The structure and spectrum of Heisenberg odometers
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- by Samuel Lightwood, Ayşe Şahi̇n and Ilie Ugarcovici PDF
- Proc. Amer. Math. Soc. 142 (2014), 2429-2443 Request permission
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.References
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Additional Information
- Samuel Lightwood
- Affiliation: Department of Mathematics, Western Connecticut State University, Danbury, Connecticut 06810
- Email: lightwoods@wcsu.edu
- Ayşe Şahi̇n
- Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
- Email: asahin@depaul.edu
- Ilie Ugarcovici
- Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
- Email: iugarcov@depaul.edu
- Received by editor(s): December 28, 2011
- Received by editor(s) in revised form: June 12, 2012, and July 25, 2012
- Published electronically: March 28, 2014
- Communicated by: Bryna R. Kra
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 2429-2443
- MSC (2010): Primary 37A15, 37A30; Secondary 20E34
- DOI: https://doi.org/10.1090/S0002-9939-2014-11963-0
- MathSciNet review: 3195765