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On the profinite rigidity of free and surface groups

Nov 2022

Let $S$ be a finitely generated free or surface group. We establish a Titsalternative for groups in the finite, soluble and $p$-genus of $S$. Thisgeneralises (and gives a new proof of) the analogous result of Baumslag forparafree groups. Then we study applications to profinite rigidity. A well-knownquestion of Remeslennikov asks whether a finitely generated residually finite$G$ with profinite completion $\widehat G\cong \widehat S$ is necessarily$G\cong S$. We give a positive answer when $G$ belongs to a class of groups$\mathcal{H}_{ab}$ that has a finite abelian hierarchy starting with finitelygenerated residually free groups. This strengthens previous results of Bridson,Corner, Reid and Wilton. Then we show that one-relator groups in the genus of$S$ are hyperbolic and virtually special. Lastly, we prove that $S\times{\mathbb Z}^n$ is profinitely rigid within finitely generated residually freegroups.

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