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Mathematics, Geometric Topology, Algebraic Geometry, Combinatorics, Dynamical Systems
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.
