On the vanishing cohomology problem for cocycle actions of groups on II$_1$ factors

Abstract: We prove that any free cocycle action of a countable amenable group $\Gamma$ on any II$1$ factor $N$ can be perturbed by inner automorphisms to a genuine action. This {\em vanishing cohomology} property, that we call $\mathcal V\mathcal C$, is also closed to free products with amalgamation over finite groups. But beyond this no other examples of $\mathcal V\mathcal C$-groups are known. In turn, by considering special cocycle actions $\Gamma \curvearrowright N$ in the case $N$ is the hyperfinite II$_1$ factor $R$, respectively the free group factor $N=L(\Bbb F\infty)$, we exclude many groups from being $\mathcal V\mathcal C$. We also show that any free action $\Gamma \curvearrowright R$ gives rise to a free cocycle $\Gamma$-action on the II$_1$ factor $R'\cap R^\omega$ whose vanishing cohomology is equivalent to Connes' Approximate Embedding property for the II$_1$ factor $R\rtimes \Gamma$.