Hyperbolicity in Teichm\"uller space

Abstract

We review and organize some results describing the behavior of a Teichm\"uller geodesic and draw several applications: 1) We show that Teichm\"uller geodesics do not back track. 2) We show that a Teichm\"uller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpoints are not necessarily in the thick part. 3) We show that if an edge of a Teichm\"uller geodesic triangle passes through the thick part, then it is close to one of the other egdes.