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 $\mathrm{\Gamma }$ 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 $\Gamma \curvearrowright N$inthecase$N$isthehyperfiniteII$_1$factor$R$, respectively the free group factor $N=L({\mathbb{F}}_{\mathrm{\infty }})$, we exclude many groups from being $\mathcal V\mathcal C$\mathrm{.}Wealsoshowthatanyfreeaction$\Gamma \curvearrowright R$givesrisetoafreecocycle$\Gamma$-actionontheII$_1$factor$R’\cap R^\omega$ whose vanishing cohomology is equivalent to Connes’ Approximate Embedding property for the II${}_1$ factor $R⋊\mathrm{\Gamma }$.