Rigidity of graph products of groups

大类型Artin群的刚性及分类结果

Rigidity and classification results for large-type Artin groups

本文计算了许多大类型Artin群交集图的自同构群，并以此为应用推导出包括外自同构群、可公度性分类及拟等距、测度等价、轨道等价、格嵌入和交叉积von Neumann代数刚性等一系列分类与刚性结果。

Rigidity of amalgamated free products in measure equivalence

A discrete countable group Γis said to be ME rigid if any discrete countable group that is measure equivalent to Γis virtually isomorphic to Γ. In this paper, we construct ME rigid groups by amalgamating two groups satisfying rigidity in a sense of measure equivalence. A class of amalgamated free products is introduced, and discrete countable groups which are measure equivalent to a group in that class are investigated.

Measure Equivalence Rigidity and Bi-exactness of Groups

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…

Rigidity of graph products of abelian groups

We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.

Rigidity of high dimensional graph manifolds

We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic manifold with toric cusps. The various pieces are attached together via affine maps of the boundary tori. We require all the hyperbolic factors in the pieces to have dimension at least 3. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory. We show that, in high dimensions, the Borel conjecture holds for our graph manifol…

Rigidity for von Neumann algebras given by locally compact groups and their crossed products

We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S.

Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that aren't quadratically hanging. Our main result is that any group quasi-isometric to $G$ is abstractly commensurable to $G$. In particular, our result applies to certain "generic" HNN extensions of a free group over cyclic subgroups.

酉群与矩阵代数约化积的迹范数刚性

Trace-norm rigidity for reduced products of unitary groups and matrix algebras

研究有限维酉群和矩阵代数的迹度量约化积间同态，证明酉群相关稳定性变体等，表明乘积形式同构由特定方式诱导，还得到坐标识别及自同构群的刚性与分类结果，矩阵代数在更一般背景下有类似刚性结果。

Some rigidity results for II$_1$ factors arising from wreath products of property (T) groups

We show that any infinite collection $(Γ_n)_{n\in \mathbb N}$ of icc, hyperbolic, property (T) groups satisfies the following von Neumann algebraic \emph{infinite product rigidity} phenomenon. If $Λ$ is an arbitrary group such that $L(\oplus_{n\in \mathbb N} Γ_n)\cong L(Λ)$ then there exists an infinite direct sum decomposition $Λ=(\oplus_{n \in \mathbb N} Λ_n )\oplus A$ with $A$ icc amenable such that, for all $n\in \mathbb N$, up to amplifications, we have $L(Γ_n) \cong L(Λ_n)$ and $L(\oplus_{k\geq n} Γ_k )\cong L((\oplus_{k\geq n} Λ_k) \oplus A)$. The result is sharp and complements the pre…

图积群的冯·诺依曼代数的刚性。I. 自同构的结构

Rigidity for von Neumann algebras of graph product groups. I. Structure of automorphisms

本文结合 Popa 变形/刚性理论与图积代数新技术，刻画了特定图积群（团环结构与圈积类性质(T)顶点群）的冯·诺依曼代数及既约 C$^*$-代数的所有对称性，建立了自同构公式。

使用群图的结构刚性与柔性

Structural rigidity and flexibility using graphs of groups

本文利用群图和群论描述超图实现的铰接运动，提出一种可建模多种情况并简洁描述刚性理论对偶性的方法，并在基础群为 Lie 群时给出了无穷小运动维数的下界。

Semidirect product rigidity of group von Neumann algebras arising from class $\mathscr{S}$, inductive limits and fundamental group

In this article we study property (T) groups arising from Rips construction in geometric group theory in the spirit of \cite{CDK19} and certain inductive limit groups from this class. Using interplay between Popa's deformation/rigidity and methods in geometric group theory we are able to extend the class of groups considered in \cite{CDK19} that remembers semidirect product features while passing to the group von Neumann algebras. Combining these results with the method developed in \cite{CDHK20} we are able to produce more examples of property (T) group factors with trivial fundamental group.…

II$_1$因子的W*-相关性以及张量积与图积的刚性

W*-correlations of II$_1$ factors and rigidity of tensor products and graph products

