Rigidity of Actions of Higher Rank Lattices

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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).