Dice Question Streamline Icon: https://streamlinehq.com

Wise’s conjecture on virtually free‑by‑cyclic hyperbolic one‑relator groups

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

Information Square Streamline Icon: https://streamlinehq.com

Background

The conjecture predicts a strong hierarchical structure for hyperbolic one‑relator groups, implying deep algorithmic and geometric consequences (e.g., virtual specialness and residual finiteness).

The authors present supporting results for many cases (e.g., virtually RFRS), but a general proof remains open.

References

Wise made the following conjecture in : If $G$ is a hyperbolic one-relator group, then $G$ is virtually free-by-cyclic.

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Section 4.2 (Virtual algebraic fibring and generalisations)