Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form

Abstract

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 <http://www.numdam.org/item?id=AFST_1996_6_5_2_233_0> © 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/

Notes on a paper of Mess

“…bending lamination) on the boundary of the convex core. However, in the hyperbolic setting, Bonahon [30] proved that the two infinitesimal rigidity questions, concerning the induced metric and concerning the bending lamination, are equivalent. Bonahon [31] also gave a careful analysis of what happens near the "Fuchsian locus", showing that uniqueness does hold there.…”

Cross ratios and identities for higher Teichmüller-Thurston theory

“…Indeed recall that every hyperbolic pair of pants with totally geodesic boundary is determined up to isometry by the length of its three boundary components and hence every geometric function is a function of these three parameters. In Section 5, using Thurston's shear coordinates, described by Bonahon in [3], and elementary manipulations involving the classical cross ratio -as opposed to hyperbolic trigonometry in the original proofs -we recover Mirzakhani-McShane formulae (1) and (2) for the pant gap function.…”

Continuity of the bending map

“…this result is a consequence of results of [Bo3]. By lemma 3.1, (f ∞ , Γ ∞ , ρ ∞ ) is an even or convex pleated surface.…”