Gersten’s conjecture on hyperbolic one‑relator groups

Prove that every one‑relator group F/<<w>> containing no Baumslag–Solitar subgroup BS(1,n) (n ≠ 0) is hyperbolic.

Background

Gersten’s conjecture posits that the absence of solvable Baumslag–Solitar subgroups (the principal non‑hyperbolic obstruction) suffices for hyperbolicity in one‑relator groups. It ties subgroup structure to global negative curvature properties.

The authors survey partial progress via hierarchies and small‑cancellation, but the full conjecture remains unresolved.

References

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

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