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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)
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