Discrete Subgroups of a Locally Compact Group with Jointly Discrete Chabauty Neighborhoods

Hatem Hamrouni¹ ; Abdellatif Omri¹

¹Faculty of Sciences at Sfax, Department of Mathematics, Sfax University, 3000 Sfax, Tunisia

Journal of Lie Theory, Tome 29 (2019) no. 1, pp. 89-93

Résumé

\newcommand{\cg}[1]{{\mathcal{S\hskip-.4pt U\hskip-.9pt B}}\hskip-.6pt\left(#1\right)} Let $G$ be a locally compact group. We denote by $\cg{G}$ the space of closed subgroups of $G$ equipped with the \textit{Chabauty topology}. A discrete subgroup $\Gamma$ of $G$ is said to admit a \textit{jointly discrete Chabauty neighborhood} if there exists an identity neighborhood $U$ in $G$ and an open neighborhood $\Omega$ of $\Gamma$ in $\cg{G}$ such that every closed subgroup $L\in \Omega$ satisfies $L\cap U=\{e\}$. Recently, T.\,Gelander and A.\,Levit proved that every lattice in a semi-simple analytic group admits a jointly discrete Chabauty neighborhood. In this paper, we prove that $G$ is a Lie group if and only if the trivial subgroup $\{e\}$ admits a jointly discrete Chabauty neighborhood, if and only if every discrete subgroup of $G$ admits a jointly discrete Chabauty neighborhood.

Détail

Classification : 22D05, 54B20, 22E40
Mots-clés : Locally compact group, Lie group, pro-Lie group, discrete subgroup, Chabauty topology, jointly discrete Chabauty neighborhood