Boundaries and automorphisms of hierarchically hyperbolic spaces

TL;DR

This paper develops a boundary theory for hierarchically hyperbolic spaces, generalizing known boundaries, and applies it to study geometrical finiteness, automorphisms, and dynamics of groups acting on these spaces.

Contribution

It introduces a compatible bordification for hierarchically hyperbolic spaces, generalizes the Gromov boundary, and extends key theorems to this setting, including boundary embeddings and automorphism classifications.

Findings

Constructed a boundary compatible with the hierarchically hyperbolic structure.

Extended the Gromov boundary concept to these spaces.

Generalized the Handel-Mosher subgroup theorem and rank-rigidity results.

Abstract

Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichm\"uller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathcal X$ is proper, we get a compactification of $\mathcal X$; 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 Teichm\"uller 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 \emph{irreducible axial} elements of a group acting on a hierarchically hyperbolic space (which generalize pseudo-Anosov elements of the mapping class group and rank-one isometries of a cube complex not virtually stabilizing a hyperplane).