Ergodicity and type of nonsingular Bernoulli actions

Abstract

We determine the Krieger type of nonsingular Bernoulli actions G↷ ∏ g∈G({ 0 , 1 } , μg). When G is abelian, we do this for arbitrary marginal measures μg. We prove in particular that the action is never of type II∞ if G is abelian and not locally finite, answering Krengel’s question for G= Z. When G is locally finite, we prove that type II∞ 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, II1 or III1. When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of Vaes and Wahl (Geom Funct Anal 28:518–562, 2018) by proving that a group G admits a Bernoulli action of type III1 if and only if G has nontrivial first L2-cohomology.