Generalizing Stretch Lines for Surfaces with Boundary

Abstract

Abstract In 1986, William Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmüller space of closed or punctured surfaces. We extend his theory to the Teichmüller space of surfaces with boundary endowed with the arc distance. We construct a large family of geodesics for the Teichmüller space of a surface with boundary, generalizing Thurston’s stretch lines. We prove that the Teichmüller space of a surface with boundary is a geodesic and Finsler metric space with respect to the arc distance. As a corollary, we find a new class of geodesics in the Teichmüller space of a closed surface that are not stretch lines in the sense of Thurston.