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arXiv · 1605.01660

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

Christopher H. Cashen
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Abstract

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.

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Christopher H. Cashen. 2016-05-05. Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology. https://doi.org/10.1515/agms-2016-0011

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