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 $A\neq 0$.