This intermediate course is designed as a comprehensive introduction to one of the most active and attractive fields of contemporary mathematics with a remarkable range of applications to Lie groups, geometry, physics and coding theory.

Prerequisites: basic algebra, linear algebra, elementary hyperbolic geometry and an interest in learning new things by solving a variety of problems.

Curriculum:

1. Discrete transformation groups.Fundamental domain. Quotients by discrete groups.
2. Cocompact discrete groups. Some instructive classical examples.
3. Discrete reflection groups in spherical,euclidean and hyperbolic geometries.
4. Polytopes with nonobtuse angles. Coxeter polytopes as fundamental domains.
5. Classification of the finite reflection groups.
6. Abstract Coxeter groups and their braid -friends.
7. Basics on the hyperbolic geometry in the Klein model.
8. Finite Coxeter polytopes in hyperbolic space and associated cocompact discrete reflection groups.
9. Circle packings and Andreev`s theorem (after Thurston)
10. Any cocompact reflection group is a Coxeter group.
11. Tits representation .
12. The root system attached to any Coxeter group.
13. Classification of finite and affine root systems.
14. Combinatorial theory of Coxeter groups.

Textbooks

• S.Lang, Undergraduate algebra, Springer 1989 (first three chapters).
• S.Lang, Algebra,Addis.-Wesley ,1970 (chapters 13,14).
• H.Coxeter, Introduction to geometry, Wiley&Sons, 1970.