Rigidity Aspects of Ergodic Actions of Lattices in Semisimple Lie Groups

（高阶）半单李群中的Katok/Furman格具有强刚性（Mostow）和超刚性（Margulis, Corlette）等显著性质，它们用环境李群的线性表示来描述这些离散群的线性表示。该提案的主题之一是理解强刚性和超刚性到一般局部紧（而不是线性或李）群中取值的表示框架的可能推广。另一个研究方向是研究有限测度保持群作用的轨道结构（但后来证明是一个密切相关的方向）。证明了高阶格的遍历有限保测度作用具有刚性的轨道结构，使得群（即格）和作用可以仅由生成的轨道关系来重构。进一步的问题和问题出现在分析这些轨道刚性作用。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，或者更一般地说：概率和代数。这种结合，即在李群中使用遍历理论方法和在遍历理论中使用李群实例，已经证明自己在两个领域中都是非常富有成效的，通过解决问题和建立新现象，这些问题在一个领域的范围内自然出现，但解决这些问题需要另一个领域的技术和思想。提出的研究重点是所谓的某些结构的刚性方面（包括遍历理论和李群）。这里的刚性是指承载着某种结构的物体，如果不对这种结构进行实质性的改变，就不能改变它。这特别意味着对象本身可以被这个结构重构（或确定对象）。许多显著的刚性结果是已知的，并广泛用于李群中的晶格，并且该提案涉及这些结果的几个自然推广。

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.