Grothendieck’s problems concerning profinite completions and representations of groups | Annals of Mathematics

Abstract 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 u^:Γ^1→Γ^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↪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 u^:P^→Γ^ 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 clA(P) for every commutative ring A≠0.

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数学年刊(Annals of Mathematics)

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