Quasiconformal homeomorphisms and the convex hull boundary

We investigate the relationship between an open simply-connected region Ω ⊂ double struck S sign² and the boundary Y of the hyperbolic convex hull in ℍ³ of double struck S sign² / 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 < c₂ < c₁, such that no pleating map which bends more than c₁ in some interval of unit length is an embedding, and such that any pleating map which bends less than c₂ in each interval of unit length is embedded. We show that every K-quasiconformal homeomorphism double struck D sign² → double struck D sign² is a (K, a(K))-quasi-isometry, where a(K) is an explicitly computed function. The multiplicative constant is best possible and the additive constant a(K) is best possible for some values of K.