Cannon's Conjecture
Let be a word-hyperbolic group whose Gromov boundary is homeomorphic to the 2-sphere. Cannon's conjecture, stated by Cannon and Swenson (1998, Conjecture 5.1), asserts that acts properly and cocompactly by isometries on hyperbolic 3-space ; for torsion-free this would make the fundamental group of a closed hyperbolic 3-manifold. Cannon-Floyd-Parry reduced it to bounds on discrete annular moduli, and Bonk-Kleiner proved it when the Ahlfors regular conformal dimension of the boundary is attained. Does every hyperbolic group with boundary act geometrically on ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Geometric group theory, hyperbolic groups
- Posed by
- James W. Cannon; stated in J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension 3, Trans. Amer. Math. Soc. 350 (1998), Conjecture 5.1
- Year posed
- 1998
- Years open
- 28y
- 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
- 62 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every hyperbolic group with there is whose action is proper and cocompact with finite kernel; the image may contain orientation-reversing isometries. For torsion-free the quotient is a closed, possibly nonorientable, hyperbolic 3-manifold, and the paper records virtual-structure consequences. The route is the Bonk-Kleiner uniformization program: the paper proves a uniform upper bound on the combinatorial 2-modulus of curve families at a fixed small scale on the visual boundary, then applies the Bourdon-Kleiner criterion (Corollary 3.5) to get a quasi-Mobius parametrization by the round sphere.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against Cannon-Swenson Conjecture 5.1 as cited. The modulus argument was not refereed. The theorem allows a finite kernel and orientation-reversing isometries, which is the standard form of the conjecture. Lean-checked on the release's own Comparator challenge CannonGeometricAction together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here.