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Finite simple groups acting with fixity 4 and their occurrence as groups of automorphisms of Riemann surfaces (extended version)

Patrick Salfeld, Rebecca Waldecker

Abstract

In previous work, all finite simple groups that act with fixity 4 have been classified. In this article we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 with fixity four in total and in such a way that fixity 4 is exhibited on at least one orbit. This is an extended version of the submitted article, including our GAP code.

Finite simple groups acting with fixity 4 and their occurrence as groups of automorphisms of Riemann surfaces (extended version)

Abstract

In previous work, all finite simple groups that act with fixity 4 have been classified. In this article we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 with fixity four in total and in such a way that fixity 4 is exhibited on at least one orbit. This is an extended version of the submitted article, including our GAP code.

Paper Structure

This paper contains 5 sections, 15 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analysis of series of groups
  4. Proof of the main result
  5. The GAP code

Key Result

Theorem 1.1

Suppose that $G$ is a finite simple group that acts as a group of automorphisms on a compact Riemann surface $X$ of genus at least $2$. Suppose further that there is at least one orbit on which $G$ acts with fixity 4 and that $G$ acts with fixity at most 4 in total on $X$. Then $G$ acts with one of

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 20 more