Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology

21 Jul 2016  ·  Cashen Christopher H. ·

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 an explicit example. read more

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Metric Geometry