## 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 $${\mathbf H}^3$$) and let $$X=S^2\setminus \Omega$$. (To avoid special cases, we suppose that $$\Omega\subset {\mathbf C}$$ and $$\Omega\not= {\mathbf C}$$.) We can form the hyperbolic convex hull $${\mathcal CH}(X)$$, and the authors denote the relative boundary of $${\mathcal CH}(X)$$ in $${\mathbf H}^3$$ 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 $${\mathbf H}^3$$ induces a path metric on $$\text{Dome}(\Omega)$$ which makes it isometric to the hyperbolic disk $${\mathbf D}^2$$. 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 $$\Omega\not= {\mathbf C}$$, 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 $${\mathbf D}^2$$ 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 $$K$$.

### MSC:

 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 30F60 Teichmüller theory for Riemann surfaces 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 32F17 Other notions of convexity in relation to several complex variables
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