Large genus asymptotics for volumes of strata of abelian differentials

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 ν 1 ( H 1 ( m ) ) \nu _1 \big ( \mathcal {H}_1 (m) \big ) of a stratum indexed by a partition m = ( m 1 , m 2 , … , m n ) m = (m_1, m_2, \ldots , m_n) is ( 4 + o ( 1 ) ) ∏ i = 1 n ( m i + 1 ) − 1 \big ( 4 + o(1) \big ) \prod _{i = 1}^n (m_i + 1)^{-1} , as 2 g − 2 = ∑ i = 1 n m i 2g - 2 = \sum _{i = 1}^n m_i tends to ∞ \infty . This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Möller-Zagier and Sauvaget, who established these limiting statements in the special cases m = 1 2 g − 2 m = 1^{2g - 2} and m = ( 2 g − 2 ) 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.