Coboundaries and measure-preserving actions of nilpotent and solvable groups

Abstract

Let $\sigma$ and $\tau$ be two measure-preserving transformations of a non-atomic probability space, and Cob$(\sigma)$, Cob$(\tau)$ be the sets of their measurable coboundaries. We show that if the group G generated by $\sigma$ and $\tau$ is nilpotent and acts ergodically, then the inclusion Cob$(\sigma)\subseteq\text{Cob}(\tau)$ implies that $\sigma=\tau^n$ for some $n\in\mathbb Z$. This fact cannot be extended to solvable G. For G virtually solvable, a detailed description of the relationship between $\sigma$ and $\tau$ satisfying the inclusion Cob$(\sigma)\subseteq\text{Cob}(\tau)$ is given. In this case $\sigma$ is a generalized power of $\tau$ and is isomorphic to some $\tau^n,\ n\in\mathbb Z$.

The proofs require some study of non-free measure-preserving actions of elementary amenable groups and their stabilizers. In particular, a version of the Rokhlin lemma for non-free measure-preserving actions admitting maximal stabilizers is given.