Kazhdan-Margulis theorem for invariant random subgroups

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 > 0 there is an identity neighbourhood U-epsilon, subset of G which intersects trivially the stabilizers of 1 - epsilon of the points in every non-atomic probability G-space. (C) 2017 Elsevier Inc. All rights reserved.