Volumes of moduli spaces of flat surfaces

Abstract

We study the moduli spaces of flat surfaces with prescribed conical singularities. Veech showed that these spaces are diffeomorphic to the moduli spaces of marked Riemann surfaces, and endowed with a natural volume form depending on the orders of the singularities. We show that the volumes of these spaces are finite. Moreover we show that they are explicitely computable by induction on the Euler characteristics of the punctured surface for almost all orders of the singularities.

arXiv comment37 pages, 4 figures

References in corpus (5)

• Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
• Topological recursion for Masur-Veech volumes
• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

Cited by in corpus (2)

• Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
• Pixton's formula and Abel-Jacobi theory on the Picard stack