On properties of jointly ergodic action of the direct product of two groups (Q1177770)

The study of co-cycles of an ergodic automorphism with the values in a locally compact group \({\mathcal G}\) leads to the joint ergodic actions of the group \({\mathcal G}\times\mathbb{R}\). The authors build the ergodic action \(\alpha\) of the direct product \(\mathbb{Z}\times{\mathcal G}\) with \({\mathcal G}=\bigoplus^{\infty}_{n=1}\mathbb{Z}_ 2\) which is not isomorphic to the product of the non-ergodic actions of \(\mathbb{Z}\) and \({\mathcal G}\), separately. They prove the theorem that the actions of \(\mathbb{Z}\) on its ergodic components are isomorphic by metric if and only if they transform to each other by the action of \({\mathcal G}\). Then the centralizer \({\mathcal C}_{\alpha}(\mathbb{Z}\times{\mathcal G})\) of the action \(\alpha\) is such that \({\mathcal C}_{\alpha}(\mathbb{Z}\times{\mathcal G})/\alpha(\mathbb{Z}\times{\mathcal G})\thickapprox\mathbb{Z}_ 2\).