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Gersten’s conjecture on hyperbolicity of one-relator groups

Prove that every one-relator group $G = F/{w}$ that contains no Baumslag–Solitar subgroup $\mathrm{BS}(1,n)$ (for any $n \neq 0$) is Gromov hyperbolic.

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Background

Hyperbolicity imposes strong geometric and algorithmic properties. The Baumslag–Solitar subgroups constitute obstructions to hyperbolicity. Gersten’s conjecture posits that these are the only obstructions within one-relator groups.

References

\begin{conjecture}[Gersten's conjecture] If $G = F/{w}$ is a one-relator group containing no Baumslag--Solitar subgroup $\bs(1, n)$ for any $n\neq 0$, then $G$ is hyperbolic. \end{conjecture}

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Subsection 6.1 (Hyperbolic one-relator groups)