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Large genus asymptotics for Siegel-Veech constants. (English) Zbl 1427.14053

A pair \((X, \omega)\) of a compact Riemann surface and a holomorphic \(1\)-form gives a singular flat metric on the surface with cone-type singularities at the zeros of \(\omega\). A geodesic arc connecting two zeros of \(\omega\) with none in its interior is called a saddle connection. Counting saddle connections (and core curves of flat cylinders) of length at most \(R\) is a key problem in the study of translation surfaces. It is known by seminal work of H. Masur [Ann. Math. (2) 115, 169–200 (1982; Zbl 0497.28012)] and W. A. Veech [Ann. Math. (2) 115, 201–242 (1982; Zbl 0486.28014)], and A. Eskin and H. Masur [Ergodic Theory Dyn. Syst. 21, No. 2, 443–478 (2001; Zbl 1096.37501)] that these counts have quadratic asymptotics in \(R\), and that the constant is the same for almost every surface with fixed combinatorics. These constants are known as Siegel-Veech constants, and are closely related to the volumes of the underlying moduli spaces of translations surfaces. Understanding the behavior of these volumes and constants as genus grows has been the subject of conjectures of A. Eskin and A. Zorich [Arnold Math. J. 1, No. 4, 481–488 (2015; Zbl 1342.32012)].
This paper proves these conjectures by careful combinatorial analysis and recent developments in the large genus aysmptotics of volumes. A particularly beautiful result is that the area Siegel-Veech constant, which governs the count of cylinders weighted by area converges to \(1/2\) (and the rate is bounded above by \(1/g\)) as genus \(g \rightarrow \infty\). Another striking result shows that the constant associated to counting saddle connections between zeros of order \(m_1\) and \(m_2\) converges to \((m_1+1)(m_2+1)\), again at a rate \(1/g\).

MSC:

14H10 Families, moduli of curves (algebraic)
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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References:

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