Topological types of actions on curves

TL;DR

This paper presents an algorithm to classify all topological types of finite group actions on compact Riemann surfaces of genus at least 2, where the quotient surface is a sphere.

Contribution

It introduces a novel algorithm for systematically constructing all topological types of holomorphic group actions on Riemann surfaces.

Findings

Algorithm successfully classifies all topological types for given genus and group

Provides a comprehensive list of group actions on Riemann surfaces

Enhances understanding of symmetries in complex algebraic curves

Abstract

We describe an algorithm that constructs a list of all topological types of holomorphic actions of a finite group on a compact Riemann surface $C$ of genus at least $g \geq 2$ with $C/G \cong \mathbb{P}^1$.