Volume 18, issue 5 (2014)

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Hyperbolicity in Teichmüller space

Kasra Rafi

Geometry & Topology 18 (2014) 3025–3053

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.

Teichmüller space, geodesics, fellow traveling, subsurface projection, curve complex
Mathematical Subject Classification 2010
Primary: 30F60
Secondary: 32Q05
Received: 14 July 2013
Accepted: 31 January 2014
Published: 1 December 2014
Proposed: Danny Calegari
Seconded: Walter Neumann, Yasha Eliashberg
Kasra Rafi
Department of Mathematics
University of Toronto
Room 6290
40 George Street
Toronto ON M5S 2E4