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

Abstract

We express the Masur-Veech volume Vol Qg,n and the area Siegel-Veech constant carea (Qg,n) of the moduli space Qg,n of meromorphic quadratic differential with n simple poles and no other poles as polynomials in the intersection numbers of ψ-classes supported on the boundary cycles of the Deligne-Mumford compactification Mg,n. The formulae obtained in this article are derived from lattice point count involving the Kontsevich volume polynomials Ng,n(b 1 , . . . , bn) that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space Mg,n(b 1 , . . . , bn) of bordered hyperbolic Riemann surfaces.A similar formula for the Masur-Veech volume Vol Qg (though without explicit evaluation) was obtained earlier by M. Mirzakhani through completely different approach. We prove further result: up to a normalization factor depending only on g and n (which we compute explicitly), the density of the orbit Modg,n •γ 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 non-separating 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 4 g times less frequent.We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume Vol Qg in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of ψ-classes in large genera.

Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials

“…Similarly q is allowed to have ordinary marked points, i.e., when some m i = 0, which are regarded as "zeros of order 0". Moreover, in our setting all zeros are labeled (as in [CMSZ19], but contrary to [DGZZ19] where only the simple poles are labeled).…”

“…ψ 2 3g−3 and can be computed efficiently (see the end of Section 3.3 for references on this topic). In addition to the large n asymptotic, in [DGZZ19] the large g asymptotic of vol(Q g,n ) was described conjecturally, and the conjecture was further extended and refined in [YZZ19].…”

“…Another approach is to decompose half-translation surfaces into ribbon graphs and sum up the corresponding local contributions. This was first carried out for all strata in genus zero in [AEZ16] and then extended to the principal strata for all genus in [DGZZ19]. These local contributions of ribbon graphs are indeed intersection numbers of ψ-classes on M g,n that come from Kontsevich's proof of Witten's conjecture [Kon92].…”

Recursion for Masur-Veech volumes of moduli spaces of quadratic differentials

“…for g = 100 is equal to 0.9993 supporting a conjecture from [4]. According to [3], the work in progress [10] will contain a refined version of this conjecture.…”

On the large genus asymptotics of psi-class intersection numbers

“…al. [9] regarding the large genus uniform leading asymptotics of the psi-class intersection numbers. We also investigate polynomiality phenomenon in the large genera.…”