Spaces of closed subgroups of locally compact 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,} École Normale Supérieure de Lyon, June 13, 2008; and seminars at the EPFL, Lausanne, and in the Université de Rennes 1. The notes do not contain any other result than those in \cite{BrHK}, and are not intended for publication.

12 pages

References in corpus (2)

• Limit groups as limits of free groups: compactifying the set of free groups
• The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

Cited by in corpus (12)

• The space of closed subgroups of $R^n$
• Limits Under Conjugacy of the Diagonal Subgroup in SL(n,R)
• A dichotomy for bounded displacement equivalence of Delone sets
• Multiscale Substitution Tilings
• Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups
• Conjugacy Limits of the Cartan Subgroup in SL(3,R)
• Introduction to Chabauty topology and Pictures of the Chabauty space of C*
• On the Chabauty space of ${\text{PSL}}_2(\mathbb{R})$, I: lattices and grafting
• Compactification de Chabauty des espaces symétriques de type non compact
• Visual limits of maximal flats in symmetric spaces and Euclidean buildings
• Effective discreteness radius of stabilisers for stationary actions
• Limits of metabelian groups