Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaces (extended version)

TL;DR

This paper classifies finite simple groups acting on Riemann surfaces with specific fixed point constraints, providing detailed computational methods and branching data for these group actions.

Contribution

It offers a comprehensive classification of such group actions, including detailed GAP computational techniques and explicit lemmas linking calculations to theoretical results.

Findings

Classification of finite simple groups with fixity 3 on Riemann surfaces

Explicit GAP code and computational methods provided

Detailed branching data for each group action

Abstract

Motivated by the theory of Riemann surfaces, we classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case we also give information about the branching datum of the surface. There is a shorter version of this article (submitted for publication), and in this extended version we give many more details about the GAP code that we use for the calculations. We also explicitly include a lemma that we only quote in the short version, so we can explain how exactly the GAP calculations and the lemma work together.