有限随机交错与Gaboriau-Lyons问题
Finitary random interlacements and the Gaboriau-Lyons problem
摘要：
冯·诺依曼-戴问题是问是否每个非顺从群都包含一个非阿贝尔自由群。在1980年代，Ol'shanskii给出了否定的回答。可测量的版本（由Gaboriau-Lyons制定）询问是否每个非顺从的可测量的等价关系包含非顺从的可树次等价关系。本文在非顺从群上的任意Bernoulli移位的情形下得到了一个肯定的答案，推广了Gaboriau-Lyons的工作。证明使用了一个近似的随机交错过程的随机多步几何杀死随机游动路径。有两个应用程序：（1）具有正Rokhlin熵的作用的Gaboriau-Lyons问题存在正解，（2）对于任意非顺从群，所有Bernoulli移位因子相互转移.
英文摘要：
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 multistep 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
