The Thurston metric on hyperbolic domains and boundaries of convex hulls

TL;DR

This paper investigates the relationship between the Thurston metric and the boundary of convex hulls in hyperbolic domains, establishing quasi-isometry properties and confirming a conjecture about Lipschitz conditions.

Contribution

It proves the nearest point retraction is a uniform quasi-isometry between the Thurston metric and convex hull boundaries, and confirms the conjecture linking uniform perfectness to Lipschitz retraction.

Findings

Nearest point retraction is a uniform quasi-isometry.

Explicit bounds on the quasi-isometry constant are provided.

Confirmed the conjecture relating uniform perfectness to Lipschitz retraction.

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.