本文提出II$_1$因子的W*-相关性术语，证明了张量积与图积的刚性结果，并据此构造了首个非von Neumann等价的不可数离散群族。

Free products, Orbit Equivalence and Measure Equivalence Rigidity

We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ({\FI}) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first $L^2$-Betti number. We obtain Bass-Serre rigidity results, \textit{i.e.} forms of uniqueness in free product decompositions of equivalence relations with ({\FI}) components. The main features of our work are weak algebraic assumptions a…

A rigidity result for crossed products of actions of Baumslag-Solitar groups

Let BS(n_1,m_1) $\curvearrowright$ X_1 and BS(n_2,m_2) $\curvearrowright$ X_2 be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag-Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II_1 factors forces BS(n_1,m_1) $\cong$ BS(n_2,m_2) when |n_1| $\neq$ |m_1| and BS(n_1,m_1) $\cong$ BS(n_2,$\pm$m_2) when |n_1| = |m_1|. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum.

Rigidity results for wreath product II$_1$ factors

We consider II$_1$ factors of the form $M=\bar{\bigotimes}_{G}N\rtimes G$, where either i) $N$ is a non-hyperfinite II$_1$ factor and $G$ is an ICC amenable group or ii) $N$ is a weakly rigid II$_1$ factor and $G$ is ICC group and where $G$ acts on $\bar{\bigotimes}_{G}N$ by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting are isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.

W$^*$-Rigidity for the von Neumann Algebras of Products of Hyperbolic Groups

We show that if $Γ= Γ_1\times\dotsb\times Γ_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $Λ$ which is $W^*$-equivalent to $Γ$ decomposes as a $k$-fold direct sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group $Λ$. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II$_1$ factors.

树乘积中随机子集的拟等距刚性

Quasi-isometric rigidity for random subsets in products of trees

本文证明了从两棵正则树的乘积空间中的随机子集到其自身的拟等距嵌入的刚性结果，推广了Eskin的高秩非均匀格点的拟等距刚性，并描述了随机样本的自拟等距嵌入，同时证明两个独立样本几乎必然非拟等距。

图积群的冯·诺伊曼代数的刚性 II. 超刚性结果

Rigidity for von Neumann algebras of graph product groups II. Superrigidity results

本文证明了花状图积群Γ可从任意无穷顶点群的非平凡图积群生成的冯·诺伊曼代数范畴中被完全识别，获得了更精确的C*-代数版本，并将W*-超刚性结论推广至性质(T) wreath-like product群的直积。

由某些半直积产生的群的 von Neumann 代数的不变子代数刚性

Invariant subalgebras rigidity for von Neumann algebras of groups arising as certain semidirect products

本文研究由 $\mathbb{Z}_2$ 直积等 2-挠群上代数作用形成的半直积群的 ISR 性质，构造了首个带非平凡阿贝尔正规子群的 amenable ISR 群，并分析特殊反例及标准 lamplighter 群的不变子代数。

Baumslag-Solitar群、相对profinite完备化与测度等价刚性

Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity

本文为保测度等价关系的非周期包含引入代数不变量，借此证明了非顺从Baumslag-Solitar群直积自由pmp作用的稳定轨道等价刚性，并完成了其相对profinite完备化直积的完全分类。

horospherical products的准等距几何刚性

Geometric rigidity of quasi-isometries in horospherical products

本文证明了超几何空间的horospherical products的准等距在几何上是刚性的，即它们接近于乘积映射。研究了可解李群R ___ (N 1 x N 2)的准等距不变量和分类。

广义Furstenberg边界与中间交叉积的应用

Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products

本文研究了广义Furstenberg边界的刚性性质，并通过引入X-plump子群概念，拓展了关于交叉积的不可约包含和中间代数的结论。

子图排除与平面刚性

Minor exclusion and planar rigidity

本文证明连通局部有限拟传递图若拟等距于排除可数团子图的图则拟等距于平面图，进而得到有限生成群为虚平面的等价条件，完善了正则映射相关的群论结论。

Rigidity for von Neumann algebras and their invariants

We give a survey of recent classification results for crossed product von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II_1 factors with uncountable fundamental groups and the construction of W*-superrigid actions where the crossed product entirely remembers the initial group action that it was constructed from.

规定图灵对称性与各类刚性

Prescribed Graphon Symmetries and Flavors of Rigidity

该研究将Frucht定理推广到图灵范畴，定义并证明了紧群图灵刚性与像刚性的等价性，明确了紧李群图灵刚性的判定条件。