Ergodicity and type of nonsingular Bernoulli actions

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.

v2: minor changes, final version, to appear in Inventiones Mathematicae