Cohomological Approach to Rigidity in Geometry and Dynamics

PI计划建立几何和动力学背景下无限群的刚性和有限性结果。为了扩大Mostow，Margulis， Zimmer等人提出的刚性范围，我们的目标是非常一般（有限生成）群的结果，特别是几何中出现的基本群。然而，所提出的方法对于更经典的算术和非算术群的研究也是新的和相关的。这个项目有三个方面。首先是构造表征各种几何情形的上同不变量；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。第二步是发展一种合适的理论，以一种有效的方式处理这些不变量，特别是通过使用遍历理论中的工具。最后，我们必须将其应用于具体情况。我们特别关注非线性群的超刚性和循环超刚性以及遍历作用的轨道等价。在过去的四十年里，我们见证了所谓的“刚性理论”的诞生和显著发展。该领域的第一个发现是这样的：尽管一个平坦的空间（从古希腊到牛顿，人们认为我们生活的那种空间）可以以多种方式变形，但一些弯曲的空间（与我们的世界的相关性已被爱因斯坦发现）在某种意义上是刚性的。后来出现了一个被认为是更基本的发现：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种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简化为算术的情况。实际上，我们提出的新方法以其抽象的本质脱颖而出，使我们能够处理甚至不源于几何的情况，而是考虑动力学的研究。（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，并且可能具有混沌行为。）我们的项目提出了新的方法，以获得更多的结果，在这种动态设置和几何，这是一个统一的方式。

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.