Ergodic Theory and Semisimple Groups

This book is based on a course given at the University of Chicago in 1980-81. As with the course, the main motivation of this work is to present an accessible treatment, assuming minimal background, of the profound work of G. A. Margulis concerning rigidity, arithmeticity, and structure of lattices in semi simple groups, and related work of the author on the actions of semisimple groups and their lattice subgroups. In doing so, we develop the necessary prerequisites from earlier work of Borel, Furstenberg, Kazhdan, Moore, and others. One of the difficulties involved in an exposition of this material is the continuous interplay between ideas from the theory of algebraic groups on the one hand and ergodic theory on the other. This, of course, is not so much a mathematical difficulty as a cultural one, as the number of persons comfortable in both areas has not traditionally been large. We hope this work will also serve as a contribution towards improving that situation. While there are anumber of satisfactory introductory expositions of the ergodic theory of integer or real line actions, there is no such exposition of the type of ergodic theoretic results with which we shall be dealing (concerning actions of more general groups), and hence we have assumed absolutely no knowledge of ergodic theory (not even the definition of "ergodic") on the part of the reader. All results are developed in full detail.

1. Introduction. - 2. Moore s Ergodicity Theorem. - 3. Algebraic Groups and Measure Theory. - 4. Amenability. - 5. Rigidity. - 6. Margulis Arithmeticity Theorems. - 7. Kazhdan s Property (T). - 8. Normal Subgroups of Lattices. - 9. Further Results on Ergodic Actions. - 10. Generalizations to p-adic groups and S-arithmetic groups. - Appendices. - A. Borel spaces. - B. Almost everywhere identities on groups. - References.