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Gersten's conjecture: are Baumslag-Solitar-free one-relator groups hyperbolic?

The Baumslag-Solitar groups BS(m,n)=⟨a,t∣tamt−1=an⟩\mathrm{BS}(m,n)=\langle a,t\mid ta^mt^{-1}=a^n\rangle, m,n≠0m,n\ne0, are the basic obstructions to word-hyperbolicity among one-relator groups: a hyperbolic group contains none of them. Gersten asked whether they are the only subgroup obstructions. Known cases included proper-power relators (Newman's spelling theorem), relators of primitivity rank other than two (Louder-Wilton, Linton) and ascending HNN extensions of free groups (Mutanguha). Let G=⟨X∣r⟩G=\langle X\mid r\rangle with XX finite. If GG contains no subgroup isomorphic to BS(m,n)\mathrm{BS}(m,n) for any nonzero m,nm,n, must GG be word-hyperbolic?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Geometric group theory; one-relator groups
Posed by
S. M. Gersten, 'Problems on automatic groups' (MSRI Publications 23, p. 228)
Year posed
1992
Years open
34y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: if a one-relator group ⟨X∣r⟩\langle X\mid r\rangle, XX finite, contains no BS(m,n)\mathrm{BS}(m,n) with m,n≠0m,n\ne0, it is word-hyperbolic. The technical core is a combination theorem for a class of HNN extensions of torsion-free hyperbolic groups with free, possibly proper edge groups, applied along the Magnus-Moldavanskii hierarchy. Combined with the companion and with Kielak-Linton, Corollary: every Baumslag-Solitar-free one-relator group is virtually free-by-cyclic, with a finite-index embedding in a free-by-Z\mathbb Z group with finitely generated kernel. The paper does not treat infinitely generated or multi-relator groups. Unrelated to the K-theoretic Gersten conjecture addressed in family 209 of the same release.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The companion on virtual compact specialness (same date) uses this paper's hyperbolicity and Magnus-subgroup results at two points; this paper in turn quotes the companion only for the free-by-cyclic corollary.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Gersten's question as the paper quotes it from Gersten (1992, p. 228); it is the conjecture in full, for every finite generating set and relator, including trivial relators and proper powers, with no torsion-freeness hypothesis. The paper notes that excluding all BS(1,n)\mathrm{BS}(1,n) is equivalent. The proof rests on Linton's one-relator hierarchy and quasiconvexity theorems and a new combination theorem for HNN extensions; it was not refereed here. There is no Lean formalization for this family (no lean/docs/258.md at the pinned commit).

Sources

Changelog1 change

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