Coboundaries and measure-preserving actions of nilpotent and solvable groups

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) subset of or equal to Cob(tau) implies that sigma = tau(n) for some n epsilon 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) subset of or equal to Cob(tau) is given. In this case sigma is a generalized power of tau and is isomorphic to some tau(n), n epsilon 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.