Displaying similar documents to “Cannon-Thurston Maps, i-bounded Geometry and a Theorem of McMullen”

Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology

Christopher H. Cashen (2016)

Analysis and Geometry in Metric Spaces

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We consider a ‘contracting boundary’ of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space.We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by...