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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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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.

References
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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.
  • © Copyright 2020 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 33 (2020), 941-989
  • MSC (2010): Primary 32G15; Secondary 37P45, 05A16
  • DOI: https://doi.org/10.1090/jams/947
  • MathSciNet review: 4155217