Quasiconformal homeomorphisms and the convex hull boundary

Abstract

We investigate the relationship between an open simply-connected region $\Omega\subset \mathbb{S}^2$ and the boundary $Y$ of the hyperbolic convex hull in $\mathbb{H}^3$ of $\mathbb{S}^2\setminus\Omega$. 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 $\Omega$. We show that the best possible universal lipschitz constant for the nearest point retraction $r:\Omega\to Y$ 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 $\mathbb{D}^2\to\mathbb{D}^2$ 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 $K$.