Sets of periods for automorphisms of compact Riemann surfaces

Let G=〈f〉G=〈f〉 be a finite cyclic group of order NN that acts by conformal automorphisms on a compact Riemann surface SS of genus g≥2g≥2. Associated to this is a set AA of periods defined to be the subset of proper divisors dd of NN such that, for some x∈Sx∈S, xx is fixed by fdfd but not by any smaller power of ff. For an arbitrary subset AA of proper divisors of NN, there is always an associated action and, if gAgA denotes the minimal genus for such an action, an algorithm is obtained here to determine gAgA. Furthermore, a set AmaxAmax is determined for which gAgA is maximal.