Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces
Annales Henri Lebesgue, Volume 3 (2020) , pp. 873-899.

Keywordsweighted geodesic cellulations, hyperbolic surfaces, Weil–Petersson form, Wolpert formula

### Abstract

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.

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