Geometry & Topology, Vol. 6 (2002) Paper no. 17, pages 495--521.

Lengths of simple loops on surfaces with hyperbolic metrics

Feng Luo, Richard Stong

Abstract. 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.

Keywords. Surface, simple loop, hyperbolic metric, Teichmueller space

AMS subject classification. Primary: 30F60. Secondary: 57M50, 57N16.

DOI: 10.2140/gt.2002.6.495

E-print: arXiv:math.GT/0211433

Submitted to GT on 20 April 2002. (Revised 19 November 2002.) Paper accepted 19 November 2002. Paper published 22 November 2002.

Notes on file formats

Feng Luo, Richard Stong
Department of Mathematics, Rutgers University
New Brunswick, NJ 08854, USA
Department of Mathematics, Rice University
Houston, TX 77005, USA


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