Classification of Amenable Actions of Discrete Groups on II1 Factors and Orbit Equivalence Rigidity

Mathematical Applications of Science Fiction We present a comprehensive classification of amenable actions of non-amenable discrete groups on hyperfinite type II1 factors, establishing a new superrigidity class for generalized Bernoulli shifts. Using Popa’s deformation and rigidity theory, particularly the interaction between s-malleable deformations and spectral gap rigidity, we show that for a broad family of property (T) groups, every free ergodic measure-preserving action that is orbit equivalent to the Bernoulli shift is actually conjugate to it. We further analyze the cohomology of the associated Cartan subalgebras, deriving an obstruction to the existence of amenable quotients in the crossed product von Neumann algebra. This work resolves a long-standing conjecture concerning the stability of L2-Betti numbers under orbit equivalence for soft-core actions.