Invariant random subgroups on certain orbits

TL;DR

This paper studies the existence of invariant random subgroups within specific orbits of the conjugation action in the space of closed subgroups of a connected Lie group, using the Chabauty topology.

Contribution

It characterizes the existence of invariant random subgroups supported on particular conjugation orbits in the space of subgroups of a Lie group.

Findings

Invariant random subgroups can be supported on certain conjugation orbits.

Conditions for the existence of such subgroups are established.

The structure of the orbit space influences the properties of invariant random subgroups.

Abstract

Let $G$ be a connected Lie group and $\text{Sub}_G$ be the space of closed subgroups of $G$ equipped with the Chabauty topology. In this article, we investigate the existence of invariant random subgroups of $G$ supported on various orbits of the conjugation action of $G$ on $\text{Sub}_G$.