Distribution of branched ¢D p-coverings of surfaces

Abstract

A well-known theorem of Alexander [1] says that every orientable surface is a branched covering of the sphere S 2, and every nonorientable surface is a branched covering of the projective plane. In the study of surface branched coverings [2,3,14], we can ask naturally as a generalization of Alexander's theorem: In how many different ways can a given surface be a branched covering of another given surface? As a partial answer of this question, Kwak et al., [10] recently enumerated the equivalence classes of regular branched prime-fold coverings p : S → S for any given surfaces S and S . In this paper, we aim to enumerate the equivalence classes of the regular branched surface coverings p : S → S whose covering transformation group is the dihedral group D p of order 2 p, p prime. In some sense, this gives a classification of the pseudo-free D p-actions on a surface when a data for a quotient surface is given.