Abstract
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.
Similar content being viewed by others
References
Costa, A.F., Izquierdo, M., Ying, D.: On cyclic p-gonal Riemann surfaces with several p-gonal morphisms. Geom. Dedicata 147, 139–147 (2010)
Grondmadzki, G., Weaver, A., Wootton, A.: On gonality of Riemann surfaces. Geom. Dedicata 149, 1–14 (2010)
Gallo, D.: Characterizing hyperelliptic surfaces in terms of closed geodesics. Ann. Acad. Sci. Fenn. Math. 44, 1–8 (2019)
Maskit, B.: A new characterization of hyperellipticity. Mich. Math. J. 47(1), 3–14 (2000)
Maskit, B.: The conformal group of a plane domain. Am. J. Math. 90, 718–722 (1968)
Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton University Press, Princeton (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gallo, D.M. A geometric characterization of cyclic p-gonal surfaces. Geom Dedicata 218, 57 (2024). https://doi.org/10.1007/s10711-024-00902-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10711-024-00902-6