Math 8803

Low-dimensional Topology and

Hyperbolic Geometry

Fall 2014


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------------------------------Theodor Geisel
News: Welcome to Math 8803!

Professor

Class Meetings

Tuesday and Thursday 3:05-4:25 pm, Skiles 154.

Office Hours

By appointment.

Homework

Optional homework will be assigned throughout the semester.

Seminar

Students will be expected to give one lecture on a related topic in the Geometry-Topology Student Seminar.

Grading

Grades will be based on attendance, homework, and seminar presentations.

Weekly Schedule

Hyperbolic 3-manifolds
Week Dates Topics Reading Lectures Notes
1 Aug 19 Overview Hatcher Week 1
2 Aug 26 Sphere/torus decompositions Hatcher Week 2
3 Sep 2 Seifert manifolds Hatcher Week 3 Labor day
4 Sep 9 Hyperbolic geometry Thurston Week 4
5 Sep 16 Hyperbolic manifolds Purcell Week 5
6 Sep 23 Completeness Thurston Week 6
7 Sep 30 Dehn surgery Thurston Week 7
8 Oct 7 Mostow Rigidity Thurston Week 8
9 Oct 14 Mostow Rigidity Thurston Week 9 Fall break
Weeks 1-9

Geometry of the Complex of Curves
Week Dates Topics Reading Lectures Notes
10 Oct 21 Overview / Hyperbolicity Sisto Week 10
11 Oct 28 Axes / Free groups MMI Week 11
12 Nov 4 BGI Webb Week 12
13 Nov 11 pA Recipe Mangahas
Guirardel
Week 13
14 Nov 18 Tight geodesics Masur-Minsky II Week 14
15 Nov 25 Acylindricity Webb Week 15 Thanksgiving
16 Dec 2 RAAGs CLM Week 16
Weeks 10-16

References

---Hyperbolic 3-manifolds

---Combinatorial cubings, cusps, and the dodecahedral knots, I.R. Aitchison and J.H. Rubinstein

---Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston's vision, Mladen Bestvina

---Geometric structures on 3-manifolds, Francis Bonahon

---Seifert fibered spaces, Matthew Brin

---Notes on 3-manifolds, Danny Calegari

---Hyperbolic geometry, James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry

---The CompuTop.org Software Archive

---An introduction to 3-manifolds, Stefan Friedl

---Hyperbolic Geometry and 3-Manifold Topology, David Gabai

---Hyperbolic 3-manifolds in the 2000's, David Gabai

---Hyperbolic manifolds according to Thurston and Jorgensen, Mikhail Gromov

---Introduction to hyperbolic geometry, Subhojoy Gupta

---Notes on Basic 3-manifolds, Allen Hatcher

---The classification of 3-manifolds - a brief overview, Allen Hatcher

---The classification of 3-manifolds, Allen Hatcher

---The homeomorphism problem: classification of 3-manifolds, William Jaco

---Hyperbolic manifolds and discrete groups: Lectures on Thurston's geometrization, Misha Kapovich

---Knots and Manifolds, Louis Kauffman

---Three-dimensional manifolds, Marc Lackenby

---Hyperbolic Manifolds, Marc Lackenby

---From metrics to moduli space, Chris Leininger

---Outer circles: an introduction to hyperbolic 3-manifolds, Albert Marden

---Towards the Poincare conjecture and the classification of 3-manifolds, John Milnor

---Around the Borromean link, Jose Maria Montesinos-Amilibia

---Notes on Geometry and 3-manifolds, Walter Neumann

---Lectures on Seifert manifolds, Walter Neumann

---Hyperbolic knot theory, Jessica Purcell

---Foundations of hyperbolic manifolds, John Ratcliffe

---Three manifolds, Saul Schleimer

---The geometries of 3-manifolds, Peter Scott

---Hyperbolic geometry, Caroline Series

---The Geometry and Topology of 3-manifolds, William Thurston

---The mystery of 3-manifolds, William Thurston

---Three dimensional manifolds, Kleinian groups and hyperbolic geometry, William Thurston

---Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, William Thurston

---How to see 3-manifolds, William Thurston

---Complex of curves

---Uniform hyperbolicity of the graphs of curves, Tarik Aougab

---Quasi-homomorphisms on mapping class groups, Mladen Bestvina and Koji Fujiwara

---Intersection numbers and the hyperbolicity of the curve complex , Brian Bowditch

---Tight geodesics in the curve complex, Brian Bowditch

---Length bounds on curves arising from tight geodesics, Brian Bowditch

---Uniform hyperbolicity of the curve graphs, Brian Bowditch

---The classification of Kleinian surface groups II: the ending lamination conjecture, Jeff Brock, Dick Canary, and Yair Minsky

---The geometry of right angled Artin subgroups of mapping class groups, Matt Clay, Chris Leininger, and Johanna Mangahas

---Uniform hyperbolicity of the curve graph via surgery sequences, Matt Clay, Kasra Rafi, and Saul Schleimer

---Rotating families, Dehn fillings and small cancellation, Vincent Guirardel ---Boundary structure of the modular group, William J. Harvey

---Mapping class groups, Nikolai Ivanov

---Automorphisms of Complexes of Curves and of Teichm├╝ller Spaces, Nikolai Ivanov

---A finite set of generators for the homeotopy group of a 2-manifold, W.B.R. Lickorish

---A recipe for short-word pseudo-Anosovs, Johanna Mangahas

---Geometry of the complex of curves I: hyperbolicity, Howard Masur and Yair Minsky

---Geometry of the complex of curves II: hierarchical structure, Howard Masur and Yair Minsky

---Curve complexes, surfaces and 3-manifolds, Yair Minsky

---The classification of Kleinian surface groups, I: models and bounds, Yair Minsky

---Notes on the complex of curves, Saul Schleimer

---Combinatorial rigidity in curve complexes and mapping class groups, Ken Shackleton

---Lecture Notes on Braids and Dynamics, Jean-Luc Thiffeault

---1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, Richard Webb, Sebastian Hensel, and Piotr Przytycki

---Uniform bounds for bounded geodesic image theorems, Richard Webb

---Combinatorics of tight geodesics and stable lengths, Richard Webb

Topics for Student Seminars

---Alexander's theorem

---The uniformization theorem

---The loop theorem

---The sphere theorem

---Classification of Seifert manifolds

---JSJ decompositions of groups

---Automorphisms of the complex of curves

---Cartan--Hadamard theorem

---Fenchel--Nielsen coordinates

---SnapPy

---Nielsen-Thurston classification theorem

---Dehn-Nielsen-Baer theorem

---Quasi-morphisms on mapping class groups

---Asymptotic dimension of the mapping class group

---Bestvina-Handel algorithm

---Boundary of the complex of curves

---Ending lamination theorem

Resources

---T-Square

---Georgia Tech Honor Code