Abstract

If one is interested in a finitely generated residually finite group, it is natural to ask to what extent it is determined by its profinite completion. The extent to which this is a hard question in general is shown by the fact that it is not yet known whether a finitely generated free group is determined by its profinite completion (a question attributed to Remeslennikov). A variant of Remeslennikov’s question is whether a free factor of a group G can be detected from its profinite completion: if G has a subgroup H, whose closure in the profinite completion of G is a profinite free factor, must H be a free factor of G? This question is still hard, with positive known answer only when G itself is a free group.

I will report on joint work with A. Jaikin in which we provide a new proof of the above positive answer and extend it to the case when G is virtually free. The methods might be extended to other classes of groups if some interesting questions are answered on their completed group algebras.