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Quasiconformal homeomorphisms and the convex hull boundary. (English) Zbl 1064.30044
Let ${\Omega }$ be an open simply-connected subset of the Riemann sphere ${S}^{2}$ (regarded as the boundary of hyperbolic 3-space ${𝐇}^{3}$) and let $X={S}^{2}\setminus {\Omega }$. (To avoid special cases, we suppose that ${\Omega }\subset 𝐂$ and ${\Omega }\ne 𝐂$.) We can form the hyperbolic convex hull $𝒞H\left(X\right)$, and the authors denote the relative boundary of $𝒞H\left(X\right)$ in ${𝐇}^{3}$ by $\text{Dome}\left({\Omega }\right)$. The study of the geometry of objects such as $\text{Dome}\left({\Omega }\right)$ was initiated by Thurston who proved for instance that the hyperboloic metric of ${𝐇}^{3}$ induces a path metric on $\text{Dome}\left({\Omega }\right)$ which makes it isometric to the hyperbolic disk ${𝐃}^{2}$. For such an ${\Omega }$, Thurston defined a “nearest point retraction” $r:{\Omega }\to \text{Dome}\left({\Omega }\right)$ as follows: for each $z\in {\Omega }$, we expand a small horoball at $z$ and we call $r\left(z\right)\in \text{Dome}\left({\Omega }\right)$ the unique first point of contact. Sullivan, and Epstein-Marden analysed that construction and they proved that there exists $K$ such that for any simply connected ${\Omega }\ne 𝐂$, there is a $K$-quasiconformal homeomorphism ${\Psi }:\text{Dome}\left({\Omega }\right)\to {\Omega }$ which extends continuously to the identity map on the common boundary $\partial {\Omega }$. Thurston suggested that the best constant $K$ is 2, and this suggestion has been called later on “Thurston’s $K=2$ conjecture”. In this paper, the authors give a counterexample to that conjecture in its equivariant form, that is, in the case where the homeomorphism respects a group of Möbius transformations which preserve ${\Omega }$. Another result that the authors prove in this paper is that the nearest point retraction $r$ is 2-Lipschitz in the respective hyperbolic metrics, and that the constant 2 here is best possible. The authors also study pleating maps of hyperbolic 2-plane. They obtain explicit universal constants $0<{c}_{1}<{c}_{2}$ 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. They show that every $K$-quasiconformal homeomrophism of the unit disk ${𝐃}^{2}$ is a $\left(K,a\left(K\right)\right)$-quasi-isometry, where $a\left(K\right)$ is an explicitely computed function, where the multiplicative constant is best possible and where the additive constant $a\left(K\right)$ is best possible for some values of $K$.
##### MSC:
 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 30F60 Teichmüller theory 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) 57M50 Geometric structures on low-dimensional manifolds 32F17 Other notions of convexity