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 $N=L({\mathbb{F}}_{\infty })$, 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 $R^{\prime }\cap R^{\omega }$ whose vanishing cohomology is equivalent to Connes' Approximate Embedding property for the II${}_1$ factor $R⋊\Gamma$.

arXiv commentFinal version, to appear in Annales de l'Ec Norm. Sup