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Teichmüller mappings, quasiconformal homogeneity, and non-amenable covers of Riemann surfaces. (English) Zbl 1239.30007
Summary: We show that there exists a universal constant \(K_c\) so that every \(K\)-strongly quasiconformally homogeneous hyperbolic surface \(X\) (not equal to \(\mathbb{H}^2\)) has the property that \(K> K_c > 1\). The constant \(K_c\) is the best possible, and is computed in terms of the diameter of the \((2, 3, 7)\)-hyperbolic orbifold (which is the hyperbolic orbifold of smallest area). We further show that the minimum strong homogeneity constant of a hyperbolic surface without conformal automorphisms decreases if one passes to a non-amenable regular cover.

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30F60 Teichmüller theory for Riemann surfaces
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