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Lengths of simple loops on surfaces with hyperbolic metrics

Feng Luo and Richard Stong

Geometry & Topology 6 (2002) 495–521

arXiv: math.GT/0211433


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.

surface, simple loop, hyperbolic metric, Teichmüller space
Mathematical Subject Classification 2000
Primary: 30F60
Secondary: 57M50, 57N16
Forward citations
Received: 20 April 2002
Revised: 19 November 2002
Accepted: 19 November 2002
Published: 22 November 2002
Proposed: David Gabai
Seconded: Jean-Pierre Otal, Joan Birman
Feng Luo
Department of Mathematics
Rutgers University
New Brunswick
New Jersey 08854
Richard Stong
Department of Mathematics
Rice University
Texas 77005