论文标题
Masur-Deech卷，简单封闭的大地测量学的频率和曲线模量空间的交点数
Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves
论文作者
论文摘要
我们表达了Moduli空间的Masur-Veech量和Siegel-Veech常数$ \ MATHCAL {Q} _ {g，n} $的属$ G $ G $ Meromorormorormorormorflic Quadric差异差异，$ n $ simple Poles作为$ -CLASSES classes classecit coefficity coeffication coreftications rationals classe classe coeffitals rations coeffitals rationals的多项式。本文在本文中获得的公式是由涉及Kontsevich体积多项式的晶格计数产生的，这些多项式在Mirzakhani的递归中也出现在带有地球界边界的边缘双曲线表面模量空间的Weil-Petersson体积中。 Mirzakhani通过完全不同的方法获得了Masur-Deech量的类似公式（尽管没有明确的评估）。 此外，我们证明，米尔扎卡尼（Mirzakhani）计算出的一组整体测量的层压板的映射类组轨道的密度与所有正方形的表面相关的平方形表面的密度与所有方形的表面相关的平方表面的密度恰逢所有方形c $ quartield coptiled $ quartiled coptiled compocts $} $} $} $ \ \ \ \ \ \ \ \ \ \ \ n n n n n n n n c $} c { 我们在特殊情况下更详细地研究了所得密度（或等效地贡献）$ 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 $\sqrt{\frac{2}{3πg}}\cdot\frac{1}{4^g}$ times less frequent.
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 $ψ$-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 $γ$ 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 $\mathcal{Q}_{g,n}$. 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 $\sqrt{\frac{2}{3πg}}\cdot\frac{1}{4^g}$ times less frequent.
