Geometry & Topology

Lengths of simple loops on surfaces with hyperbolic metrics

Feng Luo and Richard Stong

Full-text: Open access


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

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

Received: 20 April 2002
Revised: 19 November 2002
Accepted: 19 November 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57N16: Geometric structures on manifolds [See also 57M50]

surface simple loop hyperbolic metric Teichmüller space


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.

Export citation


  • A Beardon, The geometry of discrete groups, Springer–Verlag, Berlin–New York (1983)
  • F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986) 71–158
  • F Bonahon, Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Amer. Math. Soc. 330 (1992) 69–95
  • P Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser, Boston (1992)
  • M Dehn, Papers on group theory and topology, J. Stillwell (editor), Springer–Verlag, Berlin–New York (1987)
  • A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66–67, Société Mathématique de France (1979)
  • J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221–274
  • Y Imayoshi, Y. and M Taniguchi, An introduction to Teichmüller spaces, Translated and revised from the Japanese by the authors, Springer–Verlag, Tokyo (1992)
  • F Luo, Simple loops on surfaces and their intersection numbers, preprint (1997)
  • F Luo, R Stong, Dehn–Thurston coordinates of curves on surfaces, preprint (2002)
  • A Papadopoulos, On Thurston's boundary of Teichmüller space and the extension of earthquakes Topology Appl. 41 (1991) 147–177
  • R Penner, J Harer, Combinatorics of train tracks, Annals of Mathematics Studies, 125, Princeton University Press, Princeton, NJ (1992)
  • W Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417–438
  • W Thurston, Geometry and topology of 3–manifolds, Princeton University lecture notes (1976) list