Beauville surfaces and finite simple groups

A Beauville surface is a rigid complex surface of the form (C ₁ × C ₂)/G, where C ₁ and C ₂ 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 A ₅, 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.