80220 -- Introduction to Arithmetic Groups (Spring 2020)

Course schedule and suggested exercises

Week 13: Margulis superrigidity, arithmeticity, and normal subgroup theorems (16.1, 16.3, 17.1); Semester Wrap-Up

Week 12: Expander graphs; quasi-isometries (10.1); Strong Rigidity (15.1-15.2) [References on expanders: "Discrete groups, expanding graphs, and invariant measures" by Lubotzky, "An introduction to expander graphs" by Kowalski]

Suggested exercises: 10.1 #1, 2, 4, 5; 15.1 #1 (also good: 3, 4, 5)

Week 11: SL_3(R) has (T), lattices inherit (T) (13.3-13.4); induced representations and reps of abelian groups (11.3-11.5)

Suggested exercises: 11.3 #3, 5; 11.5 #1, 2, 3; 13.3 #4 (also good: 1,2,3); 13.4 #1, 4

Week 10: Property (T) (13.1-13.2), unitary reps and Moore ergodicity theorem (11.1-11.2)

Suggested exercises: 11.1 #2; 11.2 #1, 3, 4, 5, 8; 13.1 #5, 6, 8

Week 9: Amenability (12.1-12.6)

Suggested exercises: 12.1 #2; 12.2 #2, 3, 4, 7; 12.3 #1, 11, 20, 23; 12.4 #2 [many good exercises in 12.3, for those who want to dig deeper than we went in class]

Week 8: [begin remote instruction] Parabolic subgroups, Iwasawa decomposition (8.4), free subgroups (4.9)

Suggested exercises: 8.4 #1, 4, 2, 7, 9; 4.9 #3, 5 (and 6 is bonus)

Week 7: Constructing symmetric spaces (1.2), real rank (1.3, 8.1-8.3)

Suggested exercises: 1.2 #5, 7, 8 (skip (c) if you haven't studied Lie algebras before); 8.1 #1, 2, 3, 4; 8.2 #4

Week 6: SL_n(Z) is a lattice (7.1-7.3), symmetric spaces (1.1)

Suggested exercise: 7.1 #5 (b optional), 6; 7.2 #9; 7.3 #6 (also #2-4 if you want to check details of Monday's class)

Week 5: Proof of Borel density (4.6), finite generation, residual finiteness, torsionfreeness (4.7, 4.8)

Suggested exercises: 4.7 #1, 3 (see definition 4.2.7); 4.8 #2, 3, 4, 5

Week 4: Compactness considerations (4.4, 5.3, Example 6.3.1), Borel density (4.5) (and bare basics of Lie algebras, A6)

Suggested exercises: 4.4 #3, 4, 9, 10 (good exercises, but on the optional side); 5.3 #4, 5, 8; Check details of Ex 6.3.1; 4.5 #3, 4, 5, 10

Week 3: Construction of arithmetic groups via restriction of scalars (5.4-5.5), properties

Suggested exercises: 5.4 #5, 6, 7, 8; 5.5 #4, 7, 8 (these prove the main proposition from Wednesday's class)

Week 2: Zariski closed groups (A4), definition of arithmetic groups (chapter 5.1)

Suggested exercises: 5.1 #1, 2, 3; A4 #1, 3, 5 (2, 6, and 7 are also good)

Week 1: Lie group basics (Appendix A1-A3), lattice definition, commensurability, and reducibility (4.1-4.3)

Suggested exercises: A1 #3-6; A3 #3; 4.1 #4, 6, 14; 4.2 #1, 2; 4.3 #1

Week 0: Introduction: binary quadratic forms and SL_2(Z)