Shattering Operator Algebras: Explicit Non-Sofic Groups and the Refutation of Connes’s Rigidity Conjecture

In abstract algebra and functional analysis, few conjectures have cast a longer shadow than those governing group approximations and operator algebras. The third and fourth manuscripts in OpenAI's collection claim to resolve two legendary open problems: the existence of an explicit non-sofic group and the invalidation of Fields Medalist Alain Connes's Rigidity Conjecture.

1. Constructing a Non-Sofic Group

Sofic groups—introduced by Mikhail Gromov in 1999—are groups whose multiplication tables can be asymptotically approximated by finite permutation matrices. The paper constructs an explicit non-sofic group using the unit group of the binary Leavitt algebra:

• Isolates a finitely generated property-(T) subgroup.
• Deploys expander decomposition theorems on candidate permutation actions.
• Leverages self-similarity to force an embedding of Thompson's group (V), yielding a direct structural contradiction.

2. The Disproof of Connes's Rigidity Conjecture

Connes conjectured that an infinite-conjugacy-class (ICC) group with Kazhdan's property (T) is uniquely determined up to isomorphism by its group von Neumann algebra (L(G)).

The paper refutes this by constructing an infinite family of pairwise nonisomorphic property-(T) groups sharing the exact same von Neumann algebra: