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We compute the asymptotics of the number of connected branched coverings of atorus as their degree goes to infinity and the ramification type stays fixed.These numbers are equal to the volumes of the moduli spaces of pairs (curve,holomorphic differential) with fixed multiplicities of zeros of thedifferential and have several applications in ergodic theory

Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials