The Schur multiplier, profinite completions and decidability

Abstract

We fix a finitely presented group $Q$ and consider short exact sequences $1\to N\to G\to Q\to 1$ with $G$ finitely generated. The inclusion $N\to G$ induces a morphism of profinite completions $\hat{N}\to \hat{G}$. We prove that this is an isomorphism for all $N$ and $G$ 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 $G$ and a finitely presentable subgroup $P\subset G$, can determine whether or not $\hat{P}\to \hat{G}$ is an isomorphism.

6 pages no figures. To appear in the Bulletin London Math Soc

References in corpus (1)

• Decision problems and profinite completions of groups

Cited by in corpus (1)

• Nilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag