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

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.

arXiv commentThe current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042

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Cited by in corpus (13)

• Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
• Topological recursion for Masur-Veech volumes
• Large Genus Asymptotics for Intersection Numbers and Principal Strata Volumes of Quadratic Differentials
• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• Enumeration of meanders and Masur-Veech volumes
• Length statistics of random multicurves on closed hyperbolic surfaces
• Volumes of moduli spaces of flat surfaces
• Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
• On the Kontsevich geometry of the combinatorial Teichmüller space
• Uniform Lower Bound for Intersection Numbers of $\psi$-Classes
• Masur--Veech volumes of quadratic differentials and their asymptotics
• Around the combinatorial unit ball of measured foliations on bordered surfaces
• An explicit formula for Witten's 2-correlators