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 of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani through completely different approach. We prove further result: up to an explicit normalization factor depending only on the genus and on the number of cusps, the density of the orbit of any simple closed multicurve computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve. We study the resulting densities in more detail in the special case when there are no cusps. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent. We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi-classes in large genera.

References in corpus (8)

• Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
• Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow
• Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves
• Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares
• Length statistics of random multicurves on closed hyperbolic surfaces
• Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata

Cited by in corpus (11)

• Topological recursion for Masur-Veech volumes
• Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves
• Large genus asymptotics for lengths of separating closed geodesics on random surfaces
• Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials
• Counting hyperbolic multi-geodesics with respect to the lengths of individual components
• Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursions
• Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata
• Continuity of the orthogeodesic foliation and ergodic theory of the earthquake flow
• The factorial growth of topological recursion
• Contribution of $n$-cylinder square-tiled surfaces to Masur-Veech volume of $\mathcal{H}(2g-2)$
• Length spectrum of large genus random metric maps