Result 246, Group theory

Cannon's conjecture

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.

Lean formalization Proof

The bigger picture

Why it matters

Can the shape seen at infinity determine the geometry behind an abstract group? The claimed resolution of Cannon's conjecture says that a spherical boundary forces a precise connection to three-dimensional hyperbolic geometry.

What changes?

The unreviewed manuscript reports that every word-hyperbolic group whose boundary is topologically a two-dimensional sphere admits a proper, cocompact isometric action on hyperbolic three-space, with finite kernel. Word-hyperbolic means the group's geometry has uniformly thin triangles; its boundary records directions to infinity. The action preserves distances. Properness makes the action discrete with finite point stabilizers; cocompactness means the quotient is compact. A finite kernel means only finitely many group elements act trivially.

What does that help mathematicians do?

For torsion-free groups, meaning those with no nonidentity elements of finite order, the reported consequence is especially concrete: each is the fundamental group of a closed hyperbolic three-manifold, a compact space without boundary. The fundamental group records how loops combine. Researchers could therefore deduce the existence of such a geometric space from the group's hyperbolicity and spherical boundary alone, rather than constructing it separately.

Are there practical applications?

The immediate value is foundational: the claim connects algebraic descriptions of groups, their geometry at infinity, and hyperbolic three-manifolds. It would let researchers move between these descriptions for precisely the groups covered by the theorem. The supplied material describes a structural existence result, not an algorithm for constructing the action or a practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Modulus Proof of Cannon’s Conjecture

September 23, 2026 31 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/246.md.

Cannon's conjecture

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 H3\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 S2S^2.

Comparator links

Result Comparator statement
Cannon's geometric-action conclusion CannonGeometricAction.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.