Nikolay Bogachev: Geometry, Arithmetic, and Dynamics of Discrete Groups

10th September, 2024

This course is currently ongoing, so not all links will be active.

This is a 22-lecture course, with each lecture being between one and two hours, given by Nikolay Bogachev.

Modern research in the geometry, topology, and group theory often combines geometric, arithmetic and dynamical aspects of discrete groups. This course is mostly devoted to hyperbolic manifolds and orbifolds, but also will deal with the general theory of discrete subgroups of Lie groups and arithmetic groups. Vinberg’s theory of hyperbolic reflection groups will also be discussed, as it provides a lot of interesting examples and methods that turn out to be very practical. One of the goals of this course is to sketch the proof of the famous Mostow rigidity theorem via ergodic methods. Another goal is to talk about very recent results, giving a geometric characterization of arithmetic hyperbolic manifolds through their totally geodesic subspaces, and their applications. Throughout the course we will consider many examples from reflection groups and low-dimensional geometry and topology. In conclusion, I am going to provide a list of open problems related to this course.

Prerequisites: Understanding the basics of group theory (groups, homomorphisms, normal subgroups, quotients, etc.), differential geometry and topology (manifolds, fundamental groups, etc.), measure theory, and number theory (algebraic number fields and rings) is desirable.

Textbooks: There is no required textbook for this course. I will be following this book quite closely, however:

  • D. V. Alekseevskij, É. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature. Geometry II, 1–138. Encyclopaedia Math. Sci., 29.
  • C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds.
  • B. Martelli, Introduction to Geometric Topology.
  • D. Morris, Introduction to Arithmetic Groups.
  • É. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature. Geometry II, 139–248. Encyclopaedia Math. Sci., 29.
  • É. B. Vinberg, V. V. Gorbatsevich, O.V. Shvartsman, Discrete subgroups of Lie groups. Lie groups and Lie algebras II, 1–123, 217–223, Encyclopaedia Math. Sci., 21.
  1. Lecture 1: Introduction
  2. Lecture 2: Manifolds, covers, fundamental groups, group actions by homeomorphisms
  3. Lecture 3: Riemannian manifolds, isometries, geodesics. Upper half-plane with hyperbolic metric
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
  9. Lecture 9
  10. Lecture 10
  11. Lecture 11
  12. Lecture 12
  13. Lecture 13
  14. Lecture 14
  15. Lecture 15
  16. Lecture 16
  17. Lecture 17
  18. Lecture 18
  19. Lecture 19
  20. Lecture 20
  21. Lecture 21
  22. Lecture 22

These videos were produced by the Fields Institute, as a graduate course (link to course page).