Cohomological Approach to Rigidity in Geometry and Dynamics

几何和动力学中刚性的上同调方法

• 批准号: 0204601
• 负责人: Nicolas Monod
• 金额: $ 7.65万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204601&HistoricalAwards=false
• 关键词: Cohomological; Approach; Rigidity; Geometry; Dynamics

The PI plans to establish rigidity and finiteness results forinfinite groups, both in geometric and dynamical contexts. In anattempt to broaden the scope of rigidity as initiated by Mostow,Margulis, Zimmer and others, we aim at results for very general(finitely generated) groups, notably fundamental groups arisingin geometry. The proposed methods are however also new andrelevant for the more classical study of arithmetic andS-arithmetic groups. There are three aspects to the project.The first is to construct cohomological invariants characterizingvarious geometric situations; a typical example is theinterpretation of negative curvature in terms of boundedcohomology that we propose with Shalom. The second step is todevelop a suitable theory to handle these invariants in anefficient way, notably by using tools from ergodic theory. Last,one has to apply this to concrete situations. We focus inparticular on superrigidity and cocycle superrigidity fornon-linear groups and on orbit equivalence of ergodic actions.The last forty years have witnessed the birth and remarkabledevelopment of what is now called ''rigidity theory''. The firstdiscovery in that field was this: Even though a flat space (whichis the kind of space one thought we live in, from the ancientGreeks to Newton) can be deformed in many ways, some curvedspaces (whose relevance to our world has been discovered byEinstein) are in a sense rigid. Later came a discovery regardedas yet much more fundamental: The most symmetric geometric spacesturn out to be in fact arithmetic objects. The project underconsideration here is to broaden the scope this rigidity theoryto many more geometric spaces, and especially to deal withsituations in which the geometry cannot be reduced toarithmetics. Actually, we propose new methods that stand out bytheir abstract nature and allow us to tackle situations which donot even stem from geometry, but rather regard the study ofdynamics. (Dynamics is the theory of systems that evolve intime, typically ocean currents or galaxies, and may have chaoticbehaviours.) Our project proposes new methods for obtaining moreresults both in this dynamical setting and in geometry, and thisin a unified way.

PI计划建立无限群的刚性和有限性结果，在几何和动力学环境中都是如此。为了扩大Mostow，Marguis，Zimmer等人开创的刚性的范围，我们的目标是非常一般的(有限生成的)群的结果，特别是几何中出现的基本群。然而，所提出的方法对于算术与算术群这一更经典的研究也是新的和相关的。这个计划有三个方面。第一个是构造上同调不变量来刻画各种几何情况；一个典型的例子是我们用Shalom提出的有界上同调来解释负曲率。第二步是开发一个合适的理论，以高效的方式处理这些不变量，特别是使用遍历理论中的工具。最后，人们必须将这一点应用于具体情况。我们特别关注非线性群的超刚性和余圈超刚性，以及遍历作用的轨道等价。在过去的四十年里，见证了现在被称为‘刚性理论’的诞生和显著的发展。该领域的第一个发现是：尽管平坦的空间(从古希腊人到牛顿，人们认为我们所居住的空间)可以在许多方面变形，但一些曲线空间(与我们的世界相关的是爱因斯坦发现的)在某种意义上是刚性的。后来有了一个更基本的发现：最对称的几何空间实际上是算术对象。这里正在考虑的项目是将这个刚性理论的范围扩大到更多的几何空间，特别是处理几何不能归结为算术的情况。实际上，我们提出了新的方法，这些方法因其抽象的性质而脱颖而出，并允许我们处理甚至不是源于几何学的情况，而是考虑到动力学的研究。(动力学是关于随时间演化的系统的理论，通常是洋流或星系，可能会有混乱的行为。)我们的项目提出了新的方法来获得更多的结果，无论是在这个动力学背景下，还是在几何上，这都是以统一的方式进行的。