Introduction to Arithmetic Groups (University Lecture) (University Lecture Series) (University Lecture Series, 73)

Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel's Introduction aux groupes arithmétiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces. The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments. This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kazhdan's property (T), and quasi-isometries. Numerous exercises enhance the book's usefulness both as a textbook for a second-year graduate course and for self-study. In addition, notes at the end of each chapter have suggestions for further reading. (Proofs in this book often consider only an illuminating special case.) Readers are expected to have some acquaintance with Lie groups, but appendices briefly review the prerequisite background. A PDF file of the book is available on the internet. This inexpensive printed edition is for readers who prefer a hardcopy. Cover Title page Preface to the English Translation Introduction Notation Chapter I. Some Classical Groups 1. Siegel sets and reduction in GL(n,R) 2. Reduction of positive-definite quadratic forms 3. Bruhat decomposition of GL(n,k) 4. The Siegel property in GL_{n} 5. Reduction of indefinite quadratic forms 6. A finiteness lemma Chapter II. Algebraic Groups 7. A review of algebraic groups. Arithmetic groups 8. Compactness criterion 9. Fundamental sets (first type) Chapter III. Fundamental Sets with Cusps 10. Algebraic tori 11. Parabolic subgroups. Bruhat decomposition 12. Siegel sets 13. Fundamental sets (second type) 14. Fundamental representations. Associated functions 15. The Siegel property 16. Fundamental sets and minima 17. Groups with rational rank one Bibliography Index Back Cover Some Classical Groups -- Algebraic Groups -- Fundamental Sets With Cusps. Armand Borel ; Translated By Lam Laurent Pham ; Translation Edited By Dave Witte Morris. Originally Published In French: Introduction Aux Groupes Arithmétiques (paris : Hermann, 1969). Includes Bibliographical References And Index. Fifty years after it was first published, Armand Borel's Introduction aux groupes arithmetiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory.