No abstract

Consider a compact orientable surface S of genus at least 2. The Teichmiiller space J-(S) is the space of isotopy classes of conformal structures on S. By the Riemann uniformization theorem, it is also the space of isotopy classes of Riemannian metrics on S of constant curvature -1, and this is the viewpoint we are going to take.As ggZf'(S)is defined by the property that mE~--(S) tends to 2e~//~e(S) if and only if, for all simple closed curves ~, fl on S, the ratio l,,, (oO/l,,,(fl ) tends to i(~, 2)/i (~, 2) where: /~(~) is the infimum of the lengths for m of all simple closed curves isotopic to ~ (namely, dm(~) is the length of the closed m-geodesic homotopic to ct); if 2e~r162 is a closed curve, say, the geometric intersection number i(~, 2) is the infimum of the number of intersection points of 2 with the curves isotopic to ~. A priori, there is no obvious connection between Riemannian metrics and simple curves, or between lengths and intersection numbers. Here we will give a "unified theory" of these notions.

Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form Annales de la faculté des sciences de Toulouse 6 e série, tome 5, n o 2 (1996), p. 233-297 © Université Paul Sabatier, 1996, tous droits réservés. L'accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/