SYMMETRIES OF REAL CYCLIC p-GONAL RIEMANN SURFACES

A closed Riemann surface X which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. A p-gonal Riemann surface is called real p-gonal if there is an anticon-formal involution (symmetry) σ of X commuting with the p-gonal morphism. If the p-gonal morphism is a cyclic regu-lar covering the Riemann surface is called real cyclic p-gonal, otherwise it is called real generic p-gonal. The species of the symmetry σ is the number of connected components of the fixed point set Fix (σ) and the orientability of the Klein sur-face X/〈σ〉. In this paper we find the species for the possible symmetries of real cyclic p-gonal Riemann surfaces by means of Fuchsian and NEC groups. 1. Introduction. A closed Riemann surface X which can be realised as a p-sheeted covering o