Grothendieck’s problems concerning profinite completions and representations of groups

Abstract

In 1970 Alexander Grothendieck [6] posed the following problem: 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 problem 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.

The same construction allows us to settle a second problem of Grothendieck by exhibiting finitely presented, residually finite groups $P$ that have infinite index in their Tannaka duality groups ${\rm cl}_A(P)$ for every commutative ring $A\neq 0$.