Large Genus Asymptotics for Volumes of Strata of Abelian Differentials

2018/04/15 by Aggarwal, Amol · 3 citations
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1804.05431

Abstract

In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume ν₁ ( H₁ (m) ) of a stratum indexed by a partition m = (m₁, m₂, … , m_n) is ( 4 + o(1) ) ∏_{i = 1}ⁿ (m_i + 1)^-1 as 2g - 2 = ∑_{i = 1}ⁿ m_i tends to ∞. This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Moeller-Zagier and Sauvaget, who established these limiting statements in the special cases m = 1^{2g - 2} and m = (2g - 2), respectively. We also include an Appendix by Anton Zorich that uses our main result to deduce the large genus asymptotics for Siegel-Veech constants that count certain types of saddle connections.

Cited by

• Distribution of lengths of closed saddle connections on moduli space of large genus translation surface
• The central heat trace on large compact classical groups
• Asymptotics of shortest filling closed multi-geodesics

Related

• The Large genus asymptotic expansion of Masur-Veech volumes
• Volumes and Siegel-Veech constants of H(2g-2) and Hodge integrals
• Large-genus asymptotics of saddle connection Siegel-Veech constants
• Large genus asymptotics of super Weil-Petersson volumes
• On the large genus asymptotics of Weil-Petersson volumes