Rigidity property of diagram groups
Victor Guba, Mark Sapir
我们展示了两个经典对象之间的刚性连接：R.Thompson群F和Dunce hat。有向2-复形的图群包含F的副本当且仅当复形包含Dunce hat的副本。
We show a rigid connection between two classical objects: the R.Thompson group F and the Dunce hat. A diagram group of a directed 2-complex contains a copy of F if and only if the complex contains a copy of the Dunce hat.
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Diagram rigidity for geometric amalgamations of free groups
证明了二维单厚双曲P-流形的一个拓扑刚性结果：基本群的同构蕴涵P-流形的同胚。一个直接的应用是一个图刚性定理的某些合并的自由群：直接限制的两个这样的图是同构的当且仅当有一个同构之间的各自的图。
An addendum to `A rigidity property for the set of all characters induced by valuations'
这纠正和放大了罗伯特·比埃里和上述标题的作者在一篇论文中提出的一些评论。
A Rigidity Property of Some Negatively Curved Solvable Lie Groups
证明了对于某些负曲可解李群，所有的自拟等距都是几乎等距的。我们证明了这一点，通过显示，所有的自拟对称映射的理想边界（的可解李群）是bilipschitz关于视觉度量。我们还定义了Gromov双曲空间理想边界上的抛物视觉度量，并将它们与视觉度量联系起来。
Almost isometric actions, property $(T)$, and local rigidity
设$Γ$是具有Kazhdan性质$(T)$的离散群.证明了紧致流形$X$上$Γ$的任意黎曼等距作用是局部刚性的.我们还证明了这个结果的一个更一般的叶状版本。本文利用这个结果证明了高秩半单李群及其格在\cite{FM2}中的标准作用的局部刚性。性质$(T)$的一个定义是，如果希尔伯特空间上的每个等距$Γ$作用都有一个不动点，则群$Γ$具有性质$(T)$。我们证明了具有性质$(T)$的群的不动点性质的各种加强。其中一些被用于证明我们的局部刚性定理。
Quasi-homomorphism rigidity with noncommutative targets
作为对局部紧群的Kazhdan性质（T）的加强，Burger和Monod引入了（TT）性质.在本文中，我们增加了更多的刚性和引入性质（TTT）。这个性质适用于研究具有非交换目标的拟同态的刚性现象。部分推广了Burger和Monod的一个结果，证明了n ≥ 3的SL（n，R）及其格具有（TTT）性质.作为一个推论，我们推广了一个众所周知的事实，即从这样的格到顺从群或双曲群的每个同态都有有限的图像，它包括一个拟同态。
We say that a countable discrete group $Γ$ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every $Γ$- invariant von Neumann subalgebra $\mathcal{M}$ in $L(Γ)$ is of the form $L(Λ)$ for some normal subgroup $Λ\lhd Γ$. We show many ``negatively curved" groups, including all torsion free non-amenable hyperbolic groups and torsion free groups with positive first $L^2$-Betti number under a mild assumption, and certain finite direct product of them have this property. We also discuss whether the torsion-free assumption can be relaxed.
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…
Profinite Non-Rigidity of Arithmetic Lattices and the Kähler Property
本文研究算术格的非有限刚性，刻画了允许非同构实形式的高秩李群绝对Dynkin类型，并证明有限呈现剩余有限群的Kähler性质不由其有限完备化决定。
Property (T) and rigidity for actions on Banach spaces
本文研究了$L^p$及其它巴拿赫空间上作用的不动点性质。我们证明了当$L^2$被$L^p$（甚至是$L^p$的子空间/商）取代时，性质（T）成立，并且事实上它独立于$1\leq p<\infty$。本文证明了当$L^p$.对于单李群及其格，我们证明了$L^p$的不动点性质对任何$1< p<\infty$成立当且仅当秩至少为2.最后，我们得到了超自反巴拿赫空间上不可约格在一般群的乘积中作用的一个超刚性结果。
Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $Γ$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely, that there are infinitely $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. Also, we show that any lattice in $G$ has the $R_\infty$-property, extending our earlier result for irreducible lattices.
