Sarah Campbell
Using properties of their Cayley graphs, specific examples of Ozawa kernels are constructed for both free and amenable groups, thus showing that these groups satisfy Property O. It is deduced both that these groups are exact and satisfy Yu's Property A.
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We get three types of results on measure equivalence rigidity; direct product groups of Ozawa's class $\mathcal{S}$ groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class $\mathcal{S}$ groups. As consequences, Monod--Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups $A \wr G$, $B \wr Γ$ of non-amenable exact direct product groups $G$, $Γ$ with amenable bases $A$, $B$ are measure equivalent, then $G$ and $Γ$ are measure equivalent. We get Bass--Serr…
Bi-exactness of relatively hyperbolic groups
本研究证明有限生成相对双曲群具有双精确性的充要条件是其所有外围子群均为双精确的，推广了Ozawa关于外围子群顺从时相对双曲群双精确的结论。
We study class $\mathcal S$ for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class $\mathcal S$ and prove unique prime factorization results for group von Neumann algebras of products of locally compact groups in this class. We also prove that class $\mathcal S$ is a measure equivalence invariant.
We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra analogue is also obtained.
We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.
Following an approach of Ozawa, we show that several semidirect products are not weakly amenable. As a consequence, we are able to characterize the simply connected Lie groups that are weakly amenable.
This paper includes a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors \cite{CS}, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance we show that every II$_1$ factor associated with a weakly amenable group in the class $\mathcal S$ of Ozawa is strongly solid, \cite{OzSolid}. There is also the following product version of this result: any maximal abelian $\star$-subal…
Exactness and the topology of the space of invariant random equivalence relations
本文通过不变随机等价关系（IREs）刻画可数群的精确性，证明精确群等价于所有弱极限有限IREs均为可算IREs，特别指出精确群的受限重根关系在理想伯努利沃洛诺伊 tessellation 中的可算性。
Given a countable residually finite group, we construct a compact group K and two elements w and u of K with the following properties: The group generated by w and the cube of u is amenable, the group generated by w and u contains a copy of the given group, and these two groups are dense in K. By combining it with a construction of non-exact groups that are LEF by Osajda and Arzhantseva--Osajda and formation of diagonal products, we construct an example for which the latter dense group is non-exact. Our proof employs approximations in the space of marked groups of LEF ("Locally Embeddable into…
We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-Čech compactification of the group we obtain a cohomologic…
Stabilizer Subgroups and the Simplicity of Reduced Crossed Products
本文研究了可数群在紧空间上的最小作用的约化交叉积的简单性，证明了若约化交叉积简单，则存在一个点其稳定子群的可解交换余部为平凡。进一步，该研究为可数线性群、双曲群等类别的约化交叉积的简单性提供了完整刻画，并回答了Ozawa (2014)提出的问题。
Bi-exact Wreath-like Product Groups
本文证明了顺从群与双精确群的花环型积群具有双精确性，进而获得群冯诺依曼代数的坚实性及麦克达夫超刚性的新例子。
We show that every finitely generated extension by $\mathbb{Z}$ of a locally normally finite group has Shalom's property $H_{\mathrm{FD}}$. This is no longer true without the normality assumption. This permits to answer some questions of Shalom, Erschler-Ozawa and Kozma. We also obtain a Neumann-Neumann embedding result that any countable locally finite group embedds into a two generated amenable group with property $H_{\mathrm{FD}}$.
Stationary boundaries on the space of amenable subgroups and C*-simplicity
本文研究了可数群 G 存在非平凡 μ-边界条件，探讨了 amenable 子群空间的 μ-stationary 性质，并讨论了 Thompson 群 F 的 C*-简单性问题。
PHP decompositions and primitivity of group rings for linear groups
本文证明具有小泽性质PHP且含非阿贝尔自由子群的可数群的群环对任意域本原，并推论非平凡可数线性群若具有平凡可和根则也满足此性质，同时建立了PHP类的一些永久性质。
We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath pr…
Consequences of the random matrix solution to the Peterson-Thom conjecture
本研究借助Peterson-Thom猜想的随机矩阵解，推导自由群因子的多项新结构性质，解决粗性猜想、推广强稳固性结果，还得出子代数缠结二分性及超积嵌入应用，推动算子代数领域相关研究。
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