The structure and spectrum of Heisenberg odometers

Abstract:

Odometer actions of discrete, finitely generated and residually finite groups $G$ have been defined by Cortez and Petite. In this paper we focus on the case where $G$ is the discrete Heisenberg group. We prove a structure theorem for finite index subgroups of the Heisenberg group based on their geometry when they are considered as subsets of $\mathbb Z^3$. We use this structure theorem to provide a classification of Heisenberg odometers and we construct examples of each class. In order to construct some of the examples we also provide necessary and sufficient conditions for a $\mathbb Z^d$ odometer to be a product odometer as defined by Cortez. It follows from work of Mackey that all such actions have discrete spectrum. Here we provide a different proof of this fact for general $G$ odometers which allows us to identify explicitly those representations of the Heisenberg group which appear in the spectral decomposition of a given Heisenberg odometer.