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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.

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