The Schur multiplier, profinite completions and decidability

Abstract. We fix a finitely presented group Q and consider short exact sequences 1 → N → Γ → Q → 1. The inclusionN ↪ → Γ induces a morphism of profinite completions N ̂ → Γ̂. We prove that this is an isomorphism for all N and Γ if and only if Q is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residually finite group Γ and a finitely presentable subgroup P ↪ → Γ, can determine whether or not P ̂ → Γ ̂ is an isomorphism