# Realizing uniformly recurrent subgroups

## TL;DR

This paper proves that every uniformly recurrent subgroup of a locally compact group corresponds to stabilizers in a minimal compact action, introducing a universal minimal flow concept and answering a longstanding question.

## Contribution

It establishes a correspondence between uniformly recurrent subgroups and stabilizers in minimal actions, and introduces the universal minimal flow relative to such subgroups.

## Key findings

- Every uniformly recurrent subgroup is a stabilizer family of a minimal compact action.
- Every closed invariant subset of the Chabauty space corresponds to stabilizers of a compact action.
- Existence and uniqueness of the universal minimal flow relative to a URS are proven.

## Abstract

We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. We also introduce the notion of a universal minimal flow relative to a uniformly recurrent subgroup and prove its existence and uniqueness.