A relative version of Connes' $\chi (M)$ invariant and existence of orbit inequivalent actions

Abstract

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, $M=L^{\infty }(X,\mu ){⋊}_{\sigma }G$. 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.

final version