Effective discreteness radius of stabilisers for stationary actions

TL;DR

This paper establishes an effective version of the Kazhdan-Margulis theorem for stationary actions of semisimple groups, providing explicit bounds on stabilizer intersections and deriving consequences for volume and subgroup compactness.

Contribution

It introduces an explicit probabilistic bound on stabilizer intersections in stationary actions, extending classical results to a broader setting with quantitative estimates.

Findings

Bound on probability of non-trivial stabilizer intersections with small neighborhoods

Finiteness of volume from vanishing injectivity radius

Compactness of the space of discrete stationary subgroups

Abstract

We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $\beta r^\delta$ for some explicit constants $\beta, \delta > 0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact.