Reduction Theory and Arithmetic Groups

Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring of integers of an algebraic number field. They are rich, diverse structures and they arise in many areas of study. This text enables you to build a solid, rigorous foundation in the subject. It first develops essential geometric and number theoretical components to the investigations of arithmetic groups, and then examines a number of different themes, including reduction theory, (semi)-stable lattices, arithmetic groups in forms of the special linear group, unipotent groups and tori, and reduction theory for adelic coset spaces. Also included is a thorough treatment of the construction of geometric cycles in arithmetically defined locally symmetric spaces, and some associated cohomological questions. Written by a renowned expert, this book is a valuable reference for researchers and graduate students.

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Reduction Theory and Arithmetic Groups

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-xiv

Part I - Arithmetic Groups in the General Linear Group
pp 1-2

1 - Modules, Lattices, and Orders
pp 3-45

2 - The General Linear Group over Rings
pp 46-69

3 - A Menagerie of Examples: A Historical Perspective
pp 70-89

4 - Arithmetic Groups
pp 90-121

5 - Arithmetically Defined Kleinian Groups and Hyperbolic 3-Space
pp 122-138

Part II - Arithmetic Groups Over Global Fields
pp 139-140

6 - Lattices: Reduction Theory for GLn
pp 141-151

7 - Reduction Theory and (Semi)-Stable Lattices
pp 152-173

8 - Arithmetic Groups in Algebraic k-Groups
pp 174-201

9 - Arithmetic Groups, Ambient Lie Groups, and Related Geometric Objects
pp 202-236

10 - Geometric Cycles
pp 237-248

11 - Geometric Cycles via Rational Automorphisms
pp 249-273

12 - Reduction Theory for Adelic Coset Spaces
pp 274-296

Part III - Appendices
pp 297-298

Appendix A Linear Algebraic Groups: A Review
pp 299-325

Appendix B Global Fields
pp 326-334

Appendix C Topological Groups, Homogeneous Spaces, and Proper Actions
pp 335-349

References
pp 350-356

Index
pp 357-360

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Reduction Theory and Arithmetic Groups

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