Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The structure and spectrum of Heisenberg odometers
HTML articles powered by AMS MathViewer

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
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37A15, 37A30, 20E34
  • Retrieve articles in all journals with MSC (2010): 37A15, 37A30, 20E34
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