Non-amenable products are not treeable

Finitary random interlacements and the Gaboriau-Lyons problem

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 …

单端 spanning 子森林与群的树形性

One-ended spanning subforests and treeability of groups

本文研究了若干新类型的群的度量强树形性，包括具有平面 Cayley 图的有限生成群、初等自由群以及双曲平面及其闭子群。作者证明了这些群是度量强树形的，并探讨了三维闭非定向流形的基群的树形性分类。

A random matrix approach to absorption in free products

This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if $\mathcal{M} = \mathcal{P} * \mathcal{Q}$, then $\mathcal{P}$ absorbs any subalgebra of $\mathcal{M}$ that intersects it diffusely …

Voronoi percolation在非可缩积积空间中的消失唯一性阈值

Vanishing uniqueness thresholds in Voronoi percolation on products

本文研究了Voronoi渗流在非可缩积积空间中的唯一性阈值，提出了一种新方法证明小强度时阈值的小性，并提供了多个具体例子，补充了对称空间的结果，同时回答了关于非可缩 Cayley 图的问题。

Factors of IID on Trees

Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. Our results include bounds on minimum and maximum bisection for ra…