On compact uniformly recurrent subgroups

2022/10/28 by Caprace, Pierre-Emmanuel, Goffer, Gil, Lederle, Waltraud +1
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2210.16297

Abstract

Let a group Γ act on a paracompact, locally compact, Hausdorff space M by homeomorphisms and let 2^M denote the set of closed subsets of M. We endow 2^M with the Chabauty topology, which is compact and admits a natural Γ-action by homeomorphisms. We show that for every minimal Γ-invariant closed subset \mathcal Y of 2^M consisting of compact sets, the union \bigcup Y⊂ M has compact closure. As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of Ušakov on compact subgroups whose normalizer is compact.

Related

• Realizing uniformly recurrent subgroups
• Property A and uniform embedding for locally compact groups
• Locally compact groups approximable by subgroups isomorphic to \mathbb Z or \mathbb R
• Characterizing maximal compact subgroups
• Compactification de Chabauty des espaces symétriques de type non compact