Some computations of 1-cohomology groups and construction of non orbit equivalent actions

Abstract

For each group $G$ having an infinite normal subgroup with the relative property (T) (for instance $G = H \times K$ where $H$ is infinite with property (T) and $K$ is arbitrary), and any countable abelian group $\Lambda$ we construct free ergodic measure preserving actions $\sigma_\Lambda$ of $G$ on the probability space such that the 1'st cohomology group of $\sigma_\Lambda$, $H^1(\sigma_\Lambda)$, is equal to Char$(G) \times \Lambda$. We deduce that $G$ has uncountably many non stably orbit equivalent actions. We also calculate 1-cohomology groups and show existence of ``many'' non stably orbit equivalent actions for free products of groups as above.