  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 $H\leq G$
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 $\widehat G\cong
\widehat S$ is necessarily $G\cong S$. We confirm this when $G$ belongs to a
class of groups $\mathcal{H}_{ab}$ 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 $S\times {\mathbb Z}^n$ is profinitely rigid within finitely
generated residually free groups.
