We determine the Krieger type of nonsingular Bernoulli actions $G \curvearrowright \prod_{g \in G} (\{0,1\},\mu_g)$. 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$_\infty$ if $G$ is abelian and not locally finite, answering Krengel's question for $G = \mathbb{Z}$. When $G$ is locally finite, we prove that type II$_\infty$ 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 [VW17] 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.

中文翻译：

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非奇异伯努利动作的遍历性和类型

我们确定非奇异伯努利动作的 Krieger 类型 $G \curvearrowright \prod_{g \in G} (\{0,1\},\mu_g)$。当 $G$ 是 abelian 时，我们对任意边际度量 $\mu_g$ 执行此操作。我们特别证明，如果 $G$ 是阿贝尔的且不是局部有限的，则该动作永远不会属于 II$_\infty$ 类型，从而回答了 Krengel 关于 $G = \mathbb{Z}$ 的问题。当 $G$ 局部有限时，我们证明类型 II$_\infty$ 确实出现。对于任意可数组，我们假设边际度量远离 $0$ 和 $1$。当$G$只有一端时，我们证明Krieger类型总是I、II$_1$或III$_1$。当 $G$ 有多个末端时，我们表明总会出现其他类型。最后，我们通过证明群 $G$ 承认类型 III$_1$ 的伯努利作用来解决 [VW17] 的猜想，当且仅当 $G$ 具有非平凡的第一 $L^2$-上同调。