Beauville surfaces and finite simple groups

TL;DR

This paper proves that almost all finite simple groups, except finitely many, can be used to construct Beauville surfaces, confirming a conjecture for a broad class of groups using advanced algebraic and probabilistic methods.

Contribution

It confirms the conjecture that most finite simple groups produce Beauville surfaces, expanding understanding of the connection between group theory and complex surfaces.

Findings

Almost all finite simple groups give rise to Beauville surfaces.

The proof utilizes structure theory, probability, and character estimates.

Finitely many exceptions remain unclassified.

Abstract

A Beauville surface is a rigid complex surface of the form (C1 x C2)/G, where C1 and C2 are non-singular, projective, higher genus curves, and G is a finite group acting freely on the product. Bauer, Catanese, and Grunewald conjectured that every finite simple group G, with the exception of A5, gives rise to such a surface. We prove that this is so for almost all finite simple groups (i.e., with at most finitely many exceptions). The proof makes use of the structure theory of finite simple groups, probability theory, and character estimates.