The Thurston metric on hyperbolic domains and boundaries of convex hulls

Martin Bridgeman; Richard Canary

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arXiv · 1002.1929

The Thurston metric on hyperbolic domains and boundaries of convex hulls

Martin Bridgeman·Richard Canary

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Abstract

We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the Poincare metric when the domain is uniformly perfect. We also establish Marden and Markovic's conjecture that a hyperbolic domain is uniformly perfect if and only if the nearest point retraction is Lipschitz with respect to the Poincare metric.

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Martin Bridgeman, Richard Canary. 2010-10-04. The Thurston metric on hyperbolic domains and boundaries of convex hulls. https://arxiv.org/abs/1002.1929

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