Spring
2018
Math 542: Topics in Advanced Topology–Arithmetic of Hyperbolic Manifolds
This class will study arithmetic aspects of hyperbolic manifolds. This will largely be confined to finite volume hyperbolic manifolds in dimensions 2 and 3, but if time allows a discussion of higher dimensions will be given. Associated to any finite volume hyperbolic 2 or 3manifold M is a pair (AM, kM) which consists of a quaternion algebra AM over a field kM which is an invariant of the commensurability class. In particular, in dimension 3, kM is a number field, and algebraic and arithmetic invariants of this field and algebra are powerful tools in the study hyperbolic 3manifolds.
The main aim of this course is to develop this. In the special setting of arithmetic hyperbolic 3manifolds (and 2manifolds) this pair underpins much of the geometric, topological and spectral properties of the manifold.
There is no set text book. Occasional HW sheets will be given out for selfstudy and group discussion. These are not to be handed in. There will be no exams.
Prerequisites: The graduate topology and algebra courses, some number theory would be helpful, but not absolutely necessary.
Class meets: 1:00PM  1:50PM MWF
Room: HBH 423
First day handout (pdf)
Homework
1. pdf
2. pdf
3. pdf
4. pdf
5. pdf
6. pdf
7. pdf
8. pdf
