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Wise’s conjecture on virtual free-by-cyclicity

Prove that every hyperbolic one-relator group is virtually free-by-cyclic.

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Background

Free-by-cyclic groups possess rich geometric and dynamical structure, and virtual free-by-cyclicity would yield strong consequences for subgroup structure, separability, and algorithmic behavior in hyperbolic one-relator groups.

References

\begin{conjecture}[Wise] If $G$ is a hyperbolic one-relator group, then $G$ is virtually free-by-cyclic. \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)