2016/04/30 by Matthew Gentry Durham, Matthew G. Durham, Mark Hagen +2 · 55 citations
Mathematics · #Advanced Combinatorial Mathematics #Automorphism #Boundary (topology) #Compactification (mathematics) #Geometric and Algebraic Topology #Geometry #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Hyperbolic 3-manifold #Hyperbolic function #Hyperbolic group #Hyperbolic manifold #Hyperbolic space #Hyperbolic tree #Mapping class group #Mathematical analysis #Mathematics #Pure mathematics #Relatively hyperbolic group #Space (punctuation) #Surface (topology) #math.GR #math.GT #math.MG
paper · pdf · open access · doi:10.2140/gt.2017.21.3659
published in Geometry & Topology 21(6), 3659-3758 (Mathematical Sciences Publishers) · 67 pages. Various corrections in light of referee's comments
arxiv created 2017/02/08 · openalex publication_date 2017/08/31 · arxiv updated 2018/03/16 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/08
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 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).