On Finite Groups of Symmetries of Surfaces

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.