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Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Victor Gerasimov, Leonid Potyagailo (2013)

Journal of the European Mathematical Society

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using the Floyd completion we further prove that the property of relative hyperbolicity is invariant under quasi-isometric maps. If a finitely generated group H admits a quasi-isometric map ϕ into a relatively hyperbolic group G then H is itself relatively hyperbolic with respect to a system of subgroups whose image under ϕ is situated within a uniformly bounded distance...

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

Christopher H. Cashen (2016)

Analysis and Geometry in Metric Spaces

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 constructing...

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