Areas of two-dimensional moduli spaces

Abstract:

Wolpert’s formula expresses the Weil-Petersson $2$-form in terms of the Fenchel-Nielsen coordinates in case of a closed or punctured surface. The area-form in Fenchel-Nielsen coordinates is invariant under the mapping class group on each 2-dimensional Teichmüller space of a surface with singularities, hence areas with respect to it can be calculated for 2-dimensional moduli spaces in cases when the Teichmüller space admits global Fenchel-Nielsen coordinates: The area of the moduli space for the signature $(0;2\theta _{1},2\theta _{2},2\theta _{3},2\theta _{4})$ is $2(\pi ^{2}-\theta _{1}^{2}-\theta _{2}^{2}-\theta _{3}^{2}-\theta _{4}^{2})$, the definition of signature is generalized to include punctures, cone points and geodesic boundary curves. In case the surface is represented by a Fuchsian group, the area is the classical Weil-Petersson area.