A Modulus Proof of Cannon’s Conjecture

Follow-up works

• Jizhou Guo, Consequences of results in the OpenAI mathematics release, §5 (GitHub research note, 2026-10-08). The note assumes the manuscript's Cannon theorem and combines it with earlier literature to derive three results: the hyperbolic case of Wall's question, the torsion-free Kapovich–Kleiner conjecture, and the toral relative Cannon conjecture. [AI-generated per source]

Comments and discussion

• Harry Hyungryul Baik, reaction in 100+ reactions to 100+ solutions (Proofs and Prompts, 2026-10-08; updated 2026-10-19). Baik writes that "The Cannon conjecture is one example" of release problems he had worked on, and he discusses it in relation to his research on laminar groups. He does not assess the proof.

Formalization

• Lean scope notes for family 246, lean/docs/246.md (OpenAI math release Lean documentation, 2026-10-06; community mirror). The formalized statement is that a group with thin Cayley-graph triangles and boundary homeomorphic to $S^2$ admits a proper, cocompact isometric action on $\mathbb H^3$ with finite kernel. The notes link to the comparator statement.
• OpenAI, CannonGeometricAction.lean (openai/math, comparator challenge file). This file contains the target statement cannon, which defines the boundary through based geodesic rays and uses the full isometry group of upper half-space, including orientation-reversing isometries. The proof body in this file is sorry.

Related prior literature mentioned by others

• Michael Kapovich and Bruce Kleiner, Hyperbolic groups with low-dimensional boundary (Annales scientifiques de l'École Normale Supérieure 33, 2000). Guo's §5 uses Theorem 5 and Corollaries 12–13 to go conditionally from the manuscript's sphere-boundary result to torsion-free groups with Sierpiński-carpet boundary.
• Daniel Groves, Jason Fox Manning and Alessandro Sisto, Boundaries of Dehn fillings (arXiv:1612.03497; Geometry & Topology 23, 2019). Guo's §5 uses Corollary 1.4, which deduces the toral relative Cannon conjecture from Cannon's conjecture.

Possibly relevant literature (found by the agent, not checked)

• M. Bonk and B. Kleiner, Rigidity for quasi-Möbius group actions (arXiv:math/0006137, J. Differential Geom. 61 (2002)). This paper shows that a hyperbolic group whose boundary is quasisymmetric to an Ahlfors regular space of matching topological dimension is a uniform lattice up to a finite normal subgroup. This relates to the paper's step from boundary uniformization to a geometric action (prop:action).
• T. Cheetham-West and A. Nolte, Characterizing candidates for Cannon's Conjecture from Geometric Measure Theory (arXiv:2207.01720, Bull. London Math. Soc. 2023). This paper characterizes, using a spherical Plateau problem, when a torsion-free hyperbolic group with $S^2$ boundary is a closed hyperbolic 3-manifold group. That is the conclusion of the paper's corollary on torsion-free groups.
• A. Pal and R. Sardar, On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension (arXiv:2608.28346, 28 August 2026). This paper extends the Bonk–Kleiner conformal-dimension criterion to relatively hyperbolic groups with 2-sphere Bowditch boundary. The paper's introduction discusses that criterion as a related route.

Digests and expositions

• OpenAI Math Digest, A Modulus Proof of Cannon's Conjecture (unofficial reading guide, accessed 2026-10-21). This illustrated guide follows the scalar-limit construction, the exponential and subexponential contradictions, and the passage from boundary uniformization to a geometric action. It states that it has not independently verified the whole proof. [AI-generated per source]

News and blogs

• Azmat, Cannon Conjecture: OpenAI's Proof That Could Rewrite Our Understanding of 3D Geometry (BinaryVerse AI, 2026-10-11). The article outlines the manuscript's modulus argument and its Lean claim. Its author says a completed independent verification has not been established.