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Conjectural large genus asymptotics of Masur-Veech volumes and of area Siegel-Veech constants of strata of quadratic differentials. (English) Zbl 1452.14026

The moduli space of quadratic differentials (with at worst simple poles) on Riemann surfaces can be stratified according to the number and multiplicities of singularities of the differentials. For each stratum of quadratic differentials, period coordinates with respect to the induced flat metric give a natural volume form, leading to a finite volume on the hyperboloid of unit-area differentials, called the Masur-Veech volume. Computing Masur-Veech volumes and analyzing their asymptotic behavior has motivated a number of interesting works in the fields of Teichmüller dynamics, intersection theory, representation theory, and combinatorics. For the case of abelian differentials (whose global squares give quadratic differentials), large genus asymptotics of Masur-Veech volumes were conjectured by A. Eskin and A. Zorich [Arnold Math. J. 1, No. 4, 481–488 (2015; Zbl 1342.32012)], and were settled respectively by A. Aggarwal [J. Am. Math. Soc. 33, No. 4, 941–989 (2020; Zbl 1452.14025)] via a combinatorial method and by D. Chen et al. [Invent. Math. 222, No. 1, 283–373 (2020; Zbl 1446.14015)] via intersection theory. In this paper the authors make conjectural descriptions on the large genus asymptotic behavior of Masur-Veech volumes of the strata of (primitive) quadratic differentials as well as their (area) Siegel-Veech constants. Numerical evidence is also provided for these conjectures.

MSC:

14H15 Families, moduli of curves (analytic)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F30 Differentials on Riemann surfaces
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