Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

TL;DR

This paper provides explicit polynomial formulas for Masur-Veech volumes and Siegel-Veech constants of moduli spaces of quadratic differentials, linking them to intersection numbers and lattice point counts, and studies geodesic frequency asymptotics.

Contribution

It derives explicit polynomial formulas for volumes and constants in moduli spaces using intersection theory and lattice counts, connecting geometric and combinatorial aspects.

Findings

Formulas for Masur-Veech volumes and Siegel-Veech constants as polynomials in intersection numbers.

Equivalence of orbit density of multicurves and square-tiled surface densities.

Asymptotic frequencies of simple closed geodesics for small and large genus.

Abstract

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent.