On the compact space of closed subgroups of locally compact groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1305.54020
author: Hatem Hamrouni
author: Bilel Kadri
publication date: 2014-11-05T00:00:00Z
full work available at URL: jlt24031.htm
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 22D05
zbMATH DE Number: 6365979
zbMATH Keywords: hyperspace
zbMATH Keywords: Chabauty topology
zbMATH Keywords: projective limit
zbMATH Keywords: strong projective limit
zbMATH Keywords: almost connected group
zbMATH Keywords: Lie projective group
zbMATH Keywords: pro-Lie group
MaRDI profile type: Publication
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title: On the compact space of closed subgroups of locally compact groups
published in: Journal of Lie Theory
review text: For a locally compact group \(G\), denote by \(\mathcal{C} (G)\) the hyperspace of all closed subgroups of \(G\) equipped with the Chabauty topology [\textit{C. Chabauty}, Bull. Soc. Math. Fr. 78, 143--151 (1950; Zbl 0039.04101)]. It is a well known fact that \(\mathcal{C} (G)\) endowed with the Chabauty topology is compact for a locally compact group \(G\). The Main Theorem states that the functor \(\mathcal{C}: G \mapsto \mathcal{C}(G)\) preserves strong projective limits on the category of locally compact topological groups and proper morphisms. For the case \(G\) compact, the analogous theorem was proved in [\textit{S. Fisher} and \textit{P. Gartside}, On the space of subgroups of a compact group. I. Topology Appl. 156, No. 5, 862--871 (2009; Zbl 1169.22003)].NEWLINENEWLINEAs an application, certain conditions under which the hyperspace \(\mathcal{C}(G)\) is connected are established: namely, the main result is used to reduce the study of connectedness of the hyperspace \(\mathcal{C}(G)\) of an arbitrary locally compact pro-Lie group \(G\) to that of Lie groups (Theorem 4.7), from which the well-known theorem by Protasov and Tsybenko about disconnectedness of \(\mathcal{C}(G)\) in case of a compact group \(G\) follows [\textit{I. V. Protasov} and \textit{Yu. V. Tsybenko}, Ukr. Mat. Zh. 35, No. 3, 382--385 (1983; Zbl 0517.22004)].
reviewed by: Kateryna Pavlyk
description: scientific article; zbMATH DE number 6365979