L'espace des sous-groupes ferm\'es de RxZ

TL;DR

This paper provides a complete description of the space of closed subgroups of the group R×Z under the Chabauty topology, revealing complex topological properties such as an uncountable fundamental group.

Contribution

It offers a detailed analysis of the topology of closed subgroups of R×Z, a non-trivial case, expanding understanding of subgroup spaces in locally compact groups.

Findings

The space of closed subgroups of R×Z is fully characterized.

The fundamental group of this space is uncountably infinite.

The topology of the subgroup space exhibits complex features.

Abstract

The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.