The von Neumann–Day problem asks whether every non-amenable group contains a non-abelian free group. It was answered in the negative by Ol’shanskii in the 1980s. The measurable version (formulated by Gaboriau–Lyons) asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau–Lyons. The proof uses an approximation to the random interlacement process by random multisets of geometrically-killed random walk paths. There are two applications: (1) the Gaboriau–Lyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all Bernoulli shifts factor onto each other.

中文翻译：

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有限随机交错和Gaboriau–Lyons问题

冯·诺依曼·戴（von Neumann-Day）问题询问每个不满足的群体是否都包含一个非阿贝尔自由群体。奥尔尚斯基（Ol'shanskii）在1980年代否定了这个答案。可测量的版本（由Gaboriau–Lyons制定）询问每个不可满足的测得等价关系是否都包含不可适应的树可等效性关系。在任意伯努利移交给一个无法接受的群体时，本文获得了肯定的答案，从而扩展了加波罗-里昂斯的著作。该证明使用几何杀死的随机行走路径的随机多组来近似随机隔行过程。有两个应用程序：（1）对于具有正Rokhlin熵的动作的Gaboriau–Lyons问题承认一个正解；（2）对于任何不满意的群体，所有伯努利偏移因子都是彼此相关的。