Grothendieck’s problems concerning profinite completions and representations of groups

Abstract

In 1970 Alexander Grothendieck [6] posed the following problem: let 1 and 2 be finitely presented, residually finite groups, and let u : 1 2 be a homomorphism such that the induced map of profinite completions : 1 2 is an isomorphism; does it follow that u is an isomorphism?