Abstract

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 anticonformal involution (symmetry) σ of X commuting with the p-gonal morphism. If the p-gonal morphism is a cyclic regular 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 surface 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.

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