Cannon’s conjecture claimed (OpenAI, Lean); Bonk’s seminar talk

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Key facts

• Manuscript (abstract, verbatim): ‘We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon’s conjecture positively.’ 31 pages, dated 23 Sept 2026. Theorem 1.1 gives a homomorphism ρ : G → Isom(H³) with proper, cocompact action and finite kernel; ‘For torsion-free G, the quotient is a closed hyperbolic three-manifold, possibly nonorientable’
• Route (paper): Cannon’s combinatorial Riemann mapping theorem (1994), the Cannon–Swenson formulation (1998) and the Cannon–Floyd–Parry modulus criterion (1999) turned the problem into control of discrete moduli; the proof uses the Bourdon–Kleiner criterion that a uniform bound for the combinatorial 2-modulus of curves of a fixed small diameter on an approximately self-similar metric sphere yields a quasi-Möbius parametrization by the round sphere. Bonk and Kleiner had proved the conjecture when the boundary’s Ahlfors-regular conformal dimension is attained
• Lean (OpenAI scope note lean/docs/246.md): ‘The formalization proves that such a group admits an isometric action on H³ that is proper and cocompact and has finite kernel. Hyperbolicity is expressed through uniformly thin geodesic triangles in a Cayley graph, and the boundary hypothesis is a homeomorphism with S²’. Challenge CannonGeometricAction: ‘accepted’ in the oai-math-revised comparator run of 25–26 Oct (statement-to-paper fidelity not checked by that run)
• Expert talk: ‘Oct 22nd 2026, Quasiworld / Mario Bonk, The AI Proof of Cannon’s conjecture’ (YouTube, Quasiworld channel); the oai-math-revised hub lists it as a follow-up to this paper. No written summary of what Bonk concluded has been posted
• Reactions (Proofs and Prompts, 8 Oct): Harry Hyungryul Baik: ‘several of the problems or their variations were ones I had been working on right up until the day before, or were closely related to them. The Cannon conjecture is one example.’ Henry Wilton named ‘the Cannon conjecture’ first among problems he cares about and said ‘a cursory glance suggests that the write-ups vary wildly in quality. It will take months of work to go through the proofs’. Ian Agol, on the release’s geometric group theory and topology results in general: problems ‘that I might not have expected to be resolved in my lifetime (assuming they get checked to be correct)’