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 $G\to \mathrm{\Pi }(X_0,{\mu }_0{)}_g$ for $(X_0,{\mu }_0)$ an arbitrary probability space, has first cohomology group isomorphic to the character group of $G$.