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- W4309876881 abstract "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 $Hleq G$ 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 $widehat Gcong widehat S$ is necessarily $Gcong S$. 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 $Stimes {mathbb Z}^n$ is profinitely rigid within finitely generated residually free groups." @default.
- W4309876881 created "2022-11-29" @default.
- W4309876881 creator A5081570190 @default.
- W4309876881 date "2022-11-22" @default.
- W4309876881 modified "2023-09-27" @default.
- W4309876881 title "On the profinite rigidity of free and surface groups" @default.
- W4309876881 doi "https://doi.org/10.48550/arxiv.2211.12390" @default.
- W4309876881 hasPublicationYear "2022" @default.
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