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

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 whe...

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Bibliographic Details
Published in:Analysis and Geometry in Metric Spaces 2016-09, Vol.4 (1), p.278-281
Main Author: Cashen, Christopher H.
Format: Article
Language:English
Subjects:
contracting geodesic
Geodesy
Gromov boundary
Hyperbolic coordinates
Metric space
quasi-isometry
Topology
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Summary: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.
ISSN:2299-3274
2299-3274
DOI:10.1515/agms-2016-0011