Are hyperbolic one-relator groups virtually compact special?
A group is virtually compact special if a finite-index subgroup is the fundamental group of a compact nonpositively curved cube complex admitting a local isometry to the Salvetti complex of a right-angled Artin group; for hyperbolic groups this gives residual finiteness, linearity and separability of quasiconvex subgroups. Wise proved it for one-relator groups with torsion (answering Baumslag's 1967 residual-finiteness question for that class) and asked, in his question on cocompact cubulation of hyperbolic groups with hierarchies, about hyperbolic one-relator groups in general. Linton settled the case where all two-generator subgroups are free. Is every word-hyperbolic one-relator group virtually compact special?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric group theory; special cube complexes
- Posed by
- Daniel T. Wise, 'The cubical route to understanding groups', ICM 2014 (Problem 13.15)
- Year posed
- 2014
- Years open
- 12y
- 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
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: every word-hyperbolic one-relator group is virtually compact special. Theorem 1.2: every hyperbolic ascending mapping torus of an injective endomorphism of a finitely generated free group is virtually compact special. With Kielak-Linton, every hyperbolic one-relator group is virtually free-by-cyclic (kernel possibly of infinite rank in the subgroup-closed sense, with a finite-index embedding in a free-by- group with finitely generated kernel), which is Wise's Conjecture 17.8 for this class, and every hyperbolic one-relator group is residually finite and linear over . It says nothing about non-hyperbolic one-relator groups. Consistent with the release's non-residually-finite hyperbolic group (catalog entry on residual finiteness), which is not one-relator.
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 proof imports the companion hyperbolicity paper's results at two stated points (hyperbolicity of embedded primitive extension groups and the geometry of their Magnus rows); no specialness conclusion is imported from it.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the question; it asserts virtual compact specialness, with an explicit finite cube complex and local isometry to a Salvetti complex, for every hyperbolic group with finite. The argument uses Linton's reduction to primitive extension groups and a new selective-filling criterion; it was not refereed here. The entry depends on the companion hyperbolicity paper at two points, so an error there could affect it. There is no Lean formalization for this family.