2022/11/22 by Morales, Ismael
#20E06 #20E18 #20E26 (Primary) 11R34 #20F18 #20F65 #20J06 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2211.12390
Let S be either a free group or the fundamental group of a closed hyperbolic surface. We show that if G is a finitely generated residually-p group with the same pro-p completion as S, then two-generated subgroups of G 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 G is a residually-(torsion-free nilpotent) group and H≤ G is a virtually polycyclic subgroup, then H is nilpotent and the pro-p topology of G induces on H its full pro-p topology. Then we study applications to profinite rigidity. Remeslennikov conjectured that a finitely generated residually finite G with profinite completion \widehat G≅ \widehat S is necessarily G≅ S. We confirm this when G belongs to a class of groups Hab 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 S× \mathbb Zn is profinitely rigid within finitely generated residually free groups.