Grothendieck’s problems concerning profinite completions and representations of groups

Abstract

In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u :Γ 1 → Γ2 be a homomorphism such that the induced map of profinite completions ˆ u : ˆ Γ1 → ˆ Γ2 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