Lecture 3 : Property ( Τ ) and Expanders ( Preliminary

There are many excellent existing texts for the material in this lecture, starting with Lubotzky's monograph [11] and recent AMS survey paper [10]. For expander graphs and their use in theoretical computer science, check the survey by Hoory, Linial and Wigderson [6]. We give here a brief introduction. We start with a definition. Definition 0.1. (Expander graph) A finite connected k-regular graph G is said to be an ε-expander if for every subset A of vertices in G, with |A| 1 2 |G|, one has the following isoperimetric inequality: |∂A| ε|A|, where ∂A denotes the set of edges of G which connect a point in A to a point in its complement A c. The optimal ε as above is sometimes called the discrete Cheeger constant of the graph: h(G) = inf A⊂G,|A| 1 2 |G| |∂A| |A| , Just as in Lecture 1, when we discussed the various equivalent definitions of amenabil-ity, it is not a surprise that this definition turns out to have a spectral interpretation. Given a k-regular graph G, one can consider the Markov operator (also called averaging operator, or sometimes Hecke operator in reference to the Hecke graph of an integer lattice) on functions on vertices on G defined as follows: P f (x) = 1 k ∑ x∼y f (y), where we wrote x ∼ y to say that y is a neighbor of x in the graph. This operator is easily seen to be self-adjoint on ℓ 2 (G), which is a finite dimensional Euclidean space. Moreover it is a contraction, namely ||P f || 2 ||f || 2 and hence its spectrum is real and contained in [−1, 1]. We can write the eigenvalues of P in decreasing Date: July 14th 2012.