A lecture on Invariant Random Subgroups

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.

arXiv comment19 pages, 5 figures. Based on a mini course given in Ghys' birthday --- conference Geometries in Action (2015), Oberwolfach workshop on Locally Compact Groups (2014) and Ventotene's conference on Manifolds and Groups (2015)