On the profinite rigidity of free and surface groups (Q6624770)

Let \(S\) be either a free group or the fundamental group of a closed hyperbolic surface. One problem often studied is to determine the profinite rigidity of \(S\), i.e., whether given a finitely generated residually finite group \(G\) with the same profinite completion \(\widehat{G} \simeq \widehat{S}\), we necessarily have \(G \simeq S\).\N\NIn the paper under review, the author provides a partial answer to that question by proving Theorem A: Let \(p\) be a prime and let \(S\) be either a free group or hyperbolic surface group. Suppose that \(G\) is a finitely generated residually-\(p\) group with the same pro-\(p\) completion as \(S\). Then two-generated subgroups of \(G\) are free.\N\NThe author points out that \textit{A. Jaikin-Zapirain} [Res. Math. Sci. 10, No. 4, Paper No. 44, 24 p. (2023; Zbl 1523.20050)] proved that a finitely generated residually finite group \(G\) such that \(\widehat{G} \simeq \widehat{S}\) (with \(S\) as in Theorem A) is residually-\(p\) for every prime \(p\).\N\NFurthermore, in Theorem D, the author proves that if \(n \geq 1\), \(G\) is a finitely generated residually free group, \(S\) is a free or surface group and \(\widehat{G} \simeq \widehat{S} \times \widehat{\mathbb{Z}}^{n}\), then \(G \simeq S \times \mathbb{Z}^{n}\).