**AMS/IP Studies in Advanced Mathematics**

Volume: 43;
2008;
259 pp;
Softcover

MSC: Primary 11; 22;

Print ISBN: 978-0-8218-4866-1

Product Code: AMSIP/43.S

List Price: $65.00

AMS Member Price: $52.00

MAA Member Price: $58.50

**Electronic ISBN: 978-1-4704-3833-3
Product Code: AMSIP/43.E**

List Price: $65.00

AMS Member Price: $52.00

MAA Member Price: $58.50

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#### Supplemental Materials

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

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*Lizhen Ji*

A co-publication of the AMS and International Press of Boston

In one guise or another, many mathematicians are familiar with certain
arithmetic groups, such as \(\mathbf{Z}\) or
\(\mathrm{SL}(n,\mathbf{Z})\). 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.

Titles in this series are co-published with International Press of Boston, Inc., Cambridge, MA.

#### Readership

Graduate students interested in arithmetic groups and their applications to number theory, geometry and topology.

#### Reviews & Endorsements

...the author deserves credit for having done the tremendous job of encompassing every aspect of arithmetic groups visible in today's mathematics in a systematic manner; the book should be an important guide for some time to come.

-- Mathematical Reviews

# Table of Contents

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

- Cover Cover11
- Title page iii4
- Dedication v6
- Frontispiece vi7
- Contents vii8
- Preface xiii14
- Acknowledgments xv16
- Introduction 120
- General comments on references 524
- Examples of basic arithmetic groups 726
- General arithmetic subgroups and locally symmetric spaces 3756
- Discrete subgroups of Lie groups and arithmeticity of lattices in Lie groups 5776
- Different completions of โ and ๐-arithmetic groups over number fields 6988
- Global fields and ๐-arithmetic groups over function fields 7392
- Finiteness properties of arithmetic and ๐-arithmetic groups 7594
- Symmetric spaces, Bruhat-Tits buildings and their arithmetic quotients 81100
- Compactifications of locally symmetric spaces 87106
- Rigidity of locally symmetric spaces 97116
- Automorphic forms and automorphic representations for general arithmetic groups 115134
- Cohomology of arithmetic groups 127146
- ๐พ-groups of rings of integers and ๐พ-groups of group rings 135154
- Locally homogeneous manifolds and period domains 139158
- Non-cofinite discrete groups, geometrically finite groups 147166
- Large scale geometry of discrete groups 151170
- Tree lattices 165184
- Hyperbolic groups 169188
- Mapping class groups and outer automorphism groups of free groups 173192
- Outer automorphism group of free groups and the outer spaces 179198
- References 183202
- Index 245264
- Back Cover Back Cover1279