Discrete Subgroups of Semisimple Lie Groups

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Springer Science & Business Media, Feb 15, 1991 - Mathematics - 388 pages
Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. Since the notion of "Lie group" is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e.g. classification results on abstract homomorphisms of semi-simple algebraic groups over global fields. The treatise of course contains a presentation of the author's fundamental rigidity and arithmeticity theorems. The work in this monograph requires the language and basic results from fields such as algebraic groups, ergodic theory, the theory of unitary representatons, and the theory of amenable groups. The author develops the necessary material from these subjects; so that, while the book is of obvious importance for researchers working in related areas, it is essentially self-contained and therefore is also of great interest for advanced students.
 

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Contents

Introduction
1
Preliminaries
8
Density and Ergodicity Theorems
79
Property T
107
Factor Groups of Discrete Subgroups
145
Characteristic Maps
168
Discrete Subgroups and Boundary Theory
195
Rigidity
214
Normal Subgroups and Abstract Homomorphisms of Semisimple Algebraic Groups Over Global Fields
258
Arithmeticity
288
Appendices
346
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