A view on Invariant Random Subgroups and Lattices

TL;DR

This paper explores the relationship between lattices and invariant random subgroups (IRS) in semisimple analytic groups over local fields, aiming to extend lattice theory results to the broader context of IRS.

Contribution

It provides an analysis of how lattice properties can be generalized and studied through the space of all IRS in semisimple groups over local fields.

Findings

IRS generalize lattices in Lie groups

The space of IRS offers new insights into lattice structures

Connections between IRS and classical lattice results are established

Abstract

For more than half a century lattices in Lie groups played an important role in geometry, number theory and group theory. Recently the notion of Invariant Random Subgroups (IRS) emerged as a natural generalization of lattices. It is thus intriguing to extend results from the theory of lattices to the context of IRS, and to study lattices by analyzing the compact space of all IRS of a given group. This article focuses on the interplay between lattices and IRS, mainly in the classical case of semisimple analytic groups over local fields.