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

TL;DR

This paper generalizes Wolpert's formula for the Weil-Petersson symplectic form on Teichmüller space from simple closed geodesics to balanced geodesic graphs, broadening the understanding of infinitesimal deformations.

Contribution

It introduces a new class of infinitesimal deformations based on balanced geodesic graphs and extends Wolpert's formula to this broader setting.

Findings

Generalized Wolpert's formula for balanced geodesic graphs

Recovered classical Wolpert's formula for simple closed curves

Provided a new framework for representing tangent vectors in Teichmüller space

Abstract

A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichm\"uller 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 Teichm\"uller 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.