Title: Boomerang subgroups


Abstract: (Based on a joint work with Waltraud Lederle) A subgroup \Gamma < G in a locally compact topological group G is called a lattice if it is discrete, and G/\Gamma carries a G-invariant probability measure. Typical examples of lattices are \mathbb{Z} < \mathbb{R} as well as SL_n(\mathbb{Z}) < SL_n(\mathbb{R}). I will focus in my talk on the latter example. 

The lattice often mimics and reflects the properties of its ambient group in non-trivial ways. For example the group {\rm SL}_n(\mathbb{R}) is simple (modulu a possible finite center). Our lattice is far from being simple as it admits many obvious finite quotients of the form SL_n(\mathbb{Z}) \rightarrow SL_n(\mathbb{Z}/N\mathbb{Z}). In the case n > 2 two very deep theorems: Margulis normal subgroup theorem and the congruence subgroup property of Lazzard and Bass-Milonr-Serre combine to say that these account for all normal subgroups of \Gamma. On the other hand for n = 2 this breaks down completely as SL_2(\mathbb{Z}) is virtually a free group and admits many normal subgroups. 

This anomaly can be rectified by introducing the notion of an invariant random subgroup (IRS). This is by definition a G invariant probability measure on the (closed compact) space Sub(G) of all closed subgroups of G. G acts on this space by conjugation. Thus a normal subgroup is just a fixed point on this space, and the notion of an IRS generalizes this. With this terminology in place everything works. For n=2 both \Gamma,G have an abundance of IRSs. It is the content of the Nevo-Stuck-Zimmer theorem that for n \ge 2 IRSs are classified and look the same in both groups. 

I will discuss a new work of mine with Waltraud Lederle. We introduce the notion of a boomerang subgroup. This notion gives a unified proof for all of the above theorems in the case of \Gamma. We are still looking for an analogous proof for G. 
This is a joint work with Waltraud Lederle.


Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=0b110975-993b-4302-b441-b22e006a4d4a