TY - JOUR

T1 - Masur-veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves

AU - Delecroix, Vincent

AU - Goujard, Élise

AU - Zograf, Peter

AU - Zorich, Anton

N1 - Funding Information:
Goujard's work was partially supported by a Projets Exploratoires Premier Soutien (PEPS) JCJC grant. The results of Section 4 were obtained at Saint Petersburg State University under the support of Russian Science Foundation grant 19-71-30002. The work of Zograf was partially supported by the Government of Russian Federation megagrant 14.W03.31.0030. This material is based on work supported by National Science Foundation grant DMS-1440140 while Delecroix, Goujard, and Zorich were in residence at the MSRI during the Fall 2019 semester. It was also supported by Agence Nationale de la Recherche grant ANR-19-CE40-0021.
Publisher Copyright:
© 2021 Duke University Press. All rights reserved.

PY - 2021

Y1 - 2021

N2 - We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space Qg,nof genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫-Mg′,n′Ψd11···Ψdn′n′with explicit rational coefficients, where g′ 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,n·γ of any simple closed multicurve γ inside the ambient set ℳℲg,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 √2/3πg · 1/4g times less frequent.

AB - We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space Qg,nof genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ∫-Mg′,n′Ψd11···Ψdn′n′with explicit rational coefficients, where g′ 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,n·γ of any simple closed multicurve γ inside the ambient set ℳℲg,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 √2/3πg · 1/4g times less frequent.

UR - http://www.scopus.com/inward/record.url?scp=85114831202&partnerID=8YFLogxK

U2 - 10.1215/00127094-2021-0054

DO - 10.1215/00127094-2021-0054

M3 - Article

AN - SCOPUS:85114831202

VL - 170

SP - 2633

EP - 2718

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 12

ER -