A lecture on Invariant Random Subgroups

TL;DR

This paper explores invariant random subgroups (IRS) as a unifying concept extending normal subgroups and lattices, analyzing their properties and applications using two different approaches in group theory.

Contribution

It provides an overview of IRS theory, connecting it to normal subgroups and lattices, and discusses methods to derive new results in group theory.

Findings

IRS generalizes normal subgroups and lattices

Analysis of IRS as a compact G-space yields new insights

Connections established between IRS and classical subgroup theories

Abstract

Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups in a given group G. They can be regarded both as a generalization of normal subgroups as well as a generalization of lattices. As such, it is intriguing to extend results from the theories of normal subgroups and of lattices to the context of IRS. Another approach is to analyse and then use the space IRS(G) as a compact G-space in order to establish new results about lattices. The second approach has been taken in the work [7s12], that came to be known as the seven samurai paper. In these lecture notes we shall try to give a taste of both approaches.