Winter 2008 RFG Graduate Seminar:
Expanders and Amenability

The theme for this seminar will be expander graphs, amenability, and related topics.

Each week a student will present material on a topic related to the semester's theme. The talks will have lax time limits, and frequent interruptions for questions are encouraged.

• The Discrete Lapacian on a graph and Cheeger constant
• Expander graphs: both probabilistic and group theoretic constructions
• Kazhdan's Property T, the Haagerup property (a-T-menability), and amenability
• Examples of groups with Property T: combinatorial and Lie-theoretic
• Obstructions to uniform embeddings of graphs in Hilbert spaces

Schedule:

1. linear algebra on graphs, with examples (Shinpei)
2. expander graphs, Ramanujan graphs, and examples (Pamela)
3. existence of expanders (Gabe)
4. amenability (James)
5. Property (T): T is For Trivial (Joshua)
6. Property (T) and the construction of expanders (Matt)
7. Haagerup property (a-T-menability) (Lucas)
8. Geometry of groups and the Novikov conjecture (Jerry)
   
   
9. Randomness and groups (Moon, next quarter)

Resources:

• (Chapter 1, 2, 3, 4) Discrete groups, expanding graphs and invariant measures, by Alexander Lubotzky.
  This book is the canonical choice for expanders from a group-theoretic viewpoint. Chapter 4 discusses the discrete Laplacian and its eigenvalues. Chapter 1 is an introduction to expanding graphs, and contains a proof of the existence of expanders. Chapter 2 introduces amenability. Chapter 3 talks about Property (T), and explicitly constructs families of expanders. Anyone interested is encouraged to buy the book, although relevant sections may be photocopied and distributed in the seminar.
• Expander graphs and their applications, by Shlomo Hoory, Nathan Linial, and Avi Wigderson (2006).
  This is an online source, and so makes a good reference. Its emphasis is on applications of expander graphs, particularly to computer science, and so does not quite match the group-theoretic focus of this seminar. Nonetheless, it is well-written and useful.
• Notes on linear algebra and applications to graphs based on a course by Laszlo Babai, written by Mikhail Belkin and Moon Duchin.
  These notes are very exercise-oriented, and cover a large amount of introductory material.
• (First 20 pages) Elementary number theory, group theory, and Ramanujan graphs, by Giuliana Davidoff, Peter Sarnak, and Alain Valette.
  This book is the basis of a course for undergraduates on expanders. Its goal is to present a self-contained and detailed presentation of a couple of families of Ramanujan graphs, covering the necessary graph theory, group theory, number theory, and representation theory as they go.
• Spectral graph theory, by Chung.
  Another good introduction to combinatorial spectra and expanders.
• The Banach-Tarski Paradox, by Stan Wagon.
  The definition of amenability was intially motivated by questions similar to the Banach-Tarski Paradox. Chapter 10 of this book in particular focuses on amenability, and was one of James's main references for his talk.
• Kazhdan's Property (T), by Bekka, de la Harpe, and Valette.
  This book is dedicated to Kazhdan's Property (T) and as well as the related Property (FH). It also contains some exposition on a number of background topics, including for instance unitary group representations and amenability.
• Groups with the Haagerup Property (Gromov's a-T-menability), by Cherix, Cowling, Jolissaint, Julg, and Valette.
  This book is all about the Haagerup property, including a classification of Lie groups with the Haagerup property, the relationship with graphs of groups, and open questions.
• Spectral theory and geometry, edited by Brian Davies and Yuri Safarov.
  This LMS Lecture Note Series volume covers the non-discrete version of spectral theory and its relation to Riemannian manifolds and their geometry. It begins with succinct but starting-from-scratch introductory lecture notes on Basic Riemannian geometry, by F E Burstall, and on The Laplacian on Riemannian manifolds, by I Chavel.
• The Laplacian on a Riemannian manifold, by Rosenberg.
  This book covers what the title implies; look here for background on the regular Laplacian.
• Coarse embeddings into a Hilbert space, Haagerup Property and Poincare inequalities, by Romain Tessera.
  This recent paper is an example of active research in this area.
• Group C*-algebras and K-theory, by Higson and Guentner.
  Jerry's talk will cover material from this reference, which talks about the Baum-Connes conjecture.