A rigidity result for actions of property T groups by Bernoulli shifts

Sorin Popa

Let $\sigma$ be the action of an infinite property T group $G$ on the hyperfinite type II$_1$ factor $R=\ {\overline{\underset g \in G \to \otimes}} (M_{2\times 2}(\Bbb C), tr)_g$, by Bernoulli shifts. We prove that the cocycle actions obtained by reducing $\sigma$ to the algebras $pRp$, for $p$ non-trivial projections in $R$, cannot be perturbed to actions. We also prove that any $1$-cocycle for $\sigma$ vanishes. More generally, we calculate all 1-cocycles for actions of property T groups $G$ by Bernoulli shifts of Connes-St\o rmer type and use this to provide an invariant that distinguishes these actions up to cocycle conjugacy.