  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.
