Bernoulli actions of type III_1 and L^2-cohomology

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.

arXiv commentv4: minor changes, final version, to appear in Geometric and Functional Analysis (GAFA)