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Realization of primitive branched coverings over closed surfaces following the Hurwitz approach. (English) Zbl 1039.57001

This paper studies the existence and classification of connected branched coverings of a closed surface. The approach taken here is purely algebraic following the ideas of Hurwitz. Specifically, let \(V\) be a closed surface, \(H\subset\pi_1(V)\) a subgroup of finite index and \(\{A_1,\dots, A_m\}\) a collection of partitions of an integer \(d\geq 2\). One wants conditions under which there is a connected branched covering \(f: W\to V\) of order \(d\) with branch data the given partitions and having \(f_{\#}\pi_1(W)= H\). Then one wishes to know how many different such coverings there are.

MSC:

57M12 Low-dimensional topology of special (e.g., branched) coverings
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