# Ergodicity and type of nonsingular Bernoulli actions

## TL;DR

This paper classifies the Krieger type of nonsingular Bernoulli actions for various groups, revealing new results about when these actions are of certain types, including answering longstanding questions and solving a conjecture.

## Contribution

It provides a comprehensive classification of the Krieger type for nonsingular Bernoulli actions across different group structures, including abelian, locally finite, and groups with various end properties.

## Key findings

- Actions are never of type II$_
fty$ for non-locally finite abelian groups.
- Type II$_
fty$ can occur for locally finite groups.
- A group admits a type III$_1$ Bernoulli action iff it has nontrivial first $L^2$-cohomology.

## Abstract

We determine the Krieger type of nonsingular Bernoulli actions $G \curvearrowright \prod_{g \in G} (\{0,1\},\mu_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $\mu_g$. We prove in particular that the action is never of type II$_\infty$ if $G$ is abelian and not locally finite, answering Krengel's question for $G = \mathbb{Z}$. When $G$ is locally finite, we prove that type II$_\infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$_1$. When $G$ has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group $G$ admits a Bernoulli action of type III$_1$ if and only if $G$ has nontrivial first $L^2$-cohomology.