Free factors and profinite completions

TL;DR

This paper proves that free factors in virtually free groups can be detected through their profinite completions, extending previous results for free groups using new homological and decomposition methods.

Contribution

It establishes a criterion for identifying free factors in virtually free groups via profinite completions, generalizing earlier results for free groups.

Findings

Free factors correspond to profinite free factors in virtually free groups.

The characterization uses homological properties of profinite groups.

The approach differs from previous methods, combining homology and decomposition theory.

Abstract

Can one detect free products of groups via their profinite completions? We answer positively among virtually free groups. More precisely, we prove that a subgroup of a finitely generated virtually free group $G$ is a free factor if and only if its closure in the profinite completion of $G$ is a profinite free factor. This generalises results by Parzanchevski and Puder (later also proved by Wilton) for free groups. Our methods are entirely different to theirs, combining homological properties of profinite groups and the decomposition theory of Dicks and Dunwoody.