Algebraic & Geometric Topology

L'espace des sous-groupes fermés de $\mathbb{R} \times \mathbb{Z}$

Thomas Haettel

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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 subgroups of the group × (and of its dual ×), which is highly nontrivial : for example, its fundamental group contains the fundamental group of the Hawaiian earrings, hence is uncountable.

L’espace des sous-groupes fermés d’un groupe topologique localement compact est muni d’une topologie naturelle, appelée topologie de Chabauty. Nous décrivons complètement l’espace des sous-groupes fermés du groupe × (et de son dual ×), lequel est hautement non trivial : par exemple, son groupe fondamental contient le groupe fondamental des anneaux Hawaïens, et est donc non dénombrable.

Article information

Algebr. Geom. Topol., Volume 10, Number 3 (2010), 1395-1415.

Received: 19 December 2008
Revised: 20 May 2009
Accepted: 8 July 2009
First available in Project Euclid: 19 December 2017

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Zentralblatt MATH identifier

Primary: 22B99: None of the above, but in this section 22D05: General properties and structure of locally compact groups
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57M07: Topological methods in group theory

Chabauty space of closed subgroups Hawaiian earrings espace des sous-groupes fermés


Haettel, Thomas. L'espace des sous-groupes fermés de $\mathbb{R} \times \mathbb{Z}$. Algebr. Geom. Topol. 10 (2010), no. 3, 1395--1415. doi:10.2140/agt.2010.10.1395.

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