CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 12, Pages 188–198 (December 8, 2008) S 1088-4173(08)00189-6 QUASICONFORMALLY HOMOGENEOUS PLANAR DOMAINS PETRA BONFERT-TAYLOR AND EDWARD C. TAYLOR Abstract. In this paper we explore the ambient quasiconformal homogeneity of planar domains and their boundaries. We show that the quasiconformal ho- mogeneity of a domain D and its boundary E implies that the pair (D,E) is in fact quasiconformally bi-homogeneous. We also give a geometric and topolog- ical characterization of the quasiconformal homogeneity of D or E under the assumption that E is a Cantor set captured by a quasicircle. A collection of examples is provided to demonstrate that certain assumptions are the weakest possible. 1. Introduction Recall that a hyperbolic manifold is uniformly quasiconformally homogeneous if there exists a constant K ≥ 1 so that any two points on the manifold can be paired by a K-quasiconformal automorphism. Owing to quasiconformal rigidity phenomena there exists a complete topological and geometric understanding of uniformly quasiconformally homogeneous hyperbolic manifolds [2] in dimensions three and above. The situation in dimension two is richer (e.g. see [1], [4], and [3]), and our purpose in this paper is to explore various notions of quasiconformal homogeneity in the plane. In doing so we are returning to the origins of the theory of quasiconformal homogeneity as initiated by Gehring and Palka in [6], and continued by MacManus, N¨akki and Palka in [12] as well as [13]. Indeed, our work in this paper builds upon their work. We now focus our discussion on sets in the Riemann sphere ˆ C. The basic def- inition is: A set A ⊂ ˆ C is K-quasiconformally homogeneous if for every x, y ∈ A there exists a K-quasiconformal homeomorphism f : ˆ C → ˆ C such that f (A)= A and f (x)= y. If such a K exists for A, we also call A uniformly quasiconformally homogeneous. In this paper A will always be either a domain D (i.e. an open and connected set) with at least three boundary points or A will be the complement E of such a domain. We equip the domain D with the hyperbolic metric. If D = Ω(Γ) is the regular set associated to a Kleinian Schottky group Γ, then it is uniformly quasiconformally homogeneous via an application of Theorem 1.3 in [2]. Observe as well that it is a result of MacManus, N¨ akki and Palka [12] that the limit set E = Λ(Γ) of such a group Γ is an example of a compact set that is uniformly quasiconformally homogeneous. The pair (Ω(Γ), Λ(Γ)) exhibits Received by the editors June 19, 2008. 2000 Mathematics Subject Classification. Primary 30C62; Secondary 30F45. Both authors were supported in part by NSF grant DMS 0706754. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 188