Expansion in Finite Simple Groups of Lie Type (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 164)

Expander Graphs Are An Important Tool In Theoretical Computer Science, Geometric Group Theory, Probability, And Number Theory. Furthermore, The Techniques Used To Rigorously Establish The Expansion Property Of A Graph Draw From Such Diverse Areas Of Mathematics As Representation Theory, Algebraic Geometry, And Arithmetic Combinatorics. This Text Focuses On The Latter Topic In The Important Case Of Cayley Graphs On Finite Groups Of Lie Type, Developing Tools Such As Kazhdan's Property (t), Quasirandomness, Product Estimates, Escape From Subvarieties, And The Balog-szemeredi-gowers Lemma. Applications To The Affine Sieve Of Bourgain, Gamburd, And Sarnak Are Also Given. The Material Is Largely Self-contained, With Additional Sections On The General Theory Of Expanders, Spectral Theory, Lie Theory, And The Lang-weil Bound, As Well As Numerous Exercises And Other Optional Material. Part 1: Expansion In Cayley Graphs -- Chapter 1. Expander Graphs: Basic Theory -- Chapter 2. Expansion In Cayley Graphs, And Kazhdan's Property (t) -- Chapter 3. Quasirandom Groups -- Chapter 4. The Balog-szemeredi-gowers Lemma, And The Bourgain-gamburd Expansion Machine -- Chapter 5. Product Theorems, Pivot Arguments, And The Larsen-pink Noncentration Inequality -- Chapter 6. Nonconcentration In Subgroups -- Chapter 7. Sieving And Expanders -- Part 2: Related Articles -- Chapter 8. Cayley Graphs And The Algebra Of Groups -- Chapter 9. The Lang-weil Bound -- Chapter 10. The Spectral Theorem And Its Converses For Unbounded Self-adjoint Operators -- Chapter 11. Notes On Lie Algebra -- Chapter 12. Notes On Groups Of Lie Type. Terence Tao. Includes Bibliographical References (pages 293-300) And Index. Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog–Szemerédi–Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang–Weil bound, as well as numerous exercises and other optional material.