Distribution of branched \(\mathbb{D}_ p\)-coverings of surfaces (Q1382827)

The basic question of the paper is the following: In how many different ways can a given surface (a compact 2-manifold without boundary) be a branched covering of another given surface? In previous work, the authors with \textit{S.-G. Kim} [Discrete Math. 156, No. 1-3, 141-170 (1996; Zbl 0861.57003)] enumerated the equivalence classes of regular branched prime-fold coverings of a given surface. In the present work, this is generalized to the case of dihedral branched coverings, or more precisely to the regular branched coverings where the covering group is a dihedral group \(\mathbb{D}_p\) of order \(2p\) where \(p\) is prime. Equivalently, this gives the number of \(\mathbb{D}_p\)-actions on surfaces with a given quotient surface.