Lengths of simple loops on surfaces with hyperbolic metrics

Details Lengths of simple loops on surfaces with hyperbolic metrics Lengths of simple loops on surfaces with hyperbolic metrics

2002-11-27

arxiv.org/abs/math/0211421v1

Geom. Topol. 6(2002) 495-521

Mathematics

Geometric Topology

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper17.abs.html

Scientific paper

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.

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