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Uniformity from Gromov hyperbolicity. (English) Zbl 1189.30055

The authors show that, in a metric space X with annular convexity, the uniform domains are precisely those Gromov hyperbolic domains whose quasiconformal structure on the boundary agrees with that on the boundary of X.

As an application it is shown that quasi-Möbius maps between geodesic spaces with annular convexity preserve uniform domains.

Theorem 1: Let (X i ,d i ), i=1,2, be proper metric spaces, and let Ω i X i be open subsets with Ω 1 . Let h:Ω 1 Ω 2 be an η-quasi-Möbius homeomorphism. If Ω 1 is c 1 -uniform and (X 2 ,d 2 ) is c 2 -quasiconvex and c 2 -annular convex, then Ω 2 is c-uniform with c=c(η,c 1 ,c 2 ).

30C65Quasiconformal mappings in n and other generalizations
53C23Global geometric and topological methods; differential geometric analysis on metric spaces