Kazhdan-Margulis theorem for Invariant Random Subgroups

TL;DR

This paper extends the Kazhdan-Margulis theorem by demonstrating that lattices in simple Lie groups are weakly uniformly discrete using invariant random subgroups and compactness arguments.

Contribution

It introduces a new proof technique for the Kazhdan-Margulis theorem by considering invariant random subgroups instead of just lattices.

Findings

Lattices in simple Lie groups are weakly uniformly discrete.

A new proof approach using invariant random subgroups and compactness.

For every epsilon, there exists an identity neighborhood intersecting trivially with stabilizers in non-atomic G-spaces.

Abstract

Given a simple Lie group $G$, we show that the lattices in $G$ are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward --- considering general IRS rather than lattices allows us to apply a compactness argument. In terms of p.m.p. actions, we show that for every $\epsilon$ there is an identity neighbourhood $U$ which intersects trivially the stabilizers of $1-\epsilon$ of the points in every non-atomic $G$-space.