We express the Masur–Veech volume and the area Siegel–Veech constant of the moduli space of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers with explicit rational coefficients, where and . The formulas obtained in this article are derived 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 of lengths . 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 of any simple closed multicurve γ 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 γ among all square-tiled surfaces in . We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when . 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 times less frequent.
"Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves." Duke Math. J. 170 (12) 2633 - 2718, 1 September 2021. https://doi.org/10.1215/00127094-2021-0054