A relative version of Connes' $χ(M)$ 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, $σ:G\to$ Aut$(X,μ)$, denoted $χ_0(σ;G)$ and defined as the "intersection" of the 1-cohomology group, H$^1(σ,G)$, with Connes' $χ(M)$ invariant of the cross product von Neumann algebra, $M=L^\infty(X,μ)\rtimes_σG$. We calculate $χ_0(σ;G)$ 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.