On the profinite rigidity of free and surface groups.

Let be either a free group or the fundamental group of a closed hyperbolic surface. We show that if is a finitely generated residually- group with the same pro- completion as , then two-generated subgroups of are free. This generalises (and gives a new proof of) the analogous result of Baumslag for parafree groups. Our argument relies on the following new ingredient: if is a residually-(torsion-free nilpotent) group and is a virtually polycyclic subgroup, then is nilpotent and the pro- topology of induces on its full pro- topology. Then we study applications to profinite rigidity. Remeslennikov conjectured that a finitely generated residually finite with profinite completion is necessarily . We confirm this when belongs to a class of groups that has a finite abelian hierarchy starting with finitely generated residually free groups. This strengthens a previous result of Wilton that relies on the hyperbolicity assumption. Lastly, we prove that the group is profinitely rigid within finitely generated residually free groups.