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

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 $\mathrm{\Lambda }$ we construct free ergodic measure preserving actions ${\sigma }_{\mathrm{\Lambda }}$ of $G$ on the probability space such that the 1’st cohomology group of ${\sigma }_{\mathrm{\Lambda }}$, $H^1({\sigma }_{\mathrm{\Lambda }})$, is equal to Char$(G)\times \mathrm{\Lambda }$. 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.