2010/02/09 by Martin Bridgeman, Bridgeman, Martin, Richard Canary +1 · 1 citation
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #math.CV #math.GT
paper · pdf · doi:10.48550/arxiv.1002.1929
arxiv created 2010/10/04 · arxiv updated 2010/10/05
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.