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

TL;DR

This paper proves that free cocycle actions of amenable groups on II$_1$ factors can be perturbed to genuine actions, introduces the $ ext{VC}$ property, and explores its implications for group actions and von Neumann algebra properties.

Contribution

It establishes the $ ext{VC}$ property for amenable groups acting on II$_1$ factors and investigates its limitations and connections to Connes' Approximate Embedding property.

Findings

Any free cocycle action of an amenable group on a II$_1$ factor can be perturbed to a genuine action.

The $ ext{VC}$ property is preserved under free products with amalgamation over finite groups.

Many groups are excluded from being $ ext{VC}$-groups based on specific cocycle actions.

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 $R\rtimes \Gamma$.