Arithmetic Groups and Their Generalizations: What, Why, and How

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Overview

In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as $\mathbf{Z}$ or $\textrm{SL}(n,\mathbf{Z})$. Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known. Indeed, arithmetic groups admit many natural and important generalizations. The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry. It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics.
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Product Details

  • ISBN-13: 9780821846759
  • Publisher: American Mathematical Society
  • Publication date: 6/12/2008
  • Series: AMS/IP Studies in Advanced Mathematics
  • Pages: 259
  • Product dimensions: 7.10 (w) x 10.20 (h) x 0.80 (d)

Table of Contents

Ch. 1 Introduction 1

Ch. 2 General comments on references 5

Ch. 3 Examples of basic arithmetic groups 7

Ch. 4 General arithmetic subgroups and locally symmetric spaces 37

Ch. 5 Discrete subgroups of Lie groups and arithmeticity of lattices in Lie groups 57

Ch. 6 Different completions of Q and S-arithmetic groups over number fields 69

Ch. 7 Global fields and S-arithmetic groups over function fields 73

Ch. 8 Finiteness properties of arithmetic and S-arithmetic groups 75

Ch. 9 Symmetric spaces, Bruhat-Tits buildings and their arithmetic quotients 81

Ch. 10 Compactifications of locally symmetric spaces 87

Ch. 11 Rigidity of locally symmetric spaces 97

Ch. 12 Automorphic forms and automorphic representations for general arithmetic groups 115

Ch. 13 Cohomology of arithmetic groups 127

Ch. 14 K-groups of rings of integers and K-groups of group rings 135

Ch. 15 Locally homogeneous manifolds and period domains 139

Ch. 16 Non-cofinite discrete groups, geometrically finite groups 147

Ch. 17 Large scale geometry of discrete groups 151

Ch. 18 Tree lattices 165

Ch. 19 Hyperbolic groups 169

Ch. 20 Mapping class groups and outer automorphism groups of free groups 173

Ch. 21 Outer automorphism group of free groups and the outer spaces 179

References 183

Index 245

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