Connes' rigidity conjecture for lattices in higher-rank and other property (T) local-field groups (W*-superrigidity of lattice group factors)
Connes asked whether nonisomorphic ICC groups with Kazhdan's property (T) have nonisomorphic group von Neumann algebras, and what the fundamental groups of these factors are (Problems 1-2). Popa (ICM 2006) displayed the stronger stable form: an isomorphism , with any ICC group, should force and come from a group isomorphism up to a character and inner conjugacy. The motivating examples are lattices such as ; for general property (T) groups the conjecture was refuted in 2026 by wreath-like counterexamples. For ICC groups commensurable with lattices in products of higher-rank or rank-one property (T) groups over local fields, is W*-superrigid in Popa's stable sense?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Von Neumann algebras; rigidity of lattices
- Posed by
- Alain Connes, Noncommutative Geometry (1994), Ch. V, Appendix B.epsilon, Problems 1-2 (rigidity question from about 1980); stable form by Sorin Popa (ICM 2006)
- Year posed
- 1980
- Years open
- 46y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 44 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Principal (Theorem 1.2): for in the class (ICC groups commensurable with lattices in finite products of noncompact property (T) groups over , or -adic fields), any ICC and arbitrary scalar 2-cocycles, every bifinite - correspondence is a summand of finite sums of models built from finite-index subgroup isomorphisms and finite-dimensional projective representations. Companion: for and ICC iff and the groups are isomorphic with cohomologous cocycles, every such isomorphism being implemented canonically; these factors have trivial fundamental group (e.g. ). Not shown: lattices in rank-one groups without property (T), or general property (T) groups.
What the AI did
The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 23, 2026: the geometric arithmetic exhaustion theorem for finite correspondences (principal) and an algebraic companion that derives stable canonical recovery, trivial fundamental groups and all finite-index factor neighbors from it.
Verification
No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against Connes' problems and Popa's formulation as the manuscripts cite them; the cited texts themselves were not read. The headline theorem rests on a long geometric argument (boundary measurements, local-field branches, locality, gluing) that was not refereed. There is no Lean formalization for this family (lean/docs/286.md does not exist at the pinned commit). Resolution is partial because Connes' question concerns all ICC property (T) groups, where the answer is negative, and this proves it for the lattice class only. Given the claim's weight in operator algebras, it warrants expert review before anything else.