Shortening all the simple closed geodesics on surfaces with boundary

Athanase Papadopoulos (1, 2) , Guillaume Théret (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 MPI - Max-Planck-Institut für Mathematik

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple closed geodesics have are shorter. Furthermore, we show that we can do the shortening in such a way that it is bounded below by a positive constant. This improves a recent result obtained by Parlier. We include this result in a study of the weak metric theory of the Teichmueller space of surfaces with nonempty boundary. The weak metrics that we consider are defined using lengths of closed geodesics and lengths of geodesic arcs. We prove an equality between two such weak metrics.

Mots clés: Thurston's asymmetric metric; Teichmueller space; surface with boundary; weak metric; length spectrum metric; Thurston's asymmetric metric.

Domaines: Topologie géométrique [math.GT]

Athanase Papadopoulos, Guillaume Théret. Shortening all the simple closed geodesics on surfaces with boundary. 2009. ⟨hal-00389344v1⟩