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Ambient quasiconformal homogeneity of planar domains. (English) Zbl 1198.30017
An open set $$\Omega \subset {\overline{\mathbb C}}$$ is ambiently $$K$$-quasiconformally homogeneous if, for all $$x, y \in \Omega$$, there exists a $$K$$-quasiconformal homeomorphism $$f : {\overline{\mathbb C}} \to {\overline{\mathbb C}}$$ such that $$f(x) = y$$ and $$f(\Omega) =\Omega$$.
The authors prove that the ambient quasiconformal homogeneity constant of a hyperbolic planar domain, which is not simply connected, is uniformly bounded away from $$1$$.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
##### Keywords:
quasiconformal homogeneity
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