# Bernoulli actions of type III_1 and L^2-cohomology

## TL;DR

This paper explores the relationship between the existence of certain nonsingular Bernoulli actions of groups and the nonvanishing of their first L^2-cohomology, providing proofs for groups with infinite order elements and examples for specific groups.

## Contribution

It conjectures a characterization of groups admitting type III_1 Bernoulli actions based on their L^2-cohomology and proves this for groups with infinite order elements, also providing explicit examples.

## Key findings

- Proves the conjecture for groups with at least one infinite order element.
- Provides explicit examples of type III_1 Bernoulli actions for integers and free groups.
- Establishes a link between Bernoulli actions and L^2-cohomology of groups.

## Abstract

We conjecture that a countable group $G$ admits a nonsingular Bernoulli action of type III$_1$ if and only if the first $L^2$-cohomology of $G$ is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III$_1$ Bernoulli actions of the group of integers and the free groups, with different degrees of ergodicity.