This book is a slightly expanded version of a series of lectures given by Robert J. Zimmer at a CBMS conference at the University of Minnesota. The notes provide an introduction to the study of the actions of a semisimple Lie group G on a manifold M. In particular, one would like to understand the relations between the algebraic structure of G and the topology of M. The major developments and open problems in the field are discussed in the book, in a concise and relatively accessible fashion.
The actions of G on M can be classified in linear and non-linear. Since the linear ones are quite well explained by the theory of highest weights, the main theme in the book is the study of the non-linear actions.
The authors assume familiarity with the fundamentals of differential geometry, and perhaps a bit more; however, a short appendix provides some background material on more advanced notions, such as the definition of ergodic measure or the Margulis superrigidity theorem . This book is, I think, suitable for graduate students in the field; it is not an introduction to ergodic theory.
Each lecture is a short survey on a topic, for instance: Gromov representation or superrigidity. In each case, the main results are presented (and the proofs of theorems are usually sketched) and an extensive bibliography concludes the chapter. Each lecture is in fact intended as an invitation to further reading (note that the average length of a lecture is 6.3 pages).
I think this book would be very stimulating reading for students starting a PhD in the field, providing a quick and fairly complete overview of how measure-theoretic techniques can be applied to gain geometric understanding of non-linear actions of Lie groups on differential manifolds.
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at firstname.lastname@example.org.