Are all word-hyperbolic groups residually finite?
A group is residually finite if every nonidentity element survives in some finite quotient. Is every word-hyperbolic group residually finite? The question goes back to Gromov's theory of hyperbolic groups (1987) and is formulated, with its consequences, by Agol, Groves and Manning (2009, Section 5): a positive answer would make every quasiconvex subgroup of every hyperbolic group separable. Kapovich and Wise showed it is equivalent to every nontrivial hyperbolic group having a nontrivial finite quotient, and to every hyperbolic group being virtually torsion-free. Agol's theorem gives residual finiteness for cubulated hyperbolic groups, and non-residually-finite nonpositively curved examples (Wise, Burger-Mozes) were known only outside the hyperbolic class. Is every word-hyperbolic group residually finite?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Geometric group theory; hyperbolic groups
- Posed by
- Mikhail Gromov's program on hyperbolic groups; formulated by Ian Agol, Daniel Groves and Jason Fox Manning
- Year posed
- 1987
- Years open
- 39y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 66 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is a torsion-free word-hyperbolic group , the fundamental group of a finite Euclidean triangle complex, that is not residually finite; one member of a fixed finite family of nontrivial elements lies in the finite residual, without identifying which. Corollary 1.2: every finite-dimensional representation of over every commutative field kills the finite residual, so is linear over no field. With Kapovich-Wise this gives a nontrivial hyperbolic group with no nontrivial finite quotient and a hyperbolic group that is not virtually torsion-free (curator's reading of the paper's stated consequences). It says nothing about skew fields or infinite-dimensional representations, and does not settle whether all hyperbolic groups are sofic.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. 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). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the question as posed. The construction was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator TorsionFreeHyperbolic, declaration OAI.Release075.main). Its statement, read here: there exist a group that is torsion-free, word-hyperbolic (a finite set whose Cayley graph is connected and has uniformly -thin geodesic triangles) and not Group.ResiduallyFinite. That is the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The release's own Lean scope note (lean/docs/252.md) says nonlinearity over every field is not part of the formal statement; Corollary 1.2 is a short deduction via Malcev's theorem in the paper only.