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 $\Gamma=\Bbb Z$. 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 IDPFT systems of each possible Krieger's type.