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.
@article{bwmeta1.element.doi-10_1515_agms-2016-0011,
author = {Christopher H. Cashen},
title = {Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology},
journal = {Analysis and Geometry in Metric Spaces},
volume = {4},
year = {2016},
zbl = {06630896},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_agms-2016-0011}
}
Christopher H. Cashen. Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology. Analysis and Geometry in Metric Spaces, Tome 4 (2016) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_agms-2016-0011/