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Remeslennikov’s conjecture (profinite rigidity of free groups)

Prove that if a residually finite group has the same set of finite quotients as a free group of rank at least two, then it is isomorphic to that free group.

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Background

Remeslennikov’s conjecture is a central open problem in profinite group theory and rigidity. The authors note recent progress showing such a group must be parafree, but full rigidity (freedom) remains open.

Its resolution would impact classification and profinite completions across finitely presented groups, including one‑relator groups.

References

A major open problem in group theory is Remeslennikov's conjecture: if $G$ is a residually finite group with the same finite quotients as a free group, is $G$ necessarily free?

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Section 7.2 (Residually nilpotent and parafree one‑relator groups)