Poissonian Actions of Polish Groups
Dynamical Systems
2024-09-23 v2 Probability
Abstract
We define and study Poissonian actions of Polish groups as a framework to Poisson suspensions, characterize them spectrally, and provide a complete characterization of their ergodicity. We further construct 'spatial' Poissonian actions, answering partially a question of Glasner, Tsirelson & Weiss about L\'evy groups. We also construct for every diffeomorphism group an ergodic free spatial probability preserving actions. This constitutes a new class of Polish groups admitting non-essentially countable orbit equivalence relations, obtaining progress on a problem of Kechris.
Keywords
Cite
@article{arxiv.2403.19567,
title = {Poissonian Actions of Polish Groups},
author = {Nachi Avraham-Re'em and Emmanuel Roy},
journal= {arXiv preprint arXiv:2403.19567},
year = {2024}
}
Comments
43 pages. The statement of Theorem 6 is corrected (r \neq d+1), and a post-publication note regarding arXiv:2406.02401 was added