In the ergodic theory of non-locally compact Polish groups, there is an important distinction between spatial actions (a Borel action with an invariant measure) and near or Boolean actions (each group element corresponds to a measure preserving transformation that is defined almost everywhere). In this talk, I will present the Poisson suspension construction, in which a probability preserving action is constructed out of an infinite measure preserving action, and in light of the above distinction, I will present a construction of the classical Poisson point process in such a way that spatial continuous actions admit spatial Poisson suspension.
Two applications are: (1) a partial answer to an open question of Glasner, Tsirelson & Weiss about spatial actions of Lévy groups; (2) the diffeomorphisms group of a smooth manifold admits an ergodic, free, spatial action on a standard probability space.
This is a joint work with Emmanuel Roy (Université Paris 13, LAGA).