Discretely approximable locally compact groups and Jessen's theorem for nilmanifolds

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• 作者：Hatem Hamrouni ; ^{hatemhhamrouni@voila.fr" class="auth_mail" title="E-mail the corresponding author} ; Bilel Kadri ^{bilel.kadri@yahoo.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：22D05
• 刊名：Bulletin des Sciences Mathematiques
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：140
• 期：1
• 页码：1-13
• 全文大小：328 K

文摘

A topological group G   is said to be approximable by discrete subgroups, if there exists a sequence of discrete subgroups (H_n)_n∈N${(H_n)}_{n\in \mathbb{N}}$ of G such that, for any non-empty open set O of G, there exists an integer k   such that O∩H_n≠∅$O\cap H_n\ne \varnothing$, for every n≥k$n\ge k$. In this paper we shall prove that the identity component of a locally compact group approximable by discrete subgroups is nilpotent. For connected nilpotent Lie groups, explicit approximating sequences of discrete subgroups are given. As an application, we extend Jessen's theorem on Riemann sums for torus to the case of nilmanifolds.