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 for some explicit constants 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.
Dedicated to Gopal Prasad on the occasion of his 75th birthday.
"Effective Discreteness Radius of Stabilizers for Stationary Actions." Michigan Math. J. 72 389 - 438, August 2022. https://doi.org/10.1307/mmj/20217209