Grothendieck's Problems Concerning Profinite Completions and Representations of Groups

Martin R. Bridson and Fritz J. Grunewald

Preprint, 5 July 2003 --- to appear in Annals of Mathematics

In 1970 Alexander Grothendieck posed the following question: let $\Gamma_1$ and $\Gamma_2$ be finitely presented, residually finite groups, and let $u:\Gamma_1\to\Gamma_2$ be a homomorphism such that the induced map of profinite completions $\hat u :\hat\Gamma_1\to\hat\Gamma_2$ is an isomorphism; does it follow that $u$ is an isomorphism? In this paper we settle this question by exhibiting pairs of groups $u:P\hookrightarrow\Gamma$ such that $\Gamma$ is a direct product of two residually-finite hyperbolic groups, $P$ is a finitely presented subgroup of infinite index, $P$ is not abstractly isomorphic to $\Gamma$, but $\hat u:\hat P\to\hat\Gamma$ is an isomorphism. We settle a second question of Grothendieck by exhibiting finitely presented, residually finite groups $\Gamma$ that have infinite index in their Tannaka duality groups $\rm{cl}_A(\Gamma)$ for every commutative ring $A\neq 0$.