Rigidity of two-dimensional Coxeter groups
一个考克斯特系统称为二维的，如果它的相关戴维斯复形是二维的（等价地，每个球面子群的秩小于或等于2）。我们证明了给定一个二维系统（W，S）和任何其它产生相同反射的系统（W，S '），对应于这些系统的图是同构的，直到由N定义的图扭曲运算。作者：J. Muhlherr和W.诺依曼作为证明这一结果的一个步骤，我们证明了某些二维群在上述作者定义的意义下是反射刚性的，并在最后一节给出了关于二维系统强刚性的一个结果。
On quasi-homomorphism rigidity for lattices in simple algebraic groups
本文改进了Ozawa的结果，证明了局部域上秩$\geq 2$的单代数群及第二可数局部紧群中的格具有property $(TTT)$，并探讨了具有无限中心的Lie群对该性质的缺失。
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.…
Symmetry and Rigidity of Star-Shaped Coxeter Systems
本文研究了具有星形有限Coxeter图的Coxeter群的自同构群，证明了每个自同构可分解为内自同构、图自同构以及三种额外类型：转置和两个部分共轭家族，并探讨了自然的短正合列，利用Moussong的超几何性准则证明这些群具有R_∞性质，并利用已知技术建立了这些群的刚性性质，解决了星形Coxeter系统的同构问题。
Rigidity and relative hyperbolicity of real hyperbolic hyperplane complements
当n>3时，我们研究了有限体积完备实双曲n-流形通过去掉一个余维为2的紧致全测地子流形而得到的空间。证明了它们的基本群是相对双曲的、余Hopf的、双自动的、剩余双曲的、非Kähler的、与虚连通实李群中的格不同构的、不存在具有性质（T）的非平凡子群、具有有限的外自同构群、满足Mostow型刚性、具有有限的渐近维数和快速衰减性质、满足Baum-Connes猜想.我们还刻画了实李群中同构于相对双曲群的格。
We introduce and investigate the rigidity property of rank gradient in the case of the group $\mathcal G$ of intermediate growth constructed by the first author. We show that $\mathcal G$ is normally $(f,g)$-RG rigid where $f(n)=\log(n)$ and $g(n) =\log(\log(n)).$
Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions
我们研究等价关系，产生的翻译行动$Γ\curvearrowright G$这是相关的密集嵌入$Γ<G$可数群到第二可数局部紧群。假设$G$是单连通的，且作用$Γ\curvearrowright G$是强遍历的，我们证明了$Γ\curvearrowright G$与另一个这样的平移作用$Λ\curvearrowright H$是轨道等价的当且仅当存在同构$δ:G\rightarrow H$使得$δ(Γ)=Λ$ .如果$G$是一个实代数群，那么我们建立了类似的刚性结果的平移作用的$Γ$的齐性空间的形式$G/Σ$，其中$Σ<G$是一个离散或代数子群。证明了若$G$是单连通的，且作用$Γ\curvearrowright G$具有性质（T），则任何取值于可数群$Λ$的上圈$w:Γ\times G\rightarrowΛ$与一个同态$δ:Γ\rightarrowΛ$上同调.因此，我们推出的行动$Γ\curvearrowright G$是轨道等价的超刚性：任何自由的非奇异行动$Λ\curvearrowright Y$是轨道等价于$Γ\curvearrowright G$，必然共轭的诱导$Γ\curvearrowright G$。
Rigidity for equivalence relations on homogeneous spaces
研究了Popa关于齐性空间上作用诱导的等价关系的刚性概念。对具有有限中心且无紧因子的半单李群$G$中的任意格$Γ,Λ$，我们证明了作用$Γ\curvearrowright G/Λ$是刚性的.如果另外$G$具有性质（T），则我们导出von Neumann代数$L^{\infty}(G/Λ)\rtimesΓ$具有性质（T）.我们还证明了如果$G$在$G$ - $\{0\}$的李代数上的伴随作用是顺从的（例如如果$G=SL_2(\Bbb R)$），则作用$Γ\curvearrowright G/Λ$的轨道等价关系的任何遍历次等价关系要么是超有限的，要么是刚性的.
