Rigidity of Actions of Higher Rank Lattices

高阶格子作用的刚性

• 批准号: 9703770
• 负责人: Gregory Margulis
• 金额: $ 8.67万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703770&HistoricalAwards=false
• 关键词: Rigidity; Actions; Higher; Rank; Lattices

Actions of connected semi-simple Lie groups and their discrete groups on compact manifolds reflect in many ways the properties of the groups. When the groups are of higher real rank, local rigidity and superrigidity of the groups often contribute to the corresponding properties for actions of the groups. It is conjectured that to some extent, all ergodic, volume-preserving actions of higher rank lattices can be built up from a list of standard actions: (a) actions by isometries on Riemannian manifolds, (2) actions on compact nilmanifolds by automorphisms, and (3) actions on homogeneous spaces by left translations. In particular, Anatole Katok, James Lewis, and Robert Zimmer conjectured that for any higher rank lattice action, there is a smooth invariant connection on an open dense set. The objectives of this research are to establish the local rigidity of the standard actions and to establish the algebraicity of group actions under favorable conditions. The investigator will employ Zimmer's cocycle superrigidity to establish the rigidity properties of the actions in measurable tangential level. Then he will apply dynamical systems techniques to obtain the rigidity properties in topological tangential level. The transition from topological tangential level to smooth level -- which is the key step to his objectives -- will rely on the use of techniques from dynamical systems, Lie theory and representation theory, ergodic theory, harmonic analysis and differential geometry. Another goal of his research is to find smooth invariant connections for the group actions. A group action, when preserving some appropriate structures, is a part of the group of symmetries of the object on which they act. One anticipates that the object is in some sense regular if there are sufficiently many symmetries of the object. One may further imagine that if the symmetries have many relations, there is no way to deform the object and keep those symmetries. The proposed research takes the objects to be manifolds. T he researcher investigates essentially two problems. The first is to find the structure for the symmetries when there are many symmetries. The second is to identify the symmetries for which the manifolds cannot be deformed. The significance is to provide unified approaches to the research of the rigidity of group actions, develop new methods to tackle the often encountered problems in this field of research, and ultimately establish that under certain conditions, a higher rank group action must be a part of the group of symmetries for a structure (so-called connection).

连通半单李群及其离散群在紧流形上的作用在许多方面反映了群的性质。当群的实阶较高时，群的局部刚性和超刚性往往导致群的行为具有相应的性质。我们推测，在一定程度上，所有高阶格的遍历、保体积作用都可以由一系列标准作用建立起来：(a)黎曼流形上的等距作用，(2)紧零流形上的自同构作用，(3)齐次空间上的左平移作用。特别地，Anatole Katok， James Lewis和Robert Zimmer推测对于任何高阶格作用，在开密集集中存在光滑不变连接。本研究的目的是建立标准作用的局部刚性，并建立有利条件下群作用的代数性。研究者将采用齐默循环超刚性来建立作用在可测量切向水平上的刚性特性。然后，他将应用动力系统技术来获得拓扑切向水平上的刚性特性。从拓扑切向水平到平滑水平的过渡——这是实现他的目标的关键一步——将依赖于动力系统、李理论和表示理论、遍历理论、谐波分析和微分几何等技术的使用。他的另一个研究目标是寻找群体行为的光滑不变连接。群作用，当保留一些适当的结构时，是它们所作用的对象的对称群的一部分。如果物体有足够多的对称性，人们就会认为这个物体在某种意义上是规则的。人们可以进一步想象，如果对称有许多关系，就没有办法使物体变形而保持这些对称。本研究将对象视为流形。研究者主要研究两个问题。首先是在存在许多对称的情况下找到对称的结构。第二步是确定流形不能变形的对称性。其意义在于为群体行为刚性的研究提供统一的方法，发展新的方法来解决这一研究领域中经常遇到的问题，并最终确立在一定条件下，更高等级的群体行为必须是一个结构的对称群的一部分（所谓的连接）。