Contracting Boundaries of CAT(0) Spaces

Abstract

As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like properties. We prove that these properties are all equivalent and that the contracting boundary is a quasi-isometry invariant. We use this invariant to distinguish the quasi-isometry classes of certain right-angled Coxeter groups.

27 pages, 8 figures

References in corpus (1)

• The simplicial boundary of a CAT(0) cube complex

Cited by in corpus (27)

• Stability and the Morse boundary
• On strongly quasiconvex subgroups
• Topology and Dynamics of the Contracting Boundary of Cocompact CAT(0) Spaces
• A Metrizable Topology on the Contracting Boundary of a Group
• Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups
• Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, with an appendix written jointly with Christopher Cashen
• Quasi-Mobius Homeomorphisms of Morse boundaries
• Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups
• Cross ratios and cubulations of hyperbolic groups
• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• Stable Subgroups of the Genus Two Handlebody Group
• Comparing topologies on the Morse boundary and quasi-isometry invariance
• Quasi-isometrically rigid subgroups in right-angled Coxeter groups
• Sublinearly Morse Geodesics in CAT(0) Spaces: Lower Divergence and Hyperplane Characterization
• Coarse-median preserving automorphisms
• On the coarse geometry of certain right-angled Coxeter groups
• Stable subgroups and Morse subgroups in mapping class groups
• Divergence of Morse geodesics
• Divergence of finitely presented subgroups of CAT(0) groups
• Right-angled Artin subgroups of right-angled Coxeter and Artin groups
• Connected components of Morse boundaries of graphs of groups
• An embedding of the Morse boundary in the Martin boundary
• Sublinearly Morse Boundary II: Proper geodesic spaces
• Coarse cubical rigidity
• On the Čech cohomology of Morse boundaries
• Flat braid groups, right-angled Artin groups, and commensurability
• Detecting a subclass of torsion-generated groups