2002/11/27 by Feng Luo, Richard Stong
Mathematics · #math.GT #math.DG #msc:30F60 #msc:57M50 #msc:57N16
published as Geom. Topol. 6(2002) 495-521 · Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper17.abs.html
arxiv created 2002/11/27 · arxiv updated 2009/11/30
Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic in its homotopy class. We study this pairing function using the Fenchel-Nielsen coordinates on Teichmueller space and the Dehn-Thurston coordinates on the space of homotopy classes of curve systems. Our main result establishes Lipschitz type estimates for the length pairing expressed in terms of these coordinates. As a consequence, we reestablish a result of Thurston-Bonahon that the length pairing extends to a continuous map from the product of the Teichmueller space and the space of measured laminations.