Reduction Theory and Arithmetic Groups

Build a solid foundation in the area of arithmetic groups and explore its inherent geometric and number-theoretical components.

Joachim Schwermer (Author)

9781108832038, Cambridge University Press

Hardback, published 15 December 2022

374 pages
25.1 x 17.6 x 2.5 cm, 0.8 kg

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.

Part I. Arithmetic Groups in the General Linear Group: 1. Modules, lattices, and orders
2. The general linear group over rings
3. A menagerie of examples – a historical perspective
4. Arithmetic groups
5. Arithmetically defined Kleinian groups and hyperbolic 3-space
Part II. Arithmetic Groups Over Global Fields: 6. Lattices – Reduction theory for GLn
7. Reduction theory and (semi)-stable lattices
8. Arithmetic groups in algebraic k-groups
9. Arithmetic groups, ambient Lie groups, and related geometric objects
10. Geometric cycles
11. Geometric cycles via rational automorphisms
12. Reduction theory for adelic coset spaces
Appendices: A. Linear algebraic groups – a review
B. Global fields
C. Topological groups, homogeneous spaces, and proper actions
References
Index.

Subject Areas: Number theory [PBH]