Rigidity of Actions of Higher Rank Lattices

连通半单李群及其离散群在紧致流形上的作用在许多方面反映了群的性质。当群具有较高的实数阶时，群的局部刚性和超刚性往往有助于群的作用的相应性质。我们猜想，在某种程度上，高阶格的所有遍历保体积作用都可以由一系列标准作用构成：(A)黎曼流形上的等距作用，(2)紧致零流形上的自同构作用，以及(3)齐次空间上的左移作用。特别是，Anatole Katok，James Lewis和Robert Zimmer猜想，对于任何高秩格作用，在开的稠密集上存在光滑的不变联系。本研究的目的是建立标准行动的局部刚性，并在有利条件下建立群体行动的代数性。研究人员将利用齐默尔的上循环超刚性来建立作用在可测切向水平上的刚性性质。然后，他将应用动力系统技术来获得拓扑切向层次上的刚性性质。从拓扑切向光滑层的过渡--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、调和分析和微分几何中的技术的使用。他研究的另一个目标是找到群体行动的光滑不变联系。当保持适当的结构时，群作用是作用于其上的对象的对称性群的一部分。如果该对象有足够多的对称性，人们就认为该对象在某种意义上是正则的。人们可以进一步想象，如果对称性有许多关系，就没有办法使物体变形并保持这些对称性。所提出的研究将对象视为流形。研究者主要研究了两个问题。第一个问题是当对称性较多时，寻找对称性的结构。第二种是确定流形不能变形的对称性。其意义在于为群体作用的刚性研究提供统一的途径，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，较高等级的群体作用一定是结构的对称性群(所谓的联络)的一部分。

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).