## Geometry & Topology

### 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.

#### Article information

Source
Geom. Topol., Volume 6, Number 2 (2002), 495-521.

Dates
Revised: 19 November 2002
Accepted: 19 November 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513882932

Digital Object Identifier
doi:10.2140/gt.2002.6.495

Mathematical Reviews number (MathSciNet)
MR1943757

Zentralblatt MATH identifier
1029.30028

#### Citation

Luo, Feng; Stong, Richard. Lengths of simple loops on surfaces with hyperbolic metrics. Geom. Topol. 6 (2002), no. 2, 495--521. doi:10.2140/gt.2002.6.495. https://projecteuclid.org/euclid.gt/1513882932

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