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 ₍=₁^ ₆ ₆g ₙg^-1=\1G\, the projective limit of the homogeneous G -spaces G/ ₙ as n 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.