Math 569. Property (T), expander graphs, and related topics. Fall 2010

General Info | Syllabus | Grading | Announcements |

General Information

Math 569, Call #29850, Fall 2010
Classes: MWF 1-1:50 in 300 TH
Professor: Alex Furman
Office hours: TBA
Contact: furman at math.uic.edu
Main Text: Discrete groups, expanding graphs and invariant measures by A.Lubotzky, with an Appendix by J. D. Rogawski.
Additional: Kazhdan's property (T) by B.M. Bekka, P. de la Harpe, A. Valette.

Course Description

The main themes of the course are:

Kazhdan's property (T): this a property of infinite groups, defined in terms of their unitary representations. D. Kazhdan has introduced this notion in his famous thesis in 1967 in order to show (in a very indirect and amusing way) some basic facts about a class of infinite discrete subgroups of SL(n,R). Since then property (T) and related notions found a wide range of applications in such diverse areas of mathematics as combinatorics, geometric groups theory, operator algebras, ergodic theory, descriptive set theory.
Expander graphs are families of sparse (bounded valency) finite graphs with excellent connectivity properties. Such graphs have many important applications for theoretical computer science, network design, computational applications. Theoretical existence of such families was proven using probabilistic methods by Pinsker, however explicit constructions required new ideas. First expanders were constructed by Margulis using property (T), expander families with optimal constants (Ramanujan graphs) were constructed by Lubotzky-Phillips-Sarnak using deep number-theoretic tools (including Jackuet-Langlands correspondence).
Banach-Ruziewicz problem is a question about uniqueness of the Lebesgue measure as a finitely additive measure invariant under isometries. It was settled in the positive for dimensions n>4 by Margulis and Sullivan using proiperty (T), and for n=3 and n=4 by Drinfeld using a different number-theoretic construction.

In the course we intend to discuss and to connect these three themes. The actual content of the course will be determined and chosen from the following tentative list of topics:

Property (T): Definitions and basic properties, applications to expander graphs, Stenberg's theorem, property (FH) and the theorem of Delorme - Guichardet, amalgams and actions on trees.
Groups with property (T): higher rank lattices, local spectral criterion (after Gerland, Zuk, Gromov etc).
Expander graphs: Pinsker's theorem, discrete Laplacian, theorems of Buser and Cheeger, Alon-Boppana, Ramanujan graphs.
Recent developments: Bourgain - Gamburd results on SL(2,p).

Lecture plan:

Date	Topic	More details
8/23 8/25 8/27	Introduction Banach-Ruiziewicz problem Hausdorff-Banach-Tarski theorem	The problem of expander graphs . .
8/30 9/1 9/3	Hausdorff-Banach-Tarski (leftovers) Elements of algebraic groups Hilbert spaces	Ping-pong lemma, free group inside SL(2,Z), quadratic-forms SL(2,R), SO(2,1) and SO(3). Free groups in SO(3) ?
9/6 9/8 9/10	Labor Day Hilbert spaces Weak topology	Rest and relax Basic facts Weak convergence and convexity
9/13 9/15 9/17	Unitary reps The unitary dual Property (T)	Strong and weak operator topologies, measurable vs. continuous, direct integrals Spectral theorem for one unitary, general decomposition into integral of irreps, Pontryagin duality Fell topology, (K,e)-almost invariant vectors, amenability and property (T)
9/20 9/22 9/24	First fruits Banach-Ruiziewicz Lattices	Finite generation, finite Abelianization, expanders How to solve the problem Examples
9/27 9/29 9/30 10/1	. Induction of representations Student presentations Property (T) inherited by lattices	. . (1) Intro to Amenable groups. (2) Explicit Z/2*Z/3 in SO(3) .
10/4 10/6 10/7 10/8	Continuity of induction Siegel's formula and SL(n,Z) Student presentation Proof Siegel's formula	cocycles, induced representations Modular function, Haar measure on homogeneous spaces Explicit F_2 in SO(3) using quaternions (and Jacobi's thm) SL(n,R)/SL(n,Z) and zeta functions
10/11 10/13 10/14 10/15	Howe-Moore Theorem Howe-Moore Theorem (cont.) Student presentation Relative property (T)	Statement and applications, Cartan decomposition, Mautner Lemma Case of SL(2,R), general case Hahn-Banach Theorem and its applications, Banach limits Spectral theorem for lc Abelian groups
10/18 10/20 10/22	Relative property (T) Arithmetic lattices S-arithmetic lattices	The use of the spectral theorem Borel- Harish-Chandra, property (T) subgroups in SO(5) p-adic fields, quaternion algebras
10/25 10/27 10/29	Affine isometries of Hilbert spaces (FH) => (T) (FH) => (FA)	First cohomology with unitary coefficients Serre's property (FA), amalgams Embedding a tree in a Hilbert space
11/1 11/3 11/5	Pos def and cond neg def kernels Shoenberg's theorem No class	. The cone of pos def kernels .
11/8 11/10 11/12	(T) => (FH) . .	Delorme's theorem . .

Topics for presentations:

Haar measure, modular function, von Neumann's theorem
An intro to amenable groups
Random graphs are expanders (Pinsker)
Convexity: Hahn-Banach theorem and its consequences
Representations of compact groups
Navas' theorem on the circle

Grading

The course will involve student presentations. Students are expected to make one presentation during the semester and to summarize two presentations of their peers. Possible topics for presentations will be listed here:

Announcements

[Alex Furman] [CV] [Research]