The structure and spectrum of Heisenberg odometers

TL;DR

This paper classifies Heisenberg odometer actions by analyzing their subgroup structures, proves they have discrete spectrum, and explicitly describes the spectral representations, advancing understanding of these dynamical systems.

Contribution

It provides a complete classification of Heisenberg odometers based on subgroup geometry and explicitly constructs their spectral representations.

Findings

All Heisenberg odometers have discrete spectrum.

Explicit descriptions of spectral representations are provided.

Conditions for Z^d odometers to be product odometers are established.

Abstract

In recent work Cortez and Petite defined odometer actions of discrete, finitely generated and residually finite groups G. 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 Z^3. We provide a complete classification of Heisenberg odometers based on the structure of their defining subgroups and we provide examples of each class. Mackey has shown that all such actions have discrete spectrum, i.e. that the unitary operator associated to the dynamical system admits a decomposition into finite dimensional, irreducible representations of the group G. Here we provide an explicit proof of this fact for general G odometers. Our proof allows us to define explicitly those representations of the Heisenberg group which appear in the spectral decomposition of a Heisenberg odometer, as a function of the defining subgroups. Along the way we also provide necessary and sufficient conditions for a Z^d odometer to be a product odometer as defined by Cortez.