Bonfert-Taylor, Petra; Martin, Gaven; Reid, Alan W.; Taylor, Edward C. Teichmüller mappings, quasiconformal homogeneity, and non-amenable covers of Riemann surfaces. (English) Zbl 1239.30007 Pure Appl. Math. Q. 7, No. 2, 455-468 (2011). 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. Cited in 5 Documents MSC: 30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations 30F60 Teichmüller theory for Riemann surfaces Keywords:quasiconformal homogeneity; Riemann surface; hyperbolic orbifold PDFBibTeX XMLCite \textit{P. Bonfert-Taylor} et al., Pure Appl. Math. Q. 7, No. 2, 455--468 (2011; Zbl 1239.30007) Full Text: DOI