The Chabauty space of ${\mathbb{Q}}_p^{\times }$

Abstract

Let $\mathcal{C}(G)$ denote the Chabauty space of closed subgroups of the locally compact group $G$. In this paper, we first prove that $\mathcal{C}({\mathbb{Q}}_p^{\times })$ is a proper compactification of $\mathbb{N}$, identified with the set $N$ of open subgroups with finite index. Then we identify the space $\mathcal{C}({\mathbb{Q}}_p^{\times })\setminus N$ up to homeomorphism: e.g. for $p=2$, it is the Cantor space on which 2 copies of $\overline{\mathbb{N}}$ (the 1-point compactification of $\mathbb{N}$) are glued.

References in corpus (3)

• L'espace des sous-groupes fermés de RxZ
• On the Chabauty space of locally compact abelian groups
• The space of subgroups of an abelian group