Geodesic
Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology
Analysis and Geometry in Metric Spaces, Tome 4 (2016) no. 1

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Zbl  

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.
Mots-clés : Gromov boundary, quasi-isometry, contracting geodesic 
Cashen, Christopher H. Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology. Analysis and Geometry in Metric Spaces, Tome 4 (2016) no. 1. http://geodesic.mathdoc.fr/item/AGMS_2016_4_1_a17/
@article{AGMS_2016_4_1_a17,
     author = {Cashen, Christopher H.},
     title = {Quasi-Isometries {Need} {Not} {Induce} {Homeomorphisms} of {Contracting} {Boundaries} with the {Gromov} {Product} {Topology}},
     journal = {Analysis and Geometry in Metric Spaces},
     year = {2016},
     volume = {4},
     number = {1},
     zbl = {06630896},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AGMS_2016_4_1_a17/}
}
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