We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $Γ$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and $ω$ a volume form on $M$ we show that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M,ω)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show any such action has uniform subexponential growth of …
Let $Γ$ be a non-elementary Kleinian group and $H<Γ$ a finitely generated, proper subgroup. We prove that if $Γ$ has finite co-volume, then the profinite completions of $H$ and $Γ$ are not isomorphic. If $H$ has finite index in $Γ$, then there is a finite group onto which $H$ maps but $Γ$ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circleof ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, e.g. limit groups. We const…
Rigidity of graph products of abelian groups
我们证明了，如果$G$是一个组和$G$有一个图的产品分解有限生成的阿贝尔顶点组，那么$G$有两个典型的分解作为一个图的产品组：一个唯一的分解，其中每个顶点组是一个直接不可分解的循环群，和一个唯一的分解，其中每个顶点组是一个有限生成的阿贝尔群和图满足$T_0$性质。我们的结果建立在Droms，Laurence和Radcliffe的结果之上。
We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature. Our main result applies to its possible homomorphisms into other groups of homeomorphisms, and implies that arbitrary homomorphisms between the full groups of a vast class of pseudogroups must extend to continuous morphisms between pseudogroups (in particular giving rise to equivariant maps at the level of spaces). As applications, we obtain explicit obstructions to the existence of embeddings between full groups in terms…
Grothendieck rigidity and virtual retraction of higher-rank GBS groups
本文研究了高阶GBS群的格罗滕迪克刚性与虚拟退缩性质，证明了余有限的GBS群具有格罗滕迪克刚性，并刻画了满足性质（VRC）的条件。
Actions of a locally compact group G on a measure space X give rise to unitary representations of G on Hilbert spaces. We review results on the rigidity of these actions from the spectral point of view, that is, results about the existence of a spectral gap for associated averaging operators and their consequences. We will deal both with spaces X with an infinite measure as well as with spaces with an invariant probability measure. The spectral gap property has several striking applications to group theory, geometry, ergodic theory, operator algebras, graph theory, theoretical computer science…
We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $Λ$ a lattice in $G$, and $Γ$ a subgroup of the affine group Aff$(G)$ stabilizing $Λ$. Then the action of $Γ$ on $G/Λ$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $Γ$ on $\mathbb{P}(\frak{g})$ admits no $Γ$-invariant probability measure, where $\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nil…
Strong rigidity of II$_1$ factors arising from malleable actions of w-rigid groups, II
证明了群测度空间II的任意同构$θ:M_0\simeq M$，$_1$因子，$M_0=L^\infty(X_0, μ_0) \rtimes_{σ_0} G_0$，$M=L^\infty(X, μ) \rtimes_σ G$，其中$G_0$包含具有Kazhdan-Margulis的相对性质（T）的无穷正规子群（即$G_0$ { \it w-rigid}），且$G$是作用于Bernoulli移位$σ$的ICC群，本质上来自共轭作用的概率空间的同构.此外，对于$p\in M$是幂等元，$M_0$的任何同构$θ$到$M$的"角“$pMp$上都迫使$p=1$。特别地，所有与w-刚性ICC群的Bernoulli移位作用相关的群测度空间因子都有平凡的基本群，并且这些因子之间的所有同构都来自于相应群的同构。这解决了康纳斯刚性猜想的“群测度空间版本”，事实上，它比ICC性质（T）群更普遍地成立。我们应用这些结果遍历理论，建立新的强刚性和超刚性结果的轨道等价关系。
Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa)
我们调查索林波帕最近的工作伯努利行动。这篇论文是在布尔巴基研讨会上写的。使用非常原始的方法从算子代数，索林波帕已表明，轨道结构的伯努利行动的性质（T）组，完全记得集团和行动。这些信息甚至基本上包含在交叉积冯诺依曼代数中，产生了文献中的第一个冯诺依曼强刚性定理。同样的方法允许Popa获得II_1因子与规定的可数基本群。
In \cite{Kramer11} Kramer proves for a large class of semisimple Lie groups that they admit just one locally compact $σ$-compact Hausdorff topology compatible with the group operations. We present two different methods of generalising this to the group of rational points of an absolutely quasi-simple algebraic group over a non-archimedean local field (the second method only achieves this on the additional hypothesis that the group is isotropic). The first method of argument involves demonstrating that, given any topological group $G$ which is totally disconnected, locally compact, $σ$-compact,…
Poincaré inequalities and rigidity for actions on Banach spaces
将证明性质（T）的谱方法推广到自反巴拿赫空间，给出了自反Banach空间$X$上的群$G$的仿射等距作用存在不动点的条件.最后一个性质是Kazhdan性质（T）的强版本，并且等价于以下事实：对于$X$上的$G$的每个等距表示$π$，$H^1(G,π)=0$。该条件表示的$p$庞加莱常数，我们提供的例子组，满足这样的条件，其中$H^1(G,π)$消失的每一个等距表示$π$上的$L_p$空间的一些$p>2$。我们的方法允许估计这样的$p$明确，并产生几个有趣的应用。特别地，我们得到了Hilbert空间上系数为一致有界表示的1-上同调消失的定量估计。我们还给出了Gromov密度模型中双曲群边界的共形维数的下界。
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。
Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras
本文利用Rips构造研究性质(T)群的冯·诺依曼代数刚性，证明其代数特征可从代数结构识别，获得新无穷族不同构因子，支持Connes猜想，并构造了无性质(T)极大子代数的例子。
Norm rigidity for arithmetic and profinite groups
本文证明具有有界初等生成的 ${\rm{SL}}_n(A)$ 的共轭不变范数具离散或预紧二分性，并将其与算术群及profinite群的刚性结果相关联，推导出对非剩余有限中心扩张近似的约束。
We give a sufficient condition for parameter rigidity of actions of solvable Lie groups, by vanishing of (uncountably many) first cohomologies of the orbit foliations. In some cases, we can prove that vanishing of finitely many cohomologies is sufficient. For this purpose we use a rigidity property of quasiisometry. As an application we prove some actions of 2-step solvable Lie groups on mapping tori are parameter rigid. Special cases of these actions are considered in a paper of Matsumoto and Mitsumatsu. We also remark on the relation between transitive locally free actions of solvable Lie gr…
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。
Orbit equivalence rigidity
考虑可数群Gamma在概率空间（X，μ）上通过保测变换遍历作用，R_Gamma是X上对应的轨道等价关系.证明了以下刚性现象：存在群作用使得X上的等价关系R_Gamma唯一确定群Gamma和作用（X，mu，Gamma），直到有限群. SL_n（Z）在n-环面R ^n/Z ^n（n> 2）上的自然作用就是这样的例子之一。在von Neumann代数的背景下对这些结果的解释为Connes关于具有性质T的群的群代数的刚性的猜想提供了一些支持。我们的刚性结果也给出了可数的等价关系的第二类，这不能产生（模0）的任何组的自由行动的例子。这对费尔德曼和摩尔长期存在的问题给出了否定的答案。
We consider expansive group actions on a compact metric space containing a special fixed point denoted by $0$, and endomorphisms of such systems whose forward trajectories are attracted toward $0$. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to $0$ in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in p…
Profinite rigidity of fibring
本文研究了有限呈现的LERF群的代数纤维化性质，证明其为profinite属性，并探讨了极限群乘积的纤维化及庞加莱对偶群的profinite刚性。
On cube and Cremona rigidity for higher-rank lattices
研究实秩至少为2的半单李群中不可约格在CAT(0)立方体复形上的作用，通过证明上同调消失结果及利用超刚性等方法，得出相关作用可正则性的结论，绕过对FW性质的直接使用。
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