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تسجيل مستخدم جديدContribution of one-cylinder square-tiled surfaces to Masur-Veech volumes
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We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.
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er $mathbb Q$ in the moduli space of Abelian differentials. Moreover, we prove that the combinatorics of the horizontal and of the vertical decompositions are asymptotically uncorrelated. As a consequence, we prove the existence of an asymptotic distribution for the combinatorics of a random interval exchange transformation with integer lengths. We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. We conjecture that the corresponding relative contribution is asymptotically of the order $1/d$, where $d$ is the dimension of the stratum, and prove that this conjecture is equivalent to the long-standing conjecture on the large genus asymptotics of the Masur-Veech volumes. We prove, in particular, that the recent results of Chen, Moller and Zagier imply that the conjecture holds for the principal stratum of Abelian differentials as the genus tends to infinity. Our result on random interval exchanges with integer lengths allows to make empirical computation of the probability to get a $1$-cylinder pillowcase cover taking a random one in a given stratum. We use this technique to derive the approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.
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it rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhanis 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 $sqrt{frac{2}{3pi g}}cdotfrac{1}{4^g}$ times less frequent.
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aces of quadratic differentials, and propose refinements of the conjectural formulas given in [12,4] for the large genus asymptotics of ${rm Vol} , mathcal{Q}_{g,n}$ and of the associated area Siegel--Veech constants.
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