Maximal group actions on compact oriented surfaces

Abstract

Suppose S is a compact oriented surface of genus σ≥2 and Cp is a group of orientation preserving automorphisms of S of prime order p≥5. We show that there is always a finite supergroup G>Cp of orientation preserving automorphisms of S except when the genus of S/Cp is minimal (or equivalently, when the number of fixed points of Cp is maximal). Moreover, we exhibit an infinite sequence of genera within which any given action of Cp on S implies Cp is contained in some finite supergroup and demonstrate for genera outside of this sequence the existence of at least one Cp-action for which Cp is not contained in any such finite supergroup (for sufficiently large σ).