Hyperbolicity in Teichmüller space

By Kasra Rafi. 2010

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We give an inductive description of a Teichmuller geodesic, that is, we show that there is a sense in which a Teichmuller geodesic is assembled from Teichmuller geodesics in smaller subsurfaces. We then apply this description to answer various questions about the geometry of Teichmuller space, obtaining several applications: (1) We show that Teichmuller geodesics do not backtrack in any subsurface. (2) We show that a Teichmuller 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 Teichmuller geodesics. Namely, we show that if an edge of a Teichmuller geodesic triangle passes through the thick part, then it is close to one of the other edges. 30F60; 32Q05

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e179467baf28c5a8e374e581ac5c2d6b7c167709