Ergodic Theory and Semisimple Groups

1. Front Matter

   Pages i-x

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2. Introduction

   • Robert J. Zimmer

   Pages 1-7

3. Moore’s Ergodicity Theorem

   • Robert J. Zimmer

   Pages 8-31

4. Algebraic Groups and Measure Theory

   • Robert J. Zimmer

   Pages 32-58

5. Amenability

   • Robert J. Zimmer

   Pages 59-84

6. Rigidity

   • Robert J. Zimmer

   Pages 85-113

7. Margulis’ Arithmeticity Theorems

   • Robert J. Zimmer

   Pages 114-129

8. Kazhdan’s Property (T)

   • Robert J. Zimmer

   Pages 130-148

9. Normal Subgroups of Lattices

   • Robert J. Zimmer

   Pages 149-161

10. Further Results on Ergodic Actions

    • Robert J. Zimmer

    Pages 162-186

11. Generalizations to p-adic groups and S-arithmetic groups

    • Robert J. Zimmer

    Pages 187-193

12. Back Matter

    Pages 194-209

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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 a number 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.

• Arithmetic
• Identity
• Lattice
• algebra
• ergodic theory
• theorem