Teichmuller Mappings, Quasiconformal Homogeneity, and Non-amenable Covers of Riemann Surfaces

Abstract

We show that there exists a universal constant K c so that every K-strongly quasiconformally homogeneous hyperbolic surface X (not equal to 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.

Quasiconformal homogeneity and subgroups of the mapping class group

“…In a similar fashion, the authors find that a hyperbolic surface X is Aut(X) K -homogeneous for some K if and only if it is a regular cover of a hyperbolic orbifold; furthermore, there exists a constant K aut > 1 such that K ≥ K aut . A sharp bound is found for the constant K aut in [BTMRT11]. The authors in [KM11] show the existence of a lower bound K 0 > 1 for the quasiconformal homogeneity constant of genus zero surfaces, which answers a question about quasiconformal homogeneity of planar domains posed by Gehring and Palka in [GP76].…”

Quasiconformal homogeneity after Gehring and Palks

“…Bonfert-Taylor, Martin, Reid and Taylor [8] obtained a sharp version of Theorem 4.4. They begin by using the isodiametric inequality to show that the (2, 3, 7)-triangle orbifold, denoted O min , has minimal diameter among all hyperbolic two-orbifolds (see [8,Proposition 2.2]).…”

Quasiconformal Homogeneity after Gehring and Palka

“…The extremal map is unique and one demonstrates, via the line element field of this map, that it can not be realized as a quasiconformal deformation of any non-elementary Fuchsian group. The reader is referred to [8] for details.…”