  We consider a new orbit equivalence invariant for measure-preserving actions
of groups on the probability space, $\sigma:G\to$ Aut$(X,\mu)$, denoted
$\chi_0(\sigma;G)$ and defined as the "intersection" of the 1-cohomology group,
H$^1(\sigma,G)$, with Connes' $\chi(M)$ invariant of the cross product von
Neumann algebra, $M=L^\infty(X,\mu)\rtimes_\sigma G$. We calculate
$\chi_0(\sigma;G)$ for certain actions of groups of the form $G=H\times K$ with
$H$ non-amenable and $K$ infinite amenable and we deduce that any such group
has uncountably many orbit inequivalent actions.
