The geometry of the Thurston metric: a survey

TL;DR

This survey explores the Thurston metric on Teichmüller space, focusing on extremal Lipschitz maps, geometric properties, and recent developments, providing a comprehensive overview of its constructions and applications.

Contribution

It consolidates various constructions and recent advances related to the Thurston metric, highlighting its geometric properties and connections to compactification.

Findings

Thurston metric characterized by extremal Lipschitz maps.

Coarse geometry and isometry rigidity established for the Thurston metric.

Recent generalizations extend the applicability of the Thurston metric.

Abstract

This paper is a survey about the Thurston metric on the Teichm\"uller space. The central issue is the constructions of extremal Lipschitz maps between hyperbolic surfaces. We review several constructions, including the original work of Thurston. Coarse geometry and isometry rigidity of the Thurston metric, relation between the Thurston metric and the Thurston compactification are discussed. Some recent generalizations and developments of the Thurston metric are sketched.