Effective Discreteness Radius of Stabilizers for Stationary Actions

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 nontrivial intersection with a small r-neighborhood of the identity is at most βr^δ for some explicit constants β, δ > 0 depending only on 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.