Finite odometer factors of rank one group actions

Abstract

In this paper, we give explicit conditions characterizing the Følner rank one group actions that factor onto a finite odometer; those that factor onto an arbitrary, but specified odometer, and those that factor onto an unspecified odometer. We also give explicit conditions describing the Følner rank one actions that are conjugate to a specific odometer, and those that are conjugate to some odometer. These conditions are based on cutting and stacking procedures used to generate the action, and generalize results given by Foreman, Gao, Hill, Silva and Weiss for rank one $Z$-actions.