On real trigonal Riemann surfaces

Abstract

A closed Riemann surface $X$ which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) $\sigma$ of $X$ commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry $\sigma $ is the number of connected components of the fixed point set $\mathrm{Fix}(\sigma)$ and the orientability of the Klein surface $X/\langle\sigma\rangle$. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.