#### 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

and

Department of Mathematics, Rice University

Houston, TX 77005, USA

Email: fluo@math.rutgers.edu, stong@math.rice.edu

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