Locally Compact Groups with Compact Open Subgroups Having Open Chabauty Spaces

Hatem Hamrouni¹ ; Zouhour Jlali¹

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

Journal of Lie Theory, Tome 30 (2020) no. 1, pp. 1-8

Résumé

\newcommand{\cg}[1]{{\mathcal{S\hskip-.5pt 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}; this is a compact space. The topological space $\cg{G}$ is called the \textit{Chabauty space} of $G$. For a closed subgroup $H$ of $G$ the subspace $\{L\in \cg{G} \mid L\subseteq H\}$ of $\cg{G}$ is homeomorphic to the Chabauty space $\cg{H}$ of $H$ and so $\cg{H}$ is a compact subspace of $\cg{G}$. The paper discusses the scope of validity of an assertion having appeared recently in the book of Herfort-Hofmann-Russo about the openness of the subspace $\cg{H}$ in $\cg{G}$. We study the class $\mathfrak{X}$ of locally compact groups $G$ such that the subspace $\cg{H}$ is open in $\cg{G}$ for any compact open subgroup $H$ of $G$. We show that a locally compact abelian group $A$ is in $\mathfrak{X}$ if and only if $A$ contains a compact open subgroup $U$ such that $A/U$ is a finite direct sum of subgroups each of which is either cyclic or is a Pr\"{u}fer group isomorphic to $\mathbb{Z}(p^\infty)$.

Détail

Classification : 22D05, 54B20
Mots-clés : Locally compact group, Chabauty topology, finitely cogenerated group, Pruefer group