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 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 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 \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).

67 pages. Various corrections in light of referee's comments

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Cited by in corpus (17)

• Hierarchically hyperbolic spaces II: Combination theorems and the distance formula
• On hierarchical hyperbolicity of cubical groups
• Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity
• A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups
• Stable cubulations, bicombings, and barycenters
• Hyperbolic HHS I:Factor Systems and Quasi-convex subgroups
• Hyperbolic HHS II: Graphs of hierarchically hyperbolic groups
• Hierarchical hyperbolicity of hyperbolic-2-decomposable groups
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• Non-existence of boundary maps for some hierarchically hyperbolic spaces
• Affine diffeomorphism groups are undistorted
• Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity
• Automorphisms of contact graphs of $\mathrm{CAT}(0)$ cube complexes
• Free products of abelian groups in mapping class groups
• Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups
• On super-rigidity of Gromov's random monster group
• A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces