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 at most n simple poles and no other poles as polynomials in the intersection numbers ${\int }_{{\bar{\mathcal{M}}}_{g^{\prime },n^{\prime }}}{\mathit{\psi }}_1^{d_1}\cdots {\mathit{\psi }}_{n^{\prime }}^{d_{n^{\prime }}}$ with explicit rational coefficients, where $g^{\prime }<g$ and $n^{\prime }<2g+n$. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials $N_{g^{\prime },n^{\prime }}(b_1,\dots ,b_{n^{\prime }})$ that also appear in Mirzakhani’s recursion for the Weil–Petersson volumes of the moduli spaces ${\mathcal{M}}_{g^{\prime },n^{\prime }}(b_1,\dots ,b_{n^{\prime }})$ of bordered hyperbolic surfaces with geodesic boundaries of lengths $b_1,\dots ,b_{n^{\prime }}$. 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 ${\mathrm{Mod}}_{g,n}\cdot \mathit{\gamma }$ of any simple closed multicurve γ inside the ambient set ${\mathcal{ML}}_{g,n}(\mathbb{Z})$ 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 ${\mathcal{Q}}_{g,n}$. 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 $\sqrt{\frac{2}{3\mathrm{\pi }g}}\cdot \frac{1}{4^g}$ times less frequent.