Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves
| Autor: | Vincent Delecroix, Elise Goujard, Peter Zograf, Anton Zorich |
|---|---|
| Přispěvatelé: | Groupe Sociétés, Religions, Laïcités (GSRL), Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), École Pratique des Hautes Études (EPHE), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-19-CE40-0021,Phymath,physique mathématique(2019) |
| Rok vydání: | 2021 |
| Předmět: |
Teichmüller space
Surface (mathematics) Pure mathematics Geodesic General Mathematics [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] Dynamical Systems (math.DS) Algebraic geometry 01 natural sciences Mathematics - Geometric Topology Mathematics - Algebraic Geometry Mathematics::Algebraic Geometry [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] 0103 physical sciences FOS: Mathematics Mathematics - Combinatorics Mathematics - Dynamical Systems 0101 mathematics Algebraic Geometry (math.AG) Quadratic differential Mathematics Meromorphic function 010102 general mathematics Geometric Topology (math.GT) Mathematics::Geometric Topology [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] Mapping class group Moduli space [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Combinatorics (math.CO) [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] 010307 mathematical physics |
| Zdroj: | Duke Mathematical Journal Duke Mathematical Journal, Duke University Press, 2021, 170 (12), pp.2633-2718. ⟨10.1215/00127094-2021-0054⟩ Duke Mathematical Journal, Duke University Press, In press Duke Mathematical Journal, 2021, 170 (12), pp.2633-2718. ⟨10.1215/00127094-2021-0054⟩ |
| ISSN: | 0012-7094 1547-7398 |
| DOI: | 10.1215/00127094-2021-0054 |
| Popis: | 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. Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042 |
| Databáze: | OpenAIRE |
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