Hyperbolicity in Teichmüller space

Abstract

We give an inductive description of a Teichmüller geodesic, that is, we show that there is a sense in which a Teichmüller geodesic is assembled from Teichmüller geodesics in smaller subsurfaces. We then apply this description to answer various questions about the geometry of Teichmüller space, obtaining several applications: (1) We show that Teichmüller geodesics do not backtrack in any subsurface. (2) We show that a Teichmüller geodesic segment whose endpoints are in the thick part has the fellow traveling property and that this fails when the endpoints are not necessarily in the thick part. (3) We prove a thin-triangle property for Teichmüller geodesics. Namely, we show that if an edge of a Teichmüller geodesic triangle passes through the thick part, then it is close to one of the other edges.