2016/06/30 by Matthew Cordes, David Hume · 1 citation
Mathematics · #Boundary (topology) #Combinatorics #Countable set #Geodesic #Geometric and Algebraic Topology #Geometry #Handlebody #Homotopy and Cohomology in Algebraic Topology #Hyperbolic group #Hyperbolic manifold #Hyperbolic space #Invariant (physics) #Linear subspace #Mapping class group #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Morse theory #Pure mathematics #Surface (topology) #math.GR #math.GT #math.MG #msc:20F65 #msc:20F67
paper · pdf · doi:10.1112/jlms.12042
Updated with comments from the referee. To appear in the Journal of the London Mathematical Society. 28 pages, 1 figure
arxiv created 2017/02/24 · openalex publication_date 2017/03/28 · arxiv updated 2017/06/14 · openalex created_date 2021/02/01 · openalex updated_date 2026/08/06
Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. In this paper we unify and generalise these strategies by viewing any geodesic metric space as a countable union of stable subspaces: we show that every stable subgroup is a quasi-convex subset of a set in this collection and that the Morse boundary is recovered as the direct limit of the usual Gromov boundaries of these hyperbolic subspaces. We use this approach, together with results of Leininger–Schleimer, to deduce that there is no purely geometric obstruction to the existence of a non-virtually-free convex cocompact subgroup of a mapping class group. In addition, we define two new quasi-isometry invariant notions of dimension: the stable dimension, which measures the maximal asymptotic dimension of a stable subset; and the Morse capacity dimension, which naturally generalises Buyalo's capacity dimension for boundaries of hyperbolic spaces. We prove that every stable subset of a right-angled Artin group is quasi-isometric to a tree; and that the stable dimension of a mapping class group is bounded from above by a multiple of the complexity of the surface. In the case of relatively hyperbolic groups we show that finite stable dimension is inherited from peripheral subgroups. Finally, we show that all classical small cancellation groups and certain graphical small cancellation groups — including some Gromov monster groups — have stable dimension at most 2.