Krieger’s type of nonsingular Poisson suspensions and IDPFT systems

Abstract:

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