A geometric characterization of cyclic p-gonal surfaces

A closed Riemann surface S of genus \(g\ge 2\) is called cyclic p-gonal if it has an automorphism \(\rho \) of order p, where p is a prime, such that \(S/\langle \rho \rangle \) has genus 0. For \(p=2\), the surface is called hyperelliptic and \(\rho \) is an involution with \(2g+2\) fixed points. Classicaly, cyclic p-gonal surfaces can be characterized using Fuchsian groups. In this paper we establish a geometric characterization of cyclic p-gonal surfaces. Specifically, this is determined by collections of simple geodesic arcs on the surfaces and graphs associated to these arcs. In previous work, the author has given a geometric characterization of hyperelliptic surfaces in terms of simple, closed geodesics and graphs associated to these. The present work may be seen as an extension. Involutions, however, have properties that do not generalize to arbitrary automorphisms of order p. Hence, the number of vertices needed in the graphs used here is larger than that of the hyperelliptic case.