Abstract
After describing the integration procedure on quotient spaces, we derive Mirzakhani's first integration formula, which expresses the integral of functions of the type " length of a multi-curve " in terms of the volume of the moduli space of the multi-curve's complement. We then prove Bers' theorem and deduce that the Weil-Petersson volume of the moduli space is finite. Finally, we begin the proof of McShane's identity, generalized by Mirzakhani, and which constitutes the first step towards topological recursion formulas.
References :
M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Inventiones Math. 2006 (sections 3 and 4 and personal version of Theorem 8.1).
P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Modern Birkhäuser Classics, Section 5.1.
