Bernoulli actions of type III1 and L2-cohomology

Abstract

We conjecture that a countable group G admits a nonsingular Bernoulli action of type III1 if and only if the first L2-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 III1 Bernoulli actions of the groups $${\mathbb{Z}}$$ and the free groups $${\mathbb{F}_n}$$ , with different degrees of ergodicity.