Spaces of closed subgroups of locally compact groups

TL;DR

This paper explores the topology of the space of closed subgroups of various locally compact groups, providing examples including the complex plane and the Heisenberg group, revealing complex geometric structures.

Contribution

It describes the topology of the space of closed subgroups for specific groups, including elementary cases, the complex plane, and the Heisenberg group, expanding understanding of these spaces.

Findings

The space of closed subgroups of the complex plane is a 4-sphere.

The space of closed subgroups of the Heisenberg group is a 6-dimensional singular space.

The topology of these spaces varies significantly across different groups.

Abstract

The set $\Cal C(G)$ of closed subgroups of a locally compact group $G$ has a natural topology which makes it a compact space. This topology has been defined in various contexts by Vietoris, Chabauty, Fell, Thurston, Gromov, Grigorchuk, and many others. The purpose of the talk was to describe the space $\Cal C(G)$ first for a few elementary examples, then for $G$ the complex plane, in which case $\Cal C(G)$ is a 4--sphere (a result of Hubbard and Pourezza), and finally for the 3--dimensional Heisenberg group $H$, in which case $\Cal C(H)$ is a 6--dimensional singular space recently investigated by Martin Bridson, Victor Kleptsyn and the author \cite{BrHK}. These are slightly expanded notes prepared for a talk given at several places: the Kortrijk workshop on {\it Discrete Groups and Geometric Structures, with Applications III,} May 26--30, 2008; the {\it Tripode 14,} \'Ecole Normale Sup\'erieure de Lyon, June 13, 2008; and seminars at the EPFL, Lausanne, and in the Universit\'e de Rennes 1. The notes do not contain any other result than those in \cite{BrHK}, and are not intended for publication.