Abstract

This is a survey of some recent applications of the character theory of finite groups to the theory of surfaces. The emphasis is on compact Riemann surfaces, together with their associated automorphism groups, coverings, homology groups, combinatorial structures, fields of definition, and length spectra.

Introduction

With the publication of the atlasof Finite Groups [16], and subsequently of its companion, the atlasof Brauer Characters [44], there is a now considerable wealth of information available about the character theory of finite simple groups and related groups. This means that the great apparatus of representation theory developed by Frobenius, Schur, Brauer and others can be applied more effectively than ever to construct examples, to solve specific problems and to test general conjectures. Most of these applications have been within finite group theory itself, but mathematicians in other areas are also beginning to make use of these methods. My aim here is to give a survey of some of the ways in which character theory can contribute to the study of surfaces, with particular emphasis on compact Riemann surfaces. I have not attempted to be comprehensive: the applications I have chosen are simply those which I have recently found useful or interesting. Nevertheless, I hope that these brief comments, together with the references, will provide the interested reader with at least a sketch-map for further exploration.

Counting solutions of equations

One of the most effective character-theoretic techniques is the enumeration of the solutions of an equation in a finite group.