Effective Discreteness Radius of Stabilizers for Stationary Actions

T. Gelander, A. Levit, G. A. Margulis

Research output: Contribution to journalArticlepeer-review


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 βrδ 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.

Original languageEnglish
Pages (from-to)389-438
Number of pages50
JournalMichigan Mathematical Journal
StatePublished - Aug 2022


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