex not virtually stabilizing a hyperplane).

Largest acylindrical actions and Stability in hierarchically hyperbolic groups

“…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 [DHS17]. )…”

“…In [DHS17], Durham, Hagen, and Sisto introduced a boundary for any hierarchically hyperbolic space. We collect the relevant properties we need in the following theorem:…”

Retraction: Hierarchical hyperbolicity of all cubical groups

“…The import of Theorem A is that X , equipped with this set of quasitrees, and coarse projections π F : X → CF explained in [2], satisfies a list of axioms reminiscent of theorems about mapping class groups and curve graphs. Results about G, X which previously required one to hypothesize a factor system but which, in view of Theorem A, do not really need that hypothesis, include: • Theorem A combines with [2, Theorem 9.1] to provide a Masur-Minsky style distance estimate in G: up to quasi-isometry, the distance in X from x to gx, where g ∈ G, is given by summing the distances between the projections of x, gx to a collection of uniform quasi-trees associated to the elements of the factor system; • Theorem A combines with [9,Corollary 9.24] to prove that either G stabilizes a convex subcomplex of X splitting as the product of unbounded subcomplexes, or G contains an element acting loxodromically on the contact graph of X . This is a new proof of the cocompact version of the Caprace-Sageev rank-rigidity theorem [7].…”

Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups

“…In many examples of interest, is coarsely surjective, so that Corollary yields a quasi‐isometry . Moreover, if is an HHS, then (as described in ), admits an HHS structure obtained by replacing each with a hyperbolic space quasi‐isometric to , so in particular becomes quasi‐isometric to the space obtained by coning off each parallel copy of each . If, as is the case for hierarchically hyperbolic groups , the parallel copies of the various coarsely cover , this provides a hierarchically hyperbolic structure in which is a coarse intersection graph of the set of with for which there is no with .…”

“…Remark In order to apply Corollary , we must assume that for each with a ‐hyperbolic space, the projection is uniformly coarsely surjective. By the proof of [, Proposition 1.16], we can always assume that this holds (see also [, Remark 1.3]). Hence, in the proof of Theorem , we can make this coarse surjectivity assumption and thus apply Corollary .…”