Lengths of simple loops on surfaces with hyperbolic metrics

Abstract

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmüller 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 Teichmüller 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 Teichmüller space and the space of measured laminations.

Locations

• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF
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• Geometry & Topology - View - PDF