Profinite detection of free products and free factors

TL;DR

This paper investigates how the algebraic structure of certain groups, especially free products and free factors, can be detected from their profinite completions, with implications for profinite rigidity.

Contribution

It establishes conditions under which free product decompositions and free factors can be identified from the profinite completion of the group.

Findings

Cohomological goodness of G when residually finite.

Equivalence of free product splitting between G and its profinite completion for LERF groups.

Detection of one-ended free factors from the profinite completion.

Abstract

Let $G$ be the fundamental group of a graph of finitely generated virtually free groups with virtually cyclic edge groups. We shaw that $G$ is cohomologically good if $G$ is residually finite. If $G$ is LERF, we prove that G splits non-trivially as a free product if and only if its profinite completion $\widehat{G}$ splits non-trivially as a free profinite product. Moreover, we are able to detect one-ended free factors of $G$ from $\widehat{G}$. As an application, we deduce that any profinitely rigid word in a finitely generated free group is universally profinitely rigid.