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Mel’nikov’s conjecture on groups whose finite-index subgroups are one-relator

Classify residually finite groups $G$ such that every finite-index subgroup of $G$ is a one-relator group; determine whether $G$ must be either free, a Baumslag–Solitar group $\mathrm{BS}(1,n)$, or a closed surface group.

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Background

Mel’nikov conjectured a strong rigidity property for residually finite groups whose finite-index subgroups are all one-relator. Some counterexamples prompted a refined formulation excluding trivial cases, but the complete classification remains open.

References

\begin{conjecture}[Mel'nikov] If $G$ is a residually finite group in which all finite index subgroups are one-relator, then $G$ is either free, isomorphic to $\bs(1, n)$ for some $n$ or the fundamental group of a closed surface. \end{conjecture}

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Subsection 7.5 (Mel'nikov’s conjecture, finite index subgroups and infinite index subgroups)