Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces

Fillastre, François; Seppi, Andrea

Annales Henri Lebesgue, Tome 3 (2020), pp. 873-899.

Un célèbre résultat de Wolpert dit que si on calcule la forme symplectique de Weil–Petersson sur l’espace de Teichmüller de deux twists infinitésimaux le long de géodésiques fermées simples sur une surface hyperbolique donnée, on obtient la somme des cosinus des angles d’intersection. Nous définissons des déformations infinitésimales à partir d’objets plus généraux qui sont des graphes géodésiques pondérés. Ils peuvent représenter n’importe quel vecteur tangent de l’espace de Teichmüller. Nous démontrons une formule qui généralise la formule de Wolpert à ces nouveaux objets. Dans le cas de courbes fermées simples, nous retrouvons exactement le résultat de Wolpert.

A well-known theorem of Wolpert shows that the Weil–Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert’s formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.

Classification : 32G15, 53C15, 53D30
Mots clés : weighted geodesic cellulations, hyperbolic surfaces, Weil–Petersson form, Wolpert formula

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