Connes rigidity for property-(T) group factors
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Does the tracial group von Neumann factor determine a finitely generated ICC property-(T) group up to isomorphism? Connes’s rigidity question also has a weaker finite-to-one form, asking whether one factor can arise from only finitely many such groups.
Why it matters and what counts as progress
Why it matters: An infinite family sharing one factor would refute both group reconstruction and the weaker finite-to-one classification principle.
What counts as progress: Construct nonisomorphic groups with the same factor, build infinite such families or prove rigidity for further group classes.
How the problem developed
- 1980 · Conjecture [1]
Alain Connes Proved countability of the outer automorphism and fundamental groups of ICC property-(T) group factors, motivating the expectation that such factors remember their groups.
Shuoxing Zhou and OpenAI Independently reported nonisomorphic ICC property-(T) groups with isomorphic group von Neumann algebras, refuting Connes’ original conjecture in its full generality.
AI contributions 2
ICC property (T) groups without W*-superrigidity
Zhou constructs two nonisomorphic countable ICC property-(T) groups whose group von Neumann algebras are isomorphic. A quadratic fiber shear identifies the dual actions, while the groups have distinct module structures.
Ten Advances in Mathematics and Theoretical Computer Science
OpenAI constructs infinitely many pairwise nonisomorphic finitely generated ICC property-(T) groups whose tracial group von Neumann factors are mutually isomorphic. The family refutes Connes’s rigidity conjecture and a finite-to-one strengthening.
Sources
- Zhou, ICC Property (T) Groups without W*-Superrigidity
- OpenAI, Ten advances in mathematics and theoretical computer science
- Zhou, ICC Property (T) Groups Without W*-Superrigidity
- OpenAI, Ten advances: research account
- OpenAI ten-proofs formalization manifest
- Lean proof of the Connes rigidity theorem
- Sienicki et al., chapter-by-chapter mathematical review
- Zhou, separate Connes-rigidity counterexample
Website timeline
Publication dates and editorial changes.
OpenAI reports an infinite-family counterexample to Connes rigidity
AI-assisted construction gives a distinct Connes rigidity counterexample