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

100%

Cashen C. H.

Analysis and Geometry in Metric Spaces

2016

tom 4

nr 1

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.

Weak Capacity and Modulus Comparability in Ahlfors Regular Metric Spaces

63%

Lindquist J.

Analysis and Geometry in Metric Spaces

2016

tom 4

nr 1

Let (Z, d, μ) be a compact, connected, Ahlfors Q-regular metric space with Q > 1. Using a hyperbolic filling of Z,we define the notions of the p-capacity between certain subsets of Z and of theweak covering p-capacity of path families Γ in Z.We show comparability results and quasisymmetric invariance.As an application of our methodswe deduce a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected Ahlfors Q-regular metric spaces.

Ograniczanie wyników

2 Analysis and Geometry in Metric Spaces

1 Cashen C. H.

1 Lindquist J.

2 2016

Wyniki wyszukiwania

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

Weak Capacity and Modulus Comparability in Ahlfors Regular Metric Spaces