Grothendieck’s Problems concerning Profinite Completions and Representations of Groups

In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u : Γ1 → Γ2 be a homomorphism such that the induced map of profinite completions û : Γ̂1 → Γ̂2 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 ↪→ Γ such that Γ 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 Γ, but û : P̂ → Γ̂ is an isomorphism. The same construction also 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 clA(P) for every commutative ring A 6= 0.