CATALOGUE 246
Cannon's conjecture
This catalogue concerns realizing a hyperbolic group with sphere boundary by isometries of hyperbolic three-space. Its official subject is Group theory. The single manuscript claims an unconditional solution through bounded combinatorial modulus on the boundary. The exposition separates the author’s claim from the editor’s checking scope.
1. Paper in this catalogue
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | A Modulus Proof of Cannon’s Conjecture →Short name: Modulus proof of Cannon’s conjectureSeptember 23, 2026 · 31 pagesOriginal paper (GitHub) ↗ | Claims a proper cocompact isometric action on hyperbolic three-space with finite kernel for a hyperbolic group with sphere boundary; in the torsion-free case, the group is a closed hyperbolic three-manifold group. Theorem 1.1 · p. 2; Corollary 8.1 · p. 29 ↗ | Extracts a function and measure from unbounded 2-modulus, excludes both growth regimes, then constructs the action by uniformization and an invariant conformal structure. Read the proof overview → |
2. Enter the argument
The branching proof overview shows where two contradictions to unbounded modulus join at uniformization. The construction of the action on hyperbolic space distinguishes topological homeomorphism, quasi-Möbius equivalence, and equivariant conjugacy, continuing to the finite-kernel conclusion for the original group.
Direct inputs from uniformization, quasiconformal mapping theory, and hyperbolic groups are collected with their checking scope in the article’s source-and-application table. Some cross-catalogue relations remain uninvestigated; this is not a declaration of no dependencies.
3. Sources and checking scope
The fixed official catalogue lists entry 246 as “Cannon’s conjecture”. Group theory is the eleventh subject in official display order, covering entries 246–259; entry 246 is on page 27 of overview.pdf. There is one manuscript in official order, dated September 23, 2026, with 31 pages including references.
OpenAI Research Catalog · Catalogue 246; Group theory · p. 27
The article self-checks the main theorem, the proof connections of the principal intermediate propositions and direct inputs, and the torsion-free manifold consequence. It is not an independent verification of the entire proof. The parallel HK95 route, virtual-structure inputs to Corollary 8.2, and systematic cross-catalogue comparisons are outside this check. Details appear in the article’s return-to-sources guide.