Compact Invariant Random Subgroups

TL;DR

This paper investigates ergodic invariant random subgroups supported on compact subgroups across various classes of locally compact groups, revealing their containment within specific normal or radical subgroups.

Contribution

It characterizes the structure of such random subgroups in real Lie groups, p-adic Lie groups, and totally disconnected groups, identifying their containment within key subgroup classes.

Findings

In real Lie groups, they are contained in a compact normal subgroup.

In p-adic Lie groups, they lie within the locally elliptic radical.

In totally disconnected groups, they are in the intersection of Levi subgroups.

Abstract

We study ergodic invariant random subgroups that give full measure to the subset of compact subgroups. We show that in real Lie groups, compactly generated $p$-adic Lie groups, locally compact hyperbolic groups and infinitely ended groups they are always contained in a compact normal subgroup. In general $p$-adic Lie groups, we show they are contained in the locally elliptic radical. In totally disconnected locally compact groups, we show they are contained in the intersection of all Levi subgroups of inner automorphisms.