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Large genus asymptotics for volumes of strata of abelian differentials


Author: Amol Aggarwal; with an appendix by Anton Zorich
Journal: J. Amer. Math. Soc. 33 (2020), 941-989
MSC (2010): Primary 32G15; Secondary 37P45, 05A16
DOI: https://doi.org/10.1090/jams/947
Published electronically: September 28, 2020
MathSciNet review: 4155217
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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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Additional Information

Anton Zorich
Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138 – and – Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027; Center for Advanced Studies, Skoltech; Institut de Mathématiques de Jussieu – Paris Rive Gauche, Bâtiment Sophie Germain, Case 7012, 8 Place Aurélie Nemours, 75205 Paris Cedex 13, France

Received by editor(s): June 4, 2018
Received by editor(s) in revised form: June 17, 2019, and October 30, 2019
Published electronically: September 28, 2020
Additional Notes: This work was partially supported by the NSF Graduate Research Fellowship under grant numbers DGE1144152 and DMS-1664619.
Article copyright: © Copyright 2020 American Mathematical Society