Bounded combinatorics and the Lipschitz metric on Teichmüller space

Abstract

Considering the Teichmüller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are stable. Our main tool is to show that one can get a good estimate for the Lipschitz distance by considering the length ratio of finitely many curves.

20 pages, 5 figures; Revised version incorporated referee's comments: expanded the introduction and background, clarified some proofs, and corrected typos