论文标题
非阳性曲率的适当近端
Proper proximality in non-positive curvature
论文作者
论文摘要
可数组的适当近端是Boutonnet，Ioana和Peterson引入的一个概念，是研究与群体或千古群体行为相关的某些von Neumann代数的刚性特性的工具。在本文中，我们建立了作用于非弯曲空间的许多群体的适当近端。 首先，其中包括许多可计数的$ g $，在适当的$ \ mathrm {cat}（0）$ space $ x $上，由ISOMERTIOS在非元素上适当地作用。更确切地说，在排名第一的等法存在的情况下，或者$ x $是本地较厚的仿射建筑，$ g $ $ $ g $ - $ g $ action时，适当的近端率保持。 As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary $\mathrm{CAT}(0)$ cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a $2$-dimensional piecewise Euclidean $ \ mathrm {cat}（0）$ Simplicial Complext。 其次，我们建立了许多层次双曲线组的适当近端。其中包括映射可连接的可定向有限型无边界表面（除了几个低复杂案例）的映射类组，从而回答了Boutonnet，Ioana和Peterson提出的一个问题。我们还证明了在曲线图上非元素作用的所有亚组的正确接近性。 鉴于Boutonnet，Ioana和Peterson的工作，我们的结果在与上述所有群体及其雄性作用相关的von Neumann代数的结构和刚性结果中都有应用。
Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups $G$ acting properly nonelementarily by isometries on a proper $\mathrm{CAT}(0)$ space $X$. More precisely, proper proximality holds in the presence of rank one isometries or when $X$ is a locally thick affine building with a minimal $G$-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary $\mathrm{CAT}(0)$ cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a $2$-dimensional piecewise Euclidean $\mathrm{CAT}(0)$ simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions.
