Math 9510: Hyperbolic 3-Manifolds
Fall 2022
| Meets: | Tue/Thu 11:00-12:20 in Wachman Hall, room 617 |
| Instructor: | David Futer |
| Office: | 1026 Wachman Hall |
| Office Hours: | Tue 3:00-4:30, Thu 9:30-11:00, or by appointment |
| E-mail: | dfuter at temple.edu |
| Phone: | (215) 204-7854 |
Course content: Continuing the theme from Math 9500, this course aims to describe a moderately detailed picture of hyperbolic 3-manifolds. We will begin with knots and links in the 3-sphere, and construct hyperbolic structures on their complements. We will then go through some structural results in the theory: completeness and incompleteness of geometric structures, the thick-thin decomposition, and Mostow rigidity.
References:
- An introduction to geometric topology, by Bruno Martelli
- Hyperbolic knot theory, by Jessica Purcell. Note: page and exercise numbers from Purcell refer to the arXiv version linked here.
Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.
Detailed schedule
| Day | Topic | Reading | Homework/Note |
|---|---|---|---|
| 1/17 | Motivating example: the figure-8 knot | Purcell, Chap. 1 | |
| 1/19 | Hyperbolic structure on the figure-8 knot | Martelli, p. 446-447 | |
| 1/24 | Hyperbolic structure on the Whitehead link | ||
| 1/26 | Gluing and completeness for surfaces | Purcell, p. 54-59 | Homework 1, due Thursday 2/2 |
| 1/31 | Edge gluing equations for 3-manifolds | Purcell, p. 69-73 | |
| 2/2 | Developing map and holonomy; completeness equations | Purcell, 50-53; p. 74-76 | |
| 2/7 | Worksheet | ||
| 2/9 | Worksheet | ||
| 2/14 | Elementary subgroups; thick & thin parts | Purcell, p. 88-92 | |
| 2/16 | The Margulis lemma | Martelli, p. 116-119; Purcell, p. 95-101 | |
| 2/21 | Thick-think decomposition in finite volume | Martelli, p. 119-122 | |
| 2/23 | Length spectra and isometry groups | Martelli, p. 123-125 | |
| 2/28 | No class | ||
| 3/2 | Isometry groups, residual finiteness | Martelli, p. 126-128 | |
| 3/14 | Limit sets, convex cores | Martelli, p. 135-137 | |
| 3/16 | Limit sets and domains of discontinuity | Martelli, p. 137-138; animations | |
| 3/21 | Quasi-isometries and pseudo-isometries | Martelli, p. 143-144 | |
| 3/23 | Extending pseudo-isometries to the boundary | Martelli, p. 145-147 | |
| 3/28 | Extending pseudo-isometries to the boundary | Martelli, p. 148-150 | |
| 3/30 | Volumes of simplices; angle structures | Purcell, Chap. 9; Martelli, p. 405-409 | |
| 4/4 | Gromov norm | Martelli, p. 410-412 | |
| 4/6 | Simplicial volume | Martelli, p. 412-414 | |
| 4/11 | Gromov norm and volume are proportional | Martelli, p. 414-416 | |
| 4/13 | Mostow rigidity | Martelli, p. 417-420 | |
| 4/18 | Thurston norm, Dehn filling | Purcell, p. 109-117 | |
| 4/20 | Hyperbolic Dehn filling theorem | Martelli, p. 444-463 | |
| 4/25 | Presentations: Rob, Brandis | ||
| 4/27 | Presentations: Andrew, Ross |
dfuter at temple edu Last modified: Fri Aug 21 13:41:22 PDT 2009