On Finite Groups of Symmetries of Surfaces

TL;DR

This paper reviews how the genus spectrum of finite groups relates to their algebraic properties, providing new results on specific groups using theoretical and computational methods.

Contribution

It offers new insights into the arithmetical properties of genus spectra and explicit results for certain groups, combining theory and computation.

Findings

Explicit genus spectrum results for 2-groups of maximal class

New findings on sporadic simple groups

Results on groups PSL(2,q) for small q

Abstract

The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the genus spectrum of $G$ to its group theoretical properties. In particular, the arithmetical properties of genus spectra are discussed, and explicit results are given on the 2-groups of maximal class, certain sporadic simple groups and a some of the groups PSL$(2,q)$, where $q$ is a small prime power. These results are partially new, and obtained through both theoretical reasoning and application of computational techniques.