Profinite properties of residually free groups

TL;DR

This paper investigates the profinite properties of residually free groups, establishing their rigidity, coherence, and subgroup separability, and generalizing key lemmas on group extensions.

Contribution

It proves profinite rigidity of certain residually free groups and extends lemmas on recognizing group splittings via profinite completions.

Findings

Groups of the form π₁M × Z^n are profinitely rigid within finitely generated residually free groups.

Residually free groups exhibit coherence and subgroup separability.

A generalized lemma on recognizing when a central extension splits profinitely.

Abstract

Henry Wilton classified when a prime three-manifold $M$ has a residually free fundamental group $\pi_1 M$. We prove that the groups $\pi_1 M\times \mathbb Z^n$ are profinitely rigid within finitely generated residually free groups. We also establish other profinite invariants of the class of residually free groups such as coherence and subgroup separability. In the course of our proofs, we generalise a lemma of Wilton and Zalesskii on profinitely recognising when a central extension of groups splits.