Rank-one non-singular actions of countable groups and their odometer factors

Abstract For an arbitrary countable discrete infinite group G , non-singular rank-one actions are introduced. It is shown that the class of non-singular rank-one actions coincides with the class of non-singular $(C,F)$ -actions. Given a decreasing sequence of cofinite subgroups in G with $\bigcap _{n=1}^\infty \bigcap _{g\in G}g\Gamma _ng^{-1}=\{1_G\}$ , the projective limit of the homogeneous G -spaces $G/\Gamma _n$ as $n\to \infty $ is a G -space. Endowing this G -space with an ergodic non-singular non-atomic measure, we obtain a dynamical system which is called a non-singular odometer. Necessary and sufficient conditions are found for a rank-one non-singular G -action to have a finite factor and a non-singular odometer factor in terms of the underlying $(C,F)$ -parameters. Similar conditions are also found for a rank-one non-singular G -action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the non-singular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the non-singular non-commutative case from the classical finite measure preserving one: odometer actions which are not of rank-one and factors of rank-one systems which are not of rank one; however, each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.