Journal of Lie Theory 30 (2020), No. 1, 001--008
Copyright Heldermann Verlag 2020
Locally Compact Groups with Compact Open Subgroups Having Open Chabauty Spaces
Hatem Hamrouni
Faculty of Sciences at Sfax, Department of Mathematics, Sfax University, 3000 Sfax, Tunisia
hatemhhamrouni@gmail.com
Zouhour Jlali
Faculty of Sciences at Sfax, Department of Mathematics, Sfax University, 3000 Sfax, Tunisia
zouhourjlali2@gmail.com
[Abstract-pdf]
\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 $\mathbb{Z}(p^\infty)$.
Keywords: Locally compact group, Chabauty topology, finitely cogenerated group, Pruefer group.
MSC: 22D05, 54B20.
[ Fulltext-pdf (96 KB)] for subscribers only.