Automorphism Groups of Cyclic p-gonal Pseudo-real Riemann Surfaces

TL;DR

This paper classifies the automorphism groups of cyclic p-gonal pseudo-real Riemann surfaces, providing conditions for their existence and describing families with maximal automorphism groups that relate to real 2-manifolds in moduli space.

Contribution

It characterizes the full automorphism groups of these surfaces and establishes criteria for their existence, including maximal cases and connections to moduli space topology.

Findings

Automorphism group is either a semidirect product or cyclic.

Necessary and sufficient conditions for existence of certain automorphism groups.

Identification of families with maximal automorphism groups related to real 2-manifolds.

Abstract

In this article we prove that the full automorphism group of a cyclic $p$-gonal pseudo-real Riemann surface of genus $g$ is either a semidirect product $C_{n}\ltimes C_{p}$ or a cyclic group, where $p$ is a prime $>2$ and $g>(p-1)^{2}$. We obtain necessary and sufficient conditions for the existence of a cyclic $p$-gonal pseudo-real Riemann surface with full\ automorphism group isomorphic to a given finite group. Finally we describe some families of cyclic $p$-gonal pseudo-real Riemann surfaces where the order of the full automorphism group is maximal and show that such families determine some real 2-manifolds embbeded in the branch locus of moduli space.