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

TLDR: In this paper, the Masur-Veech volume and area Siegel Veech constant of the moduli space Qg,n of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫M g′,n′ψ1d1⋯ψ n′dn′ with explicit rational coefficients, where g>g and n′<2g+n.

Abstract: We express the Masur–Veech volume and the area Siegel–Veech constant of the moduli space Qg,n of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫M‾ g′,n′ψ1d1⋯ψ n′dn′ with explicit rational coefficients, where g′<g and n′<2g+n. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials N g′,n′(b1,…,bn′) that also appear in Mirzakhani’s recursion for the Weil–Petersson volumes of the moduli spaces Mg′,n′(b1,…,bn′) of bordered hyperbolic surfaces with geodesic boundaries of lengths b1,…,bn′. A similar formula for the Masur–Veech volume (but without explicit evaluation) was obtained earlier by Mirzakhani through a completely different approach. We prove a further result: the density of the mapping class group orbit Modg,n⋅γ of any simple closed multicurve γ inside the ambient set MLg,n(Z) of integral measured laminations, computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in Qg,n. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n=0. In particular, we compute explicitly the asymptotic frequencies of separating and nonseparating 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 2 3πg⋅1 4g times less frequent.