Realization of primitive branched coverings over closed surfaces following the Hurwitz approach (Q1407184)

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.