Abstract

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. In this survey we discuss the groups that may be used in this way. En route we discuss several open problems, questions and conjectures.

Introduction

Roughly speaking (precise definitions will be given in the next section), a Beauville surface is a complex surface S defined by taking a pair of complex curves, i.e., Riemann surfaces, C1 and C2 and letting a finite group G act freely on their product to define S as a quotient (C1×C2)/G. These surfaces have a wide variety of attractive geometric properties: they are surfaces of general type; their automorphism groups [50] and fundamental groups [20] are relatively easy to compute (being closely related to G — see Section 7.2 and 7.3); these surfaces are rigid surfaces in the sense of admitting no nontrivial deformations [10] and thus correspond to isolated points in the moduli space of surfaces of general type [37].

Much of this good behaviour stems from the fact that the surface (C1× C2)/G is uniquely determined by a particular pair of generating sets of G known as a ‘Beauville structure’. This converts the study of Beauville surfaces to the study of groups with Beauville structures, i.e., Beauville groups.

Beauville surfaces were first defined by Catanese in [20] as a generalisation of an earlier example of Beauville [14, Exercise X.13(4)] (native English speakers may find the English translation [15] somewhat easier to read and get hold of) in which C = C’ and the curves are both the Fermat curve defined by the equation X5 +Y5 +Z5 = 0 being acted on by the group (Z/5Z)×(Z/5Z) (this choice of group may seem somewhat odd at first, but the reason will become clear later). Bauer, Catanese and Grunewald went on to use these surfaces to construct examples of smooth regular surfaces with vanishing geometric genus [11]. Early motivation came from the consideration of the ‘Friedman-Morgan speculation’ — a technical conjecture concerning when two algebraic surfaces are diffeomorphic which Beauville surfaces provide counterexamples to.

Abstract

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and practically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmetric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic simple groups.

Introduction

This article is concerned with groups that are generated by highly symmetric subsets of their elements: that is to say by subsets of elements whose set normalizer within the group they generate acts on them by conjugation in a highly symmetric manner. Rather than investigate the behaviour of known groups we turn this procedure around and ask what groups can be generated by a set of elements that possesses a certain assigned set of symmetries. This enables constructions by hand of a number of interesting groups, including many of the sporadic simple groups. Much of the emphasis of the research project to date has been concerned with using these techniques to construct sporadic simple groups, and this article will emphasize this important special case. Recent work of the author and Müller has been concerned with Coxeter groups, so we shall describe this case too.

This article is intended as an ‘update’ to the earlier survey article of Curtis [14]. Since [14] appeared several of the larger sporadic groups have succumbed to these techniques and a much wider class of reflections groups have been found to admit symmetric presentations corresponding to symmetric generating sets.