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

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##### External Resource

https://doi.org/10.1215/00127094-2021-0054

(Publisher version)

https://doi.org/10.48550/arXiv.2011.05306

(Preprint)

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##### Citation

Delecroix, V., Goujard, E., Zograf, P. G., & Zorich, A. (2021). Masur-Veech volumes,
frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves.* Duke Mathematical
Journal,* *170*(12), 2633-2718. doi:10.1215/00127094-2021-0054.

Cite as: https://hdl.handle.net/21.11116/0000-0009-490E-7

##### 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 $n$ simple poles as polynomials in the intersection numbers

of $\psi$-classes with explicit rational coefficients. The formulae obtained in

this article result 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. A similar formula for the Masur-Veech volume (though without

explicit evaluation) was obtained earlier by Mirzakhani via completely

different approach.

Furthermore, we prove that the density of the mapping class group orbit of

any simple closed multicurve $\gamma$ 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

$\gamma$ 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 $n=0$. In particular, we compute the asymptotic

frequencies of separating and non-separating simple closed geodesics on a

closed hyperbolic surface of genus $g$ for small $g$ and we show that for large

genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi

g}}\cdot\frac{1}{4^g}$ times less frequent.

moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic

differentials with $n$ simple poles as polynomials in the intersection numbers

of $\psi$-classes with explicit rational coefficients. The formulae obtained in

this article result 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. A similar formula for the Masur-Veech volume (though without

explicit evaluation) was obtained earlier by Mirzakhani via completely

different approach.

Furthermore, we prove that the density of the mapping class group orbit of

any simple closed multicurve $\gamma$ 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

$\gamma$ 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 $n=0$. In particular, we compute the asymptotic

frequencies of separating and non-separating simple closed geodesics on a

closed hyperbolic surface of genus $g$ for small $g$ and we show that for large

genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi

g}}\cdot\frac{1}{4^g}$ times less frequent.