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Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. (English) Zbl 1019.32014
Let $$\Sigma$$ be a compact Riemann surface of genus $$g>1$$ and let $$\mu$$ be a partition of $$2g-2$$ into $$l=l(\mu)$$ parts. Let $${\mathcal H}(\mu)$$ denote the moduli space of $$(l+2)$$-tuples $$(\Sigma, \omega,p_1,\cdots,p_l)$$ where $$\omega$$ is a holomorphic differential on $$\Sigma$$ with divisor $$(w)=\sum \mu_i[p_i].$$
The authors consider the subspace $${\mathcal H}_1(\mu)$$ of $${\mathcal H}(\mu)$$ defined by the equation Area$$_{\omega}(\Sigma)=1$$ (where the area is taken with respect to the metric defined by $$\omega$$), together with a measure $$\nu$$ on $${\mathcal H}_1(\mu)$$ defined as follows. For a set $$E \subset {\mathcal H}_1(\mu)$$ that lies in the domain of a coordinate chart $$\Phi,$$ let $$C\Phi(E)$$ be the cone over $$\Phi(E)$$ with vertex at the origin, and set $$\nu(E)\colon =\text{vol}(C\Phi(E)).$$ This measure is invariant under the action of $$SL(2,\mathbb R)$$ on $${\mathcal H}_1(\mu)$$ and it was known by results of Masur and Veech that $$\nu({\mathcal H}_1(\mu))<\infty.$$ The authors compute $$\nu({\mathcal H}_1(\mu))$$ showing in particular the rationality of $$\pi^{-2g}\nu({\mathcal H}_1(\mu)),$$ that had been conjectured by Kontsevich and Zorich.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 05A17 Combinatorial aspects of partitions of integers
##### Keywords:
volumes of moduli spaces; branched covering
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