Math 125: Geometry of Discrete Groups

Summer 2015


Course Info:



[PDF] Syllabus

This course will provide an introduction to the geometry of discrete groups. Specifically, we will study the action of discrete subgroups (lattices) inside groups of isometries of hyperbolic spaces, especially dimensions 2 and 3. An interesting class of such groups arises from arithmetic constructions, and the corresponding quotients give rise to orbifolds and manifolds of interest in number theory, geometry, spectral theory, combinatorics, and topology.



The homework assignments will be assigned on a weekly basis and will be posted below.

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.

[PDF] Homework Submission Guidelines

Please sign up for problems on the Homework cribsheet.

Bibliography (as of 1 July 2015)

126 Jun(F)Introduction, geodesic spaces (27.1)Chapter 27: Hyperbolic geometry
-27 Jun(S)No class, 8:00 a.m. on a Saturday, srsly?
229 Jun(M)Upper half-plane, isometries, geodesics (27.2-27.4)
31 Jul(W)Area, unit disc, Lorentz model (27.5-27.6)HW 1
42 Jul(R)Hyperbolic space (27.7)
-3 Jul(F)No class, College holiday
-6 Jul(M)No class, JV in Oberwolfach
-8 Jul(W)No class, JV in Oberwolfach
-10 Jul(F)No class, JV in Oberwolfach
513 Jul(M)Topological group actions (28.1-28.2)Chapter 28: Discrete group actions
615 Jul(W)Hausdorff, proper group actions (28.3)HW 2
716 Jul(R)Symmetric space model, Fuchsian groups (28.4-28.5)
817 Jul(F)SL2(Z) (29.1)
920 Jul(M)Binary quadratic forms (29.2)Chapter 29: Classical modular group (fixed 29.3 on Jul 23)
1022 Jul(W)Functions on lattices, complex tori (29.3)HW 3
1123 Jul(R)Modular forms (29.4)
1224 Jul(F)Kleinian groups, SL2(Z[i])
-27 Jul(M)No class, grrr, JV on tarmac
1329 Jul(W)Hyperbolic volumes (30.6)
1430 Jul(R)Dirichlet domains for Fuchsian groups (31.1)
1531 Jul(F)Ford domains (31.2)
163 Aug(M)Generators (31.4)Chapter 31: Fundamental domains (fixed typos on 5 Aug)
175 Aug(W)Side pairings (31.3)HW 4
186 Aug(R)Examples
197 Aug(F)Poincare's theorem (31.5)
-10 Aug(M)No class, JV at AIM
-12 Aug(W)No class, JV at AIM
-14 Aug(F)No class, JV at AIM
2017 Aug(M)Motivation, quaternion algebrasChapter 1: Introduction, Chapter 2: Beginnings
2119 Aug(W)Splitting, involutionsChapter 3: Involutions, Chapter 8: Orders,
Chapter 11: Quaternion algebras over local fields
2220 Aug(R)Orders, discriminantsHW 5
2321 Aug(F)Quaternionic groups: discriminant 6 (32.1)
2424 Aug(M)Quaternionic groups: discriminant 6
2526 Aug(W)Quaternionic groups
2627 Aug(R)Wrapup