Maximal group actions on compact oriented surfaces.

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