## Abstract

We determine the Krieger type of nonsingular Bernoulli actions G↷ ∏ _{g}_{∈}_{G}({ 0 , 1 } , μ_{g}). When G is abelian, we do this for arbitrary marginal measures μ_{g}. 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, 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 Vaes and Wahl (Geom Funct Anal 28:518–562, 2018) 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.

Original language | American English |
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Pages (from-to) | 573-625 |

Number of pages | 53 |

Journal | Inventiones Mathematicae |

Volume | 224 |

Issue number | 2 |

DOIs | |

State | Published - May 2021 |

### Bibliographical note

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