Quasiconformal homogeneity of hyperbolic surfaces with fixed-point full automorphisms

Abstract

Abstract. We show that any closed hyperbolic surface admitting a conformal automorphism with "many" fixed points is uniformly quasiconformally homogeneous, with constant uniformly bounded away from 1. In particular, there is a uniform lower bound on the quasiconformal homogeneity constant for all hyperelliptic surfaces. In addition, we introduce more restrictive notions of quasiconformal homogeneity and bound the associated quasiconformal homogeneity constants uniformly away from 1 for all hyperbolic surfaces.

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

“…The concept of quasiconformal homogeneity was introduced and developed by Gehring and Palka in [7]; for other work on quasiconformally homogeneous structures see [8], [9], [4], and [5]. In dimensions three and above, owing to well-known quasiconformal rigidity phenomena, the property of being uniformally quasiconformally homogeneous is a topologically restrictive one, we recall Theorem 1.3 of [4].…”

Quasiconformal Homogeneity after Gehring and Palka

“…Bonfert-Taylor, Bridgeman, Canary and Taylor [4] exhibited a lower bound on the uniform quasiconformal homogeneity constant of any hyperelliptic surface. We recall that a closed hyperbolic surface S of genus g is hyperelliptic if it admits a conformal involution with 2g + 2 fixed points.…”

“…The argument outlined above easily generalizes to show: One may also modify the question by considering more restrictive forms of quasiconformal homogeneity, where the arguments of the previous section do apply. Bonfert-Taylor, Bridgeman, Canary and Taylor [4] define a hyperbolic surface S to be K-strongly quasiconformally homogenous if for any x, y ∈ S there exists a K-quasiconformal homeomorphism f : S → S such that f (x) = y and f is homotopic to a conformal automorphism of S. Similarly S is K-extremely quasiconformally homogenous if for any x, y ∈ S there exists a K-quasiconformal homeomorphism f : S → S such that f (x) = y and f is homotopic to the identity. Gehring and Palka's Lemma 2.2 can be again used to show that every closed hyperbolic surface is both strongly and extremely quasiconformally homogeneous…”

“…Lemma 3.6 implies that a hyperbolic surface is extremely quasiconformally homogeneous if and only if it is closed (see [4,Theorem 6.4]). The proofs of Theorems 3.5 and 3.7 immediately generalize in the strongly quasiconformally homogeneous setting.…”

Ambient quasiconformal homogeneity of planar domains

“…In earlier work [6], the authors established that for any n ≥ 3 there exists K n > 1 such that if N is a K-quasiconformally homogeneous hyperbolic n-manifold, other than H n , then K ≥ K n . It is natural to ask whether or not such a constant can be found in dimension 2 (see, for example, [5]). …”