Quasiconformal homeomorphisms and the convex hull boundary

Abstract

We investigate the relationship between an open simply-connected region Ω ⊂ S^2 and the boundary Y of the hyperbolic convex hull in H^3 of S^2∖Ω. A counterexample is given to Thurston’s conjecture that these spaces are related by a 2-quasiconformal homeomorphism which extends to the identity map on their common boundary, in the case when the homeomorphism is required to respect any group of Möbius transformations which preserves Ω. We show that the best possible universal lipschitz constant for the nearest point retraction r:Ω → Y is 2. We find explicit universal constants 0 < c_2 < c_1, such that no pleating map which bends more than c_1 in some interval of unit length is an embedding, and such that any pleating map which bends less than c_2 in each interval of unit length is embedded. We show that every K-quasiconformal homeomorphism D^2 → D^2 is a (K,ɑ(K))-quasi-isometry, where ɑ(K) is an explicitly computed function. The multiplicative constant is best possible and the additive constant ɑ(K) is best possible for some values of K.