Discretely approximable locally compact groups and Jessen's theorem for nilmanifolds
type: Item
instance of: scholarly article
title: Discretely approximable locally compact groups and Jessen's theorem for nilmanifolds
zbMATH Open document ID: 1336.22006
author: Bilel Kadri
author: Hatem Hamrouni
published in: Bulletin des Sciences Mathématiques
publication date: 2016-01-28T00:00:00Z
review text: A topological group \(G\) is said to be \textit{approximable by discrete subgroups}, if there exists a sequence of discrete subgroups \((H_n)_{n\in \mathbb N}\) of \(G\) such that, for any non-empty open set \(O\) of \(G\), there exists an integer \(k\) such that \(O\cap H_n\not= \emptyset\) for every \(n\geq k\). This paper further develops investigation on approximation of Lie groups by discrete subgroups. The main result established by the authors is that the identity component \(G_0\) of a locally compact discretely approximable group \(G\) is nilpotent. In the case of connected nilpotent Lie groups, an explicit approximating sequence of discrete subgroups is given. As an application, an extension is obtained to nilmanifolds of a theorem of \textit{B. Jessen} [Ann. Math. (2) 35, 248--251 (1934; Zbl 0009.30603)] concerning generalized limits of Riemann sums on the 1-dimensional torus \(\mathbb T\).
Mathematics Subject Classification ID: 22D05
zbMATH DE Number: 6536787
zbMATH Keywords: locally compact group
zbMATH Keywords: Chabauty topology
zbMATH Keywords: Riemann sums
zbMATH Keywords: uniform subgroup
zbMATH Keywords: nilmanifold
reviewed by: Salvador Hernández
MaRDI profile type: Publication
full work available at URL: j.bulsci.2015.11.002
OpenAlex ID: W2183680209
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DOI: 10.1016/J.BULSCI.2015.11.002
description: scientific article; zbMATH DE number 6536787