On compact uniformly recurrent subgroups

TL;DR

This paper studies the structure of uniformly recurrent subgroups in locally compact groups, showing they are contained in compact normal subgroups, thus extending previous results on subgroup normalizers.

Contribution

It proves that every compact uniformly recurrent subgroup is contained in a compact normal subgroup, generalizing Ušakov's result.

Findings

Every minimal invariant closed subset of compact sets has union with compact closure

Compact uniformly recurrent subgroups are contained in compact normal subgroups

Generalizes previous results on subgroup normalizers

Abstract

Let a group $\Gamma$ 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 $\Gamma$-action by homeomorphisms. We show that for every minimal $\Gamma$-invariant closed subset $\mathcal Y$ of $2^M$ consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ 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 U\v{s}akov on compact subgroups whose normalizer is compact.