Vincent Delecroix, Élise Goujard, Peter Zograf, Anton Zorich

Duke Math. J. 170 (12), 2633-2718, (1 September 2021) DOI: 10.1215/00127094-2021-0054
KEYWORDS: Masur–Veech volume, quadratic differential, Teichmüller theory, square-tiled surfaces, multicurves, simple closed curves, hyperbolic surfaces, 32G15, 57M50

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 ${\int}_{{\stackrel{\u203e}{\mathcal{M}}}_{{g}^{\prime},{n}^{\prime}}}{\mathit{\psi}}_{1}^{{d}_{1}}\cdots {\mathit{\psi}}_{{n}^{\prime}}^{{d}_{{n}^{\prime}}}$ 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.