Contribution of \(n\)-cylinder square-tiled surfaces to Masur-Veech volume of \(\mathcal{H}(2g-2)\). (English) Zbl 1530.30040
Square-tiled surfaces are special kinds of quadrangulations of surfaces. Their asymptotic enumeration is tightly linked with the so-called Masur-Veech volumes of the strata of the moduli space of abelian and quadratic differentials as shown by A. Zorich [in: Rigidity in dynamics and geometry. Berlin: Springer. 459–471 (2002; Zbl 1038.37015)]. This link has been used to deduce many values and properties of the Masur-Veech volumes.
The simplest stratum of the moduli space of abelian differentials is \(\mathcal{H}(2g-2)\) which consists of pairs \((X, \omega)\) where \(X\) is a Riemann surface and \(\omega\) an abelian differential on \(X\) with a single zero of degree \(2g-2\). Their Masur-Veech volumes have been studied from an algebraic geometry perspective in a work of A. Sauvaget [Geom. Funct. Anal. 28, No. 6, 1756–1779 (2018; Zbl 1404.14035)]. In this article, an alternative computation of the Masur-Veech volumes of \(\mathcal{H}(2g-2)\) is provided using square-tiled surface enumeration. The main result shows that a natural refined asymptotic count of square-tiled surfaces in \(\mathcal{H}(2g-2)\) is a one-parameter deformation of the original generating series of A. Sauvaget.
The strategy of the proof follows the initial idea of J. Athreya et al. [Geom. Dedicata 170, 195–217 (2014; Zbl 1290.32012)] and the reviewer et al. [Duke Math. J. 170, No. 12, 2633–2718 (2021; Zbl 1471.14066)]. Namely, square-tiled surfaces are counted with an additional parameter taking into account the combinatorics of their cylinders. That required the author to develop a complete framework for the asymptotic enumeration of metric unicellular bipartite maps (i.e., metric graphs embedded in surface with a proper 2-coloring of its vertices and whose complement is homeomorphic to a disk). One important tool turns out to be the so-called Chapuy-Féray-Fusy bijection between unicellular maps and decorated trees [G. Chapuy et al., J. Comb. Theory, Ser. A 120, No. 8, 2064–2092 (2013; Zbl 1278.05081)].
The simplest stratum of the moduli space of abelian differentials is \(\mathcal{H}(2g-2)\) which consists of pairs \((X, \omega)\) where \(X\) is a Riemann surface and \(\omega\) an abelian differential on \(X\) with a single zero of degree \(2g-2\). Their Masur-Veech volumes have been studied from an algebraic geometry perspective in a work of A. Sauvaget [Geom. Funct. Anal. 28, No. 6, 1756–1779 (2018; Zbl 1404.14035)]. In this article, an alternative computation of the Masur-Veech volumes of \(\mathcal{H}(2g-2)\) is provided using square-tiled surface enumeration. The main result shows that a natural refined asymptotic count of square-tiled surfaces in \(\mathcal{H}(2g-2)\) is a one-parameter deformation of the original generating series of A. Sauvaget.
The strategy of the proof follows the initial idea of J. Athreya et al. [Geom. Dedicata 170, 195–217 (2014; Zbl 1290.32012)] and the reviewer et al. [Duke Math. J. 170, No. 12, 2633–2718 (2021; Zbl 1471.14066)]. Namely, square-tiled surfaces are counted with an additional parameter taking into account the combinatorics of their cylinders. That required the author to develop a complete framework for the asymptotic enumeration of metric unicellular bipartite maps (i.e., metric graphs embedded in surface with a proper 2-coloring of its vertices and whose complement is homeomorphic to a disk). One important tool turns out to be the so-called Chapuy-Féray-Fusy bijection between unicellular maps and decorated trees [G. Chapuy et al., J. Comb. Theory, Ser. A 120, No. 8, 2064–2092 (2013; Zbl 1278.05081)].
Reviewer: Vincent Delecroix (Bordeaux)
MSC:
| 30F30 | Differentials on Riemann surfaces |
| 30F60 | Teichmüller theory for Riemann surfaces |
| 05A16 | Asymptotic enumeration |
| 05E14 | Combinatorial aspects of algebraic geometry |
| 57M50 | General geometric structures on low-dimensional manifolds |
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