Abstract
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 \textit{vanishing cohomology} property, called \( \mathcal{V}\mathcal{C} \), is also closed to free products with amalgamation over finite groups. While no other examples of \( \mathcal{V}\mathcal{C} \)-groups are known, by considering special cocycle actions \( \Gamma \curvearrowright N \) in the case \( N = R \), \( N = L(\mathbb{F}_\infty) \), one can exclude many groups from being \( \mathcal{V}\mathcal{C} \).
I will also explain a connection between the vanishing cohomology problem and Connes’ Approximate Embedding conjecture.
I will also explain a connection between the vanishing cohomology problem and Connes’ Approximate Embedding conjecture.