Math Conjectures

Connes Rigidity Conjecture for Property-(T)(T) Groups

OPENMajorConjectureProposed c. 1982 · Standard version

Canonical statement

If GG and HH are countable ICC groups with Kazhdan's property (T)(T), then every isomorphism of their group von Neumann algebras,
L(G)≅L(H), L(G)\cong L(H),
implies that G≅HG\cong H. Here ICC means that every nonidentity conjugacy class is infinite.
View source LaTeX
If \(G\) and \(H\) are countable ICC groups with Kazhdan's property \((T)\), then every isomorphism of their group von Neumann algebras,
\[
  L(G)\cong L(H),
\]
implies that \(G\cong H\). Here ICC means that every nonidentity conjugacy class is infinite.
ANAL-FA-019:[Connes1982Classification][Popa2007DeformationRigidity][OpenAI2026TenAdvances]
Analysis and functional analysisAlgebra and representation theory

Notes

Connes's rigidity program asks how much of a property-(T)(T) group can be recovered from its group von Neumann algebra [Connes1982Classification]. Popa's deformation-rigidity theory proves powerful positive results for structured families [Popa2007DeformationRigidity], but not the universal statement. The August 1, 2026 bundled manuscript claims infinitely many counterexamples [OpenAI2026TenAdvances]; pending independent operator-algebra verification, the conjecture remains in the stable catalog with a linked Grade C claim.

Proof-claim watch (1)

Counterexamples to Connes rigidity

Grade C

Full-scope refutation claim awaiting independent verification.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.