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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The structure and spectrum of Heisenberg odometers


Authors: Samuel Lightwood, Ayşe Şahin and Ilie Ugarcovici
Journal: Proc. Amer. Math. Soc. 142 (2014), 2429-2443
MSC (2010): Primary 37A15, 37A30; Secondary 20E34
Published electronically: March 28, 2014
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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.


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Additional Information

Samuel Lightwood
Affiliation: Department of Mathematics, Western Connecticut State University, Danbury, Connecticut 06810
Email: lightwoods@wcsu.edu

Ayşe Şahin
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11963-0
PII: S 0002-9939(2014)11963-0
Keywords: Odometer actions, subgroups of the Heisenberg group, discrete spectrum
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
Article copyright: © Copyright 2014 American Mathematical Society