Rigidity Aspects of Ergodic Actions of Lattices in Semisimple Lie Groups

抽象的Katok/Furman （高阶）半单李群中的格具有许多显著的 性质，其中包括强刚性（Mostow）和超刚性 （Margulis，Corlette），它描述了这些离散群的线性表示的环境李群的线性表示。该提案的主要主题之一是理解强刚性和超刚性对 取值于一般局部紧的表示框架 （而不是线性或李）群。 另一个（但事实证明是一个密切相关的）研究方向是研究有限保测度群作用的轨道结构。事实证明，高阶格的遍历有限保测度作用具有这样一个刚性的轨道结构，即群（即格）和作用可以仅仅从所产生的轨道关系中重构。进一步的问题和难题出现在 分析这些轨道刚性的行为。在提出的几个问题上已经取得了一些进展。 拟议的研究涉及两个学科的各种问题： 遍历理论和李群，或者更一般地：概率和代数。 这种结合，即在李群中使用遍历理论方法 在遍历理论中，李群的例子，已经证明了自己， 在这两个领域都非常富有成效，通过解决问题和建立 新的现象，自然出现在一个领域的范围内，但 解决方案需要另一个人的技术和想法。的 建议的研究集中在，所谓的，刚性方面的某些 结构（遍历理论和李群）。这里僵硬 意味着该对象，携带一定的结构，不能改变 而不对这种结构进行实质性的改变。这尤其意味着 物体本身可以被重建（或物体被确定） 通过这种结构。许多可标记的刚度结果是已知的和广泛的 用于李群中的格，该提案解决了几个自然的 这些结果的推广。

Abstract Katok/Furman Lattices in (higher rank) semisimple Lie groups have many remarkable properties, among them strong rigidity (Mostow) and superrigidity (Margulis, Corlette), which describe linear representations of these discrete groups in terms of the linear representations of the ambient Lie groups. One of the main themes of the proposal is to understand possible generalizations of strong rigidity and superrigidity to the framework of representations taking values in general locally compact (rather than linear, or Lie) groups. Another (but as it turns out to be a closely related) line of research is the study of the orbit structure of finite measure-preserving group actions. It turns out that ergodic finite measure-preserving actions of higher rank lattices have such a rigid orbit structure, that the group (i.e. the lattice) and the action can be reconstructed just from the produced orbit relation. Further questions and problems arise in the analysis of these orbitally rigid actions. Some progress has already been obtained in several of the proposed problems. The proposed research addresses various questions which join two disciplines: ergodic theory and Lie groups, or more generally: probability and algebra. This joining, namely the use of ergodic-theoretic methods in Lie groups and Lie groups examples in ergodic theory, has already proved itself as extremely fruitful in both fields, by solving problems and establishing new phenomena, which appeared naturally within the scope of one filed, but solutions of which required techniques and ideas from another one. The proposed research focuses on, so called, rigidity aspects of certain structures (both in ergodic theory and in Lie groups). Here rigidity means that the object, carrying certain structure, cannot be changed without a substantial change in this structure. This in particular means that the object itself can be reconstructed (or the object is determined) by this structure. Many rem arkable rigidity results are known and widely used for lattices in Lie groups, and the proposal addresses several natural generalizations of these results.