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Zbl 1064.30044
Epstein, D.B.A.; Marden, A.; Markovic, V.
Quasiconformal homeomorphisms and the convex hull boundary. (English)
[J] Ann. Math. (2) 159, No. 1, 305-336 (2004). ISSN 0003-486X; ISSN 1939-8980/e

Let $\Omega$ be an open simply-connected subset of the Riemann sphere $S^2$ (regarded as the boundary of hyperbolic 3-space ${\bold H}^3$) and let $X=S^2\setminus \Omega$. (To avoid special cases, we suppose that $\Omega\subset {\bold C}$ and $\Omega\not= {\bold C}$.) We can form the hyperbolic convex hull ${\cal CH}(X)$, and the authors denote the relative boundary of ${\cal CH}(X)$ in ${\bold H}^3$ by $\text {Dome}(\Omega)$. The study of the geometry of objects such as $\text {Dome}(\Omega)$ was initiated by Thurston who proved for instance that the hyperboloic metric of ${\bold H}^3$ induces a path metric on $\text {Dome}(\Omega)$ which makes it isometric to the hyperbolic disk ${\bold D}^2$. For such an $\Omega$, Thurston defined a ``nearest point retraction" $r:\Omega\to \text {Dome}(\Omega)$ as follows: for each $z\in\Omega$, we expand a small horoball at $z$ and we call $r(z)\in\text {Dome}(\Omega)$ 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 $\Omega\not= {\bold C}$, there is a $K$-quasiconformal homeomorphism $\Psi: \text {Dome}(\Omega)\to\Omega$ 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 ${\bold D}^2$ is a $(K,a(K))$-quasi-isometry, where $a(K)$ is an explicitely computed function, where the multiplicative constant is best possible and where the additive constant $a(K)$ is best possible for some values of $K$.
[Athanase Papadopoulos (Strasbourg)]

MSC 2000:: *30F45 Conformal metrics
30F60 Teichmueller theory
37F30 Quasiconformal methods and Teichmüller theory, etc.
57M50 Geometric structures on low-dimensional manifolds
32F17 Other notions of convexity

Keywords: convex hull boundary; pleated surface; pleating map; earthquake; bending

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