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

TL;DR

This paper studies the topological structure of the space of closed subgroups of the multiplicative group of p-adic numbers, revealing it as a compactification of natural numbers with a detailed description of its topology.

Contribution

It characterizes the Chabauty space of bQ_p^ imes, showing it as a proper compactification of bN and describes its structure explicitly.

Findings

bC(bQ_p^ imes) is a proper compactification of bN.

The space bC(bQ_p^ imes) bsetminus N is homeomorphic to a space formed by gluing two copies of bNbbar.

For p=2, the space resembles a Cantor space with specific glued components.

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 $\mathcal{C}(\mathbb{Q}_p^\times) \smallsetminus N$ 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.