The Schur multiplier, profinite completions and decidability

TL;DR

This paper investigates the conditions under which the profinite completion of subgroups in finitely presented groups is an isomorphism, linking it to properties of the quotient group and establishing undecidability results.

Contribution

It characterizes when the inclusion induces an isomorphism on profinite completions and proves the undecidability of this property for finitely presented groups.

Findings

Proves the isomorphism condition for all subgroups characterizes super-perfect groups with no proper finite index subgroups.

Establishes the non-existence of an algorithm to determine isomorphism of profinite completions in general.

Links group-theoretic properties to decidability and profinite completion behavior.

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.