Finiteness Properties and Profinite Completions

In this note we show that various (geometric/homological) finiteness properties are not profinite properties. For example for every 1 \le k, \ell \le \bbn, there exist two finitely generated residually finite groups \Ga_1 and \Ga_2 with isomorphic profinite completions, such that \Ga_1 is strictly of type $F_k$ and \Ga_2 of type $F_\ell$

Beauville surfaces and finite simple groups

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.Comment: 20 page