Lengths of simple loops on surfaces with hyperbolic metrics

Luo, Feng¹ ; Stong, Richard²

¹ Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854, USA
² Department of Mathematics, Rice University, Houston, Texas 77005, USA

Geometry & topology, Tome 6 (2002) no. 2, pp. 495-521.

Résumé

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.

Détail

DOI : 10.2140/gt.2002.6.495

Keywords: surface, simple loop, hyperbolic metric, Teichmüller space

Bibliographie

[1] A F Beardon, The geometry of discrete groups, Graduate Texts in Mathematics 91, Springer (1983)

[2] F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. $(2)$ 124 (1986) 71

[3] F Bonahon, Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Amer. Math. Soc. 330 (1992) 69

[4] P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser (1992)

[5] M Dehn, Papers on group theory and topology, Springer (1987)

[6] A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284

[7] J Hubbard, H Masur, Quadratic differentials and foliations, Acta Math. 142 (1979) 221

[8] Y Imayoshi, M Taniguchi, An introduction to Teichmüller spaces, Springer (1992)

[9] F Luo, Simple loops on surfaces and their intersection numbers, Math. Res. Lett. 5 (1998) 47

[10] F Luo, R Stong, Dehn–Thurston coordinates for curves on surfaces, Comm. Anal. Geom. 12 (2004) 1

[11] A Papadopoulos, On Thurston's boundary of Teichmüller space and the extension of earthquakes, Topology Appl. 41 (1991) 147

[12] R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton University Press (1992)

[13] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417

[14] W P Thurston, Three-dimensional geometry and topology Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997)