On the profinite rigidity of free and surface groups

Let $S$ be a finitely generated free or surface group. We establish a Tits alternative for groups in the finite, soluble and $p$-genus of $S$. This generalises (and gives a new proof of) the analogous result of Baumslag for parafree groups. Then we study applications to profinite rigidity. A well-known question 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 finitely generated 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 free groups.Comment: 39 pages. There are small modifications in the statement of Theorem B, the proof of Theorem E and section 7. The introduction was also updated, correcting a citation of the article of Puder and Parzanchevsk