G-odometers and their almost 1-1 extensions

In this paper we recall the concepts of $G$-odometer and $G$-subodometer for $G$-actions, where $G$ is a discrete finitely generated group, which generalize the notion of odometer in the case $G=\ZZ$. We characterize the $G$-regularly recurrent systems as the minimal almost 1-1 extensions of subodometers, from which we deduce that the family of the $G$-Toeplitz subshifts coincides with the family of the minimal symbolic almost 1-1 extensions of subodometers.