∑math20

Group theory · Result 246

Cannon's conjecture

1 paperLean

Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.

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1

The problem

What is being asked, where it comes from, and why it is hard. No proof needed.

▶Introduction to the problemin preparation
2

The solution proposed by OpenAI

How the paper solves it, step by step.

Full explanation
▶The proposed solutionin preparation

Papers 1

First page of A Modulus Proof of Cannon’s Conjecture

A Modulus Proof of Cannon’s Conjecture

OpenAI · September 23, 2026

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.

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Cite
@misc{OAI:A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A Modulus Proof of Cannon's Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf}{OAI:A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Cannon's conjecture asks whether a hyperbolic group with boundary homeomorphic to the two-sphere acts geometrically on hyperbolic three-space. The formalization proves that such a group admits an isometric action on \(\mathbb H^3\) 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 \(S^2\).

Comparator links

ResultComparator statement
Cannon's geometric-action conclusionCannonGeometricAction.lean

Scope note on GitHub →