Hyperlinearity, essentially free actions and $L^2$-invariants. The sofic property

一族新的sofic单关系群

A new family of sofic one-relator groups

本文构造了一族无限的sofic单关系群，并证明它们既非剩余可解也非剩余有限，其证明不依赖Magnus分解，与已有方法本质不同。

Free products of sofic groups with amalgamation over monotileably amenable groups

We show that free products of sofic groups with amalgamation over monotileably amenable subgroups are sofic. Consequently, so are HNN extensions of sofic groups relative to homomorphisms of monotileably amenable subgroups. We also show that families of independent uniformly distributed permutation matrices and certain families of non-random permutation matrices (essentially, those coming from quasi--actions of a sofic group) are asymptotically *-free as the matrix size grows without bound.

On sofic approximations of Property (T) groups

We prove Bowen's conjecture that every sequence of finite graphs that locally converges to the Cayley graph of a countably infinite group with Kazhdan Property (T) is essentially a vertex-disjoint union of expander graphs. We characterize graph sequences that are essentially a vertex-disjoint union of expander graphs in terms of the Markov operator.

存在一个 von Neumann 代数不为 Connes 可嵌入的等价关系

There Is An Equivalence Relation Whose von Neumann Algebra Is Not Connes Embeddable

通过简化 Aldous-Lyons 猜想的还原过程，证明了存在一个等价关系，其 von Neumann 代数不为 Connes 可嵌入，且定义了超线性性概念，展示了非交换自由群上的非超线性 IRS。

Aldous–Lyons 猜想 I：子群检验

The Aldous--Lyons Conjecture I: Subgroup Tests

本文通过引入子群检验框架，结合非局部博弈与 MIP*=RE 归约，证明了自由群中存在非 co-sofic 不变随机子群，从而否定解决 Aldous–Lyons 猜想。

全移位自同构群中的扭曲元素

Distortion element in the automorphism group of a full shift

本文证明了全移位自同构群的有限生成子群中存在字范数呈多对数增长的无限阶扭曲元素，并以此推导出子移位熵维数下界及sofic shift自同构群的性质，揭示了Turing机群与高维Brin-Thompson群的相关特征。