ON THE VANISHING COHOMOLOGY PROBLEM FOR COCYCLE ACTIONS OF GROUPS ON II1 FACTORS

We prove that any free cocycle action of a countable amenable group Gamma on any II1 factor N can be perturbed by inner automorphisms to a genuine action. Besides being satisTHORNed by all amenable groups, this universal vanishing cohomology property, that we call VC, is also closed to free products with amalgamation over finite groups.While no other examples of VC-groups are known, by considering special cocycle actions Gamma curved right arrow N in the case N is the hyperfinite II1 factor R, respectively the free group factor N = L(F-infinity), we exclude many groups from being VC. We also show that any free action Gamma curved right arrow R gives rise to a free cocycle _-action on the II1 factor R' boolean AND R-omega whose vanishing cohomology is equivalent to ConnesO Approximate Embedding property for the II1 factor R proportional to Gamma.