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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 $\left({X}_{i},{d}_{i}\right)$, $i=1,2$, be proper metric spaces, and let ${{\Omega }}_{i}\subset {X}_{i}$ be open subsets with $\partial {{\Omega }}_{1}\ne \varnothing$. Let $h:{{\Omega }}_{1}\to {{\Omega }}_{2}$ be an $\eta$-quasi-Möbius homeomorphism. If ${{\Omega }}_{1}$ is ${c}_{1}$-uniform and $\left({X}_{2},{d}_{2}\right)$ is ${c}_{2}$-quasiconvex and ${c}_{2}$-annular convex, then ${{\Omega }}_{2}$ is $c$-uniform with $c=c\left(\eta ,{c}_{1},{c}_{2}\right)$.

##### MSC:
 30C65 Quasiconformal mappings in ${ℝ}^{n}$ and other generalizations 53C23 Global geometric and topological methods; differential geometric analysis on metric spaces