Krieger's type for ergodic nonsingular Poisson actions of non-(T) locally compact groups

TL;DR

The paper investigates ergodic nonsingular Poisson actions of non-(T) locally compact groups, classifying their Krieger's types and linking properties like amenability and the Haagerup property to the nature of these actions.

Contribution

It establishes the existence of Poisson actions with arbitrary Krieger's types for a broad class of non-(T) groups, and characterizes when these actions are amenable or of 0-type.

Findings

Existence of non-strongly ergodic Poisson actions of arbitrary Krieger's type.

Amenability of the group is equivalent to the amenability of these actions.

Groups with the Haagerup property admit 0-type Poisson actions.

Abstract

It is shown that each non-compact locally compact second countable non-(T) group $G$ possesses non-strongly ergodic weakly mixing IDPFT Poisson actions of arbitrary Krieger's type. These actions are amenable if and only if $G$ is amenable. If $G$ has the Haagerup property then (and only then) these actions can be chosen of 0-type. If $G$ is amenable and unimodular then $G$ has weakly mixing Bernoulli actions of any possible Krieger's type.