On the vanishing cohomology problem for cocycle actions of groups on II_1 factors

We prove that any free cocycle action of a countable amenable group $\Gamma$ on II$_1$ factor $N$ can be perturbed by inner automorphisms to genuine action. This {\em vanishing cohomology} property, we call $\mathcal V\mathcal C$, is also closed products with amalgamation over finite groups. But beyond this no other examples C$-groups are known. In turn, considering special actions $\Gamma \curvearrowright N$ in the case hyperfinite $R$, respectively $N=L(\Bbb F_\infty)$, exclude many groups from being C$. show R$ gives rise $\Gamma$-action $R'\cap R^\omega$ whose cohomology equivalent Connes' Approximate Embedding property for $R\rtimes \Gamma$.