Boundaries and automorphisms of hierarchically hyperbolic spaces

By Mark F. Hagen, Alessandro Sisto. 2016

72 citations recorded by the source corpus.

Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmuller 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 Teichmuller 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).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f3f8bbc42505e48ad8ec3c9a08c0d0efd3a9fb5

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Citations in the corpus

Quasiflats in hierarchically hyperbolic spaces (2017, background): “˙ lie at nite Hausdor distance, so for our purposes an arbitrary choice is ne. If ˙˘˙1PHingepSq, then d hausph ˙;h ˙1q8. Remark 5.5. Each hinge corresponds to a 0{simplex in the HHS boundary BX; see [DHS17]. The rst additional assumption holds, for example, in any hierarchically hyperbolic group for”
Asymptotic dimension and small‐cancellation for hierarchically hyperbolic spaces and groups (2015, background): “16 of [DHS16], we can always assume that this holds (see also Remark 1.”
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, methodology): “iconvexity and gates We now introduce the notion of hierarchical quasiconvexity, which is essential for the discussion of product regions, the combination theorem of Section 8, and in the forthcoming [DHS17]. Denition 5.1 (Hierarchical quasiconvexity). Let pX;Sqbe a hierarchically hyperbolic space. ”