The Chabauty space of $\mathbb{Q}_p^\times$
Antoine Bourquin, Alain Valette

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 denote the Chabauty space of closed subgroups of the locally compact group . In this paper, we first prove that is a proper compactification of , identified with the set of open subgroups with finite index. Then we identify the space up to homeomorphism: e.g. for , it is the Cantor space on which 2 copies of (the 1-point compactification of ) are glued.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research · Geometric and Algebraic Topology
