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1 September 2021 Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves
Vincent Delecroix, Élise Goujard, Peter Zograf, Anton Zorich
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Duke Math. J. 170(12): 2633-2718 (1 September 2021). DOI: 10.1215/00127094-2021-0054


We express the Masur–Veech volume and the area Siegel–Veech constant of the moduli space Qg,n of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers M g,nψ1d1ψ ndn with explicit rational coefficients, where g<g and n<2g+n. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials N g,n(b1,,bn) that also appear in Mirzakhani’s recursion for the Weil–Petersson volumes of the moduli spaces Mg,n(b1,,bn) of bordered hyperbolic surfaces with geodesic boundaries of lengths b1,,bn. A similar formula for the Masur–Veech volume (but without explicit evaluation) was obtained earlier by Mirzakhani through a completely different approach. We prove a further result: the density of the mapping class group orbit Modg,nγ of any simple closed multicurve γ inside the ambient set MLg,n(Z) 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 Qg,n. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n=0. In particular, we compute explicitly the asymptotic frequencies of separating and nonseparating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g, and we show that in large genera the separating closed geodesics are 2 3πg1 4g times less frequent.


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Vincent Delecroix. Élise Goujard. Peter Zograf. Anton Zorich. "Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves." Duke Math. J. 170 (12) 2633 - 2718, 1 September 2021.


Received: 13 February 2020; Revised: 1 October 2020; Published: 1 September 2021
First available in Project Euclid: 9 August 2021

Digital Object Identifier: 10.1215/00127094-2021-0054

Primary: 32G15
Secondary: 57M50

Keywords: hyperbolic surfaces , Masur–Veech volume , multicurves , quadratic differential , simple closed curves , square-tiled surfaces , Teichmüller theory

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 12 • 1 September 2021
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