Reduction Theory and Arithmetic Groups

Joachim Schwermer is Emeritus Professor of Mathematics at the University of Vienna, and recently Guest Researcher at the Max-Planck-Institute for Mathematics, Bonn. He was Director of the Erwin-Schrödinger-Institute for Mathematics and Physics, Vienna from 2011 to 2016. His research focuses on questions arising in the arithmetic of algebraic groups and the theory of automorphic forms.

• 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.