Instructor: Professor Nikolay Bogachev, University of Toronto

Course Dates: September 10th - December 5th, 2024
Mid-Semester Break: October 14th - 18th, 2024
Lecture Times: Tuesdays, 4:00 PM - 6:00 PM (ET) | Thursdays, 12:00 PM - 2:00 PM (ET)
Office Hours: by request

Registration Fee: PSU Students - Free | Other Students - CAD$500

Capacity Limit: 30 students

Format: Hybrid synchronous delivery

• In-Person: Room 210, Fields Institute (222 College Street, 2nd Floor, Toronto, ON M5T 3J1)
• Online: via Zoom

Course Description

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.

Course Program:

1. Introduction: what do we study? Examples and pictures. Hyperbolic background: the hyperbolic space $\mathbb{H}^{d}$ and its ideal boundary $\partial\mathbb{H}^{d}$ in different models, classification of isometries, convex polyhedra.
2. Background on discrete subgroups of Lie groups: Haar measures, lattices, and their basic properties.
3. Fundamental polyhedra for discrete subgroups of hyperbolic isometries, the Dirichlet domain, the Poincare method.
4. Vinberg’s theory of hyperbolic reflection groups: hyperbolic Coxeter polyhedra, Coxeter–Vinberg diagrams.
5. Commensurability invariants of lattices: the adjoint trace field and the ambient group. Examples from reflection groups.
6. Geometrization of surfaces, deformations in $\mathbb{H}^{2}$.
7. Sketch of proof of the Mostow Rigidity Theorem: lifting a smooth homotopy to pseudo-isometry, extension to a ball homeomorphism, quasi-conformality and differentiability on the ideal boundary, dynamics and ergodic theory, ergodicity of geodesic flows on hyperbolic manifolds, the Howe–Moore ergodicity theorem and ergodicity of group actions, conformal map on the ideal boundary $=$ hyperbolic isometry.
8. Algebraic number fields and rings, algebraic k-groups, general definition of arithmetic and quasi-arithmetic lattices, arithmetic lattices in PSL$_{2}$($\mathbb{R}$) and PSL$_{2}$($\mathbb{C}$).
9. Arithmetic hyperbolic lattices of type I via quadratic forms. Vinberg’s arithmeticity criterion for reflection groups.
10. Quaternion algebras, arithmetic hyperbolic lattices of type II via skew-Hermitian forms over quaternion algebras.
11. Totally geodesic subspaces. Two constructions of nonarithmetic hyperbolic manifolds: hybrids of Gromov and Piatetski-Shapiro and properly quasiarithmetic manifolds with arbitrarily small systole constructed by Agol, Belolipetsky–Thomson, and Bergeron–Haglund–Wise.
12. Totally geodesic subspaces and characterization of arithmeticity. Algebraic classification of geodesic immersions of arithmetic hyperbolic orbifolds into one another.

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.

Evaluation: The course grades will be based on a few written homework assignments.

Textbooks and Surveys:

1. D. V. Alekseevskij, É. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature. Geometry II, 1–138. Encyclopaedia Math. Sci., 29.
2. C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds.
3. B. Martelli, Introduction to Geometric Topology.
4. D. Morris, Introduction to Arithmetic Groups.
5. É. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature. Geometry II, 139–248. Encyclopaedia Math. Sci., 29.
6. 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.

(You can also view the course outline in a PDF file HERE.)