Boundaries and automorphisms of hierarchically hyperbolic spaces

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 $\mathcal{X}$, we build a bordification of $\mathcal{X}$ compatible with its hierarchically hyperbolic structure.

If $\mathcal{X}$ is proper, eg a hierarchically hyperbolic group such as the mapping class group, 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 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 $CAT\left(0\right)$ 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 pseudo-Anosov elements of the mapping class group and rank-one isometries of a cube complex not virtually stabilizing a hyperplane).