Hyperbolicity in Teichmüller space

Rafi, Kasra

doi:10.2140/gt.2014.18.3025

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Hyperbolicity in Teichmüller space

Kasra Rafi

Geometry & Topology 18 (2014) 3025–3053

DOI: 10.2140/gt.2014.18.3025

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.

Keywords

Teichmüller space, geodesics, fellow traveling, subsurface projection, curve complex

Mathematical Subject Classification 2010

Primary: 30F60

Secondary: 32Q05

References

Publication

Received: 14 July 2013

Accepted: 31 January 2014

Published: 1 December 2014

Proposed: Danny Calegari

Seconded: Walter Neumann, Yasha Eliashberg

Authors

Kasra Rafi
Department of Mathematics University of Toronto Room 6290 40 George Street Toronto ON M5S 2E4 Canada

Volume 18, issue 5 (2014)