On the Cohomology of Actions of Groups by Bernoulli Shifts

We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to \Pi} (X_0, \mu_0)_g$ for $(X_{0},\mu_{0})$ an arbitrary probability space, has first cohomology group isomorphic to the character group of $G$.