Course Objectives: The aim of this course is give a gentle introduction to the theory of arithmetic groups. Arithmetic groups are groups of matrices with integral entries (e.g., the group SL_d(Z)). They appear naturally in many different areas of mathematics and have remarkably rich theory. In particular, they play fundamental roles in

• number theory: in reduction of quadratic forms and in automorphic representations,
• geometry: as fundamental groups of interesting spaces,
• graph theory: in construction of expander graphs.

The course will be based on the recent book of Dave Morris, and we will only assume knowledge of algebra, analysis and some number theory.

I expect to cover the following topics

• reduction theory of arithmetic groups,
• group-theoretic properties of arithemtic groups (finite generation, finite subgroups, and subgroups of finite index),
• compactness criterion,
• congruence subgroup property and superrigidity,
• spectral gap property and construction of expander graphs.

As time permits I hope to touch on some of more advanced topics such as

• Margulis superrigidity and arithmeticity,
• amenability, property (T), and Margulis normal subgroup theorem
• Borel density theorem,
• Tits alternative on free subgroups,
• number-theoretic constructions of lattices,

Time: Friday, 12-2pm; See TCC Timetable

Lecture notes:

• Lecture 1: Arithmetic groups - fundamental questions and results
• Lecture 2: Reduction theory for GL(d,R) - Siegel sets
• Lecture 3: Reduction theory for general algebraic groups
• Lecture 4: Mahler compactness criterion and generalisations
• Lecture 5: Congruence subgroup problem and an appication to rigidity
• Lecture 6: Bounded generation property
• Lecture 7: Kazhdan property (T)
• Lecture 8: Expander graphs and product replacement algorithm

Homework Problems:

• Problem set

Course Assessment: The course will be assessed by problem sheets that will be posted here.

References:

• Dave Witte Morris, Introduction to Arithmetic Groups.
• Lizhen Ji, Arithmetic Groups and Their Generalizations: What, Why, and How, AMS 2008.
• James Humphreys, Arithmetic Groups. Lecture Notes in Mathematics, 789. Springer, Berlin, 1980.
• Christophe Soule, An Introduction to Arithmetic Groups.
• Vladimir Platonov and Andrei Rapinchuk, Algebraic Groups and Number Theory. Academic Press, Inc., 1994.