The Geometry of the Thurston Metric: A Survey

This chapter is a survey about the Thurston metric on the Teichmüller space. The central issue is the construction 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.