Quasiconformal homogeneity of genus zero surfaces

Abstract

A Riemann surface M is said to be K -quasiconformally homogeneous if, for every two points p, q ∈ M , there exists a K -quasiconformal homeomorphism f : M →M such that f (p) = q. In this paper, we show there exists a universal constant K > 1 such that if M is a K -quasiconformally homogeneous hyperbolic genus zero surface other than D 2 , then K ≥ K. This answers a question by Gehring and Palka [10]. Further, we show that a non-maximal hyperbolic surface of genus g ≥ 1 is not K -quasiconformally homogeneous for any finite K ≥ 1.

Quasiconformal homogeneity and subgroups of the mapping class group

“…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 Palka

“…Kwakkel and Markovic [22] have resolved the question for planar surfaces. (Bonfert-Taylor, Canary, Martin, Taylor and Wolf [6] had earlier produced a lower bound, greater than 1, on the ambient quasiconformal homogeneity constant of a planar hyperbolic surface.)…”

“…Theorem 4.1. (Kwakkel-Markovic [22]) There exists K planar > 1 so that if S is a planar uniformly quasiconformally hyperbolic surface, then…”

A lower bound for Torelli- $$K$$ K -quasiconformal homogeneity

“…Let c be a shortest geodesic on M . The injectivity radius ι(M ) is the infimum over all p ∈ M of the largest radius for which the exponential map at p is injective (see §2.1 of [1]). In particular, |c| ≥ 2ι(M ), where |c| denotes the length of c.…”