Large genus asymptotics for volumes of strata of abelian differentials

Aggarwal, Amol

doi:10.1090/jams/947

Large genus asymptotics for volumes of strata of abelian differentials
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by Amol Aggarwal; with an appendix by Anton Zorich HTML | PDF

J. Amer. Math. Soc. 33 (2020), 941-989 Request permission

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 $\nu _1 \big ( \mathcal {H}_1 (m) \big )$ of a stratum indexed by a partition $m = (m_1, m_2, \ldots , m_n)$ is $\big ( 4 + o(1) \big ) \prod _{i = 1}^n (m_i + 1)^{-1}$, as $2g - 2 = \sum _{i = 1}^n m_i$ 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 $m = 1^{2g - 2}$ 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.

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