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

Sorin Popa
University of California, Los Angeles (UCLA)

I will present a result showing 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. Besides containing all amenable groups, this {\it vanishing cohomology} property, called $\Cal V\Cal C$, is also closed to free products with amalgamation over finite groups. While no other examples of $\Cal V\Cal C$-groups are known, by considering special cocycle actions $\Gamma \curvearrowright N$ in the case $N=R$, $N=L(\Bbb F_\infty)$, one can be exclude many groups from being $\Cal V\Cal C$.
I will also explain a connection between the vanishing cohomology problem and Connes’ Approximate Embedding conjecture.

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