AnIntroductiontoCocycleSuper-Rigidity-Washington.PDFVIP

An Introduction to Cocycle Super-Rigidity Renato Feres Washington University, St. Louis MO 63130, USA Abstract. The cocycle super-rigidity theorem is a central result in the study of dynamics of semisimple Lie groups and lattices. We give an overview of the main ideas centered on this theorem and some of its most immediate applications. The emphasis will be on the topological and differentiable (as opposed to measurable) aspects of the theory. 1 Introduction The dynamical study of actions of semisimple Lie groups is a subject of present research whose sources and motivations come from a wide range of topics such as the geometry and topology of spaces of non-positive curva- ture, linear representation of lattice groups, the theory of random walks on groups, to cite a few. Of central importance to this field are the celebrated super-rigidity and arithmeticity theorems of Margulis. These theorems are fundamental for describing the structure and linear representations of lattice subgroups of semisimple Lie groups (see [8], as well as [9]). The super-rigidity theorem was extended into the nonlinear setting of G-spaces by Zimmer, whose cocycle super-rigidity theorem is now an essential tool for the develop- ment of the ergodic theory of actions of semisimple groups and their lattice subgroups (cf. [11]). The purpose of these notes is to provide a brief overview of some of the main ideas centered on the cocycle super-rigidity theorem and some of the connections this theorem has with differential geometry and dynamical systems. Differently from [11], which is concerned mainly with measurable ergodic theory, we will emphasize the topological and differentiable aspects of the theory, along the lines of [5]. Sections 1, 2 and 3 give some motivation for the main theorem by sh