Ergodicity and type of nonsingular Bernoulli actions

Journal article, 2020

We determine the Krieger type of nonsingular Bernoulli actions G curved right arrow Pi(g is an element of 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 infinity 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 infinity 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 L-2-cohomology.