ON COMPACT UNIFORMLY RECURRENT SUBGROUPS

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 Y of 2^M consisting of compact sets, the union S 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.