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

Representation Growth of Linear Groups

Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function \calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then \calz_\Gamma(s) has an ``Euler factorization". The "factor at infinity" is sometimes called the "Witten zeta function" counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it is bounded away from 0. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa