Krieger’s type of nonsingular Poisson suspensions and IDPFT systems

Abstract

Given an infinite countable discrete amenable group Γ \Gamma , we construct explicitly sharply weak mixing nonsingular Poisson Γ \Gamma -actions of each Krieger’s type: I I I λ III_\lambda , for λ ∈ [ 0 , 1 ] \lambda \in [0,1] , and I I ∞ II_\infty . The result is new even for Γ = Z \Gamma =\mathbb {Z} . As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli Γ \Gamma -actions and infinite direct product of finite type systems of each possible Krieger’s type.