Isometric group actions on Hilbert spaces: structure of orbits
Yves de Cornulier, Romain Tessera, Alain Valette
我们的主要结果是：在具有稠密轨道的无限维希尔伯特空间上，一个生成的幂零群没有等距作用.相反，我们构造这样一个行动与一个可生成的亚阿贝尔群。
Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.
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Isometric group actions on Hilbert spaces: growth of cocycles
本文研究了局部紧群的1-上圈的增长性，其值为酉表示。讨论了线性增长的1-上圈的存在性，对于一类包含多环群和连通顺从李群的顺从群G，得到了如下结果：要么G在希尔伯特空间中没有拟等距嵌入，要么G在某个欧几里得空间上有一个真的上紧作用.另一方面，注意到几乎coboundaries（即1-cocycles近似有界1-cocycles）有次线性增长，我们讨论了相反的，这原来是持有顺从的群体与“控制”福尔纳序列;对于一般顺从的群体，我们证明了较弱的结果，1-cocycles与足够小的增长几乎coboundaries。此外，我们还证明了在a-T-可列群上存在具有任意小增长的真上圈。
The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine…
Compact operators and algebraic $K$-theory for groups which act properly and isometrically on Hilbert space
我们证明了$K$理论Farrell-Jones猜想的群的标题系数环和$C^*$ -代数是稳定的关于紧算子。我们使用这一点和希森-卡斯帕罗夫的结果，鲍姆-康纳斯猜想与系数持有这样的群体，以表明，如果$G$是在标题然后的代数和$C^*$ -交叉产品的$G$与一个稳定的$C^*$ -代数有相同的$K$ -理论。
Isometric orbit equivalence for probability-measure preserving actions
本文通过图化研究有限生成群的保测作用，引入等距轨道等价概念，证明了其对特定群具有刚性，并构造了自由群的非平凡示例，表明混合性在该等价下非不变量。
Irreducible affine isometric actions on Hilbert spaces
本文系统地研究了希尔伯特空间上局部紧群的不可约仿射等距作用。事实证明，虽然这是一些平行的研究到现在的经典理论的不可约酉表示，这两个理论不同的几个方面（例如，直接和两个不可约仿射行动仍然可以是不可约的）。我们使用的主要工具之一是舒尔引理的仿射版本，其特征在于仿射等距群作用的不可约性。这使我们能够描述例如幂零群的不可约仿射等距作用。作为另一个应用，对Neretin的下述结果：局部紧群的不可约仿射作用对余紧格的限制仍然是不可约的，给出了一个简短的证明.给出了固定的酉表示是不可约仿射作用的线性部分的一个充要条件。特别地，当酉表示是离散群G的正则表示的倍数时，我们证明了这个问题与G的L2-贝蒂数的关系。在给出不可约仿射作用直和不可约的一个充要条件后，我们得到了如下超刚性结果：如果G是两个或多个局部紧群的乘积，则G中任一不可约余紧格的不可约仿射作用扩张为G的一个仿射作用，只要该作用的线性部分不弱包含平凡表示.
Quasi-actions whose quasi-orbits are quasi-isometric to trees
本文给出了任意群在度量空间上的拟作用可归约为有界度数树上有界等距作用的充要条件，并证明若拟作用度量正则且拟轨道与树拟等距，则该群是虚拟自由的。
Compression of uniform embeddings into Hilbert space
如果试图将一个度量空间均匀嵌入到希尔伯特空间中，这种嵌入能有多接近拟等距？我们回答这个问题的有限维CAT（0）立方复形和双曲群。特别地，我们证明了任何双曲群的希尔伯特空间压缩为1。
We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $σ$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).
We investiguate a property of affine isometric actions on Hilbert spaces called evanescence. Evanescent actions are the extreme opposite of irreducible actions. Every affine isometric action decomposes naturally into an evanescent part and an irreducible part. We study when this decomposition is unique. We also study when an action that has almost fixed points is automatically evanescent. We relate these questions to the identity excluding property for groups. We also relate them to the finiteness of the von Neumann algebras generated by the linear part of the action.
Global and local boundedness of Polish groups
我们提出了一个全面的理论，波兰群的有界性的发展，主要集中在Roelcke预紧性（下一致性的预紧性）和性质（OB）（可分度量空间上的所有等距作用的有界性）。特别是，这些属性的特点是轨道结构的等距行动的度量空间和等距或连续仿射表示可分离的巴拿赫空间。
Groups that (do not) act isometrically on hyperbolic spaces
本文研究群在良好双曲空间上的等距作用，证明非初等作用可导出 L^p 空间上的无界仿射作用，并由此推出具有不动点性质 F_∞ 的群在该类空间上的等距作用必有有界轨道。
Answering a question by Chatterji--Druţu--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0<p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occure when the linear part comes from a measure preserving action, or more generally a sta…
Fixed point-free isometric actions of topological groups on Banach spaces
通过仿射等距证明了在适当的巴拿赫空间上，每个非预紧拓扑群都存在一个无不动点的连续作用。因此，准紧群是由巴拿赫空间上仿射等距作用的不动点性质定义的。对于可分拓扑群，在上面的陈述中，考虑一个特定巴拿赫空间上的仿射作用就足够了：普适Urysohn度量空间的唯一巴拿赫空间包络，称为霍姆斯空间。同时，我们证明了波兰群不需要在巴拿赫空间上（甚至在完全度量空间上）具有拓扑真的（特别是自由的）仿射等距作用：对于具有强算子拓扑的可分无限维希尔伯特空间的酉群、无限对称群等，情况就是这样。
Actions of acylindrically hyperbolic groups on $\ell^1$
本文为具有双曲特征的群构造了 $\ell^1$ 和 $L^1$ 上的仿射一致 Lipschitz 作用，并证明对 acylindrically hyperbolic 群轨道无界，对剩余有限双曲群和映射类群轨道为真且诱导度量与字度量拟等距。
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。
A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p>1$, say that $G$ has property $BP_{L^p}$ if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and $BP_{L^p}$ and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix by Corina Ciobotaru provides new examples of groups with property PL, inc…
Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1…
We say that a finitely generated group $G$ has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the $\ell^1$-metric. As an application of the projection complex techniques, we prove that residually finite hyperbolic groups and mapping class groups have (QT).
On purely loxodromic actions
我们构造了一个例子的等距作用的$F(a,b)$上的$δ$ -双曲图$Y$，使这个行动是acylinertium，纯粹斜驶，有渐近平移长度的非平凡元素的$F(a,b)$分离远离$0$，有拟凸轨道的$Y$，但这样的轨道映射$F(a,b)\to Y$是不是一个拟等距嵌入。
Let $G$ be a compactly generated locally compact group and $(π, \mathcal H)$ a unitary representation of $G.$ The $1$-cocycles with coefficients in $π$ which are harmonic (with respect to a suitable probability measure on $G$) represent classes in the first reduced cohomology $\bar{H}^1(G,π).$ We show that harmonic $1$-cocycles are characterized inside their reduced cohomology class by the fact that they span a minimal closed subspace of $\mathcal H.$ In particular, the affine isometric action given by a harmonic cocycle $b$ is irreducible (in the sense that $\mathcal H$ contains no non-empty,…
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