Profinite detection of free products and free factors

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.