Cohomology of property T groupoids and applications

本质群胚C*-代数的理想交性质

The ideal intersection property for essential groupoid C*-algebras

本文刻画了拓扑传递étale群胚的本质群胚C*-代数具有理想交性质的若干等价条件，包括不存在本质上受限的可服从各向同性截面和等变伪期望唯一性，并引入群胚在C*-代数上作用的新概念，给出Thompson群拟正则表示在Cuntz代数中的相对Powers平均性质应用。

横截测度群胚的有界上同调归纳

Bounded cohomological induction for transverse measured groupoids

本文为横截测度群胚建立可测有界上同调的归纳同构，推广Eckmann-Shapiro同构，并证明其与群连续有界上同调的等价性，进而给出非平凡有界上同调群的首次显式计算。

几乎基本平展群胚

Almost elementary étale groupoids

本研究引入几乎基本性作为平展群胚的新有限逼近性质，证明极小几乎基本群胚的约化C*-代数为$\boldsymbol{\text{Z}}$-稳定，还探讨了其与群胚相关性质的联系。

群胚的Choquet-Deny性质

The Choquet-Deny Property for Groupoids

本文将群的Choquet-Deny性质研究拓展到离散测度群胚框架，通过迷向群与对应等价关系给出该性质的完整刻画，并分类了变换群胚的Choquet-Deny性质。

群胚$C^*$-代数的Fejér性质与伽罗瓦对应

Fejér property and Galois correspondence for groupoid $C^*$-algebras

本文引入拓扑étale群胚的Fejér性质，证明满足该性质的主étale第二可数群胚的闭双模结构特征，并建立了对应约化群胚$C^*$-代数的中间代数与开子群胚间的伽罗瓦对应。

正则近距可数可测群胚

Properly proximal countable measured groupoids

该研究为可数可测群胚引入正则近距概念，证明其与关联冯·诺依曼代数的强化相对正则近距等价，探究其持久性并确定多类正则近距群胚，同时证明内可和群胚不具备该性质。

Topological full groups of etale groupoids

This is a survey of the recent development of the study of topological full groups of etale groupoids on the Cantor set. Etale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations. Minimal Z-actions, minimal Z^N-actions and one-sided shifts of finite type are basic examples. We are interested in algebraic, geometric and analytic properties of topological full groups. More concretely, we discuss simplicity of commutator subgroups, abelianization, finite generation, cohomological finiteness properties, amenability, the Haagerup property, and so…

Splitting in orbit equivalence, treeable groups, and the Haagerup property

Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property.

A spectral sequence to compute L2-Betti numbers of groups and groupoids

We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existe…

通过叶状单纯形与对偶性研究单纯体积

Simplicial volume via foliated simplices and duality

该研究在带基本群保测作用的闭流形框架下，通过定义奇异叶状同调与有界上同调，证明叶状基本类范数等于流形单纯体积，并建立非球面情形下的等距同构与零化准则，形式化了Gromov的相关思路。