The final aim of these lectures will be to prove spectral gaps for finite groups and to turn certain Cayley graphs into expander graphs. However in order to do so it is useful to have some understanding of the analogous spectral notions of amenability and Kazhdan property (T ) which are important for infinite groups. In fact one important aspect of asymptotic group theory (the part of group the...

Polynomial calculus is a Hilbert-style proof system in which lines are polynomials modulo x = x (for each variable x) and the rules allow deriving c1P1 + c2P2 from P1, P2 and xP from P for a variable x. A polynomial calculus refutation of a set of axioms is a derivation of 1 from these axioms. Research in proof complexity tends to concentrate on the length of proofs. We will rather be intereste...

Today we examine the zig-zag product introduced by Reingold, Vadhan, and Wigderson [2] and Capalbo, Reingold, Vadhan, and Wigderson [1]. The product leads to a remarkable constructioon of expander graphs needed in the Spiser-Spielman code. In addition, their works introduce a probabilistic viewpoint of expansion. In previous literature, the spectral techniques are used to analyze the expansion ...

I have three goals for this lecture. The first is to introduce one of the most important familes of graphs: expander graphs. They are the source of much combinatorial power, and the counterexample to numerous conjectures. We will become acquainted with these graphs by examining random walks on them. To facilitate the analysis of random walks, we will examine these graphs through their adjacency...

We construct explicit generating sets Sn and S̃n of the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), Sn) and C(Sym(n), S̃n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of random walks on groups, card shuffling a...

This paper gives a new way of showing that certain constant degree graphs are graph expanders. This is done by giving new proofs of expansion for three permutations of the Gabber–Galil expander. Our results give an expansion factor of 3 16 for subgraphs of these three-regular graphs with (p− 1)2 inputs for p prime. The proofs are not based on eigenvalue methods or higher algebra. The same metho...

Today we examine the zig-zag product introduced by Reingold, Vadhan, and Wigder­ son [2] and Capalbo, Reingold, Vadhan, and Wigderson [1]. The product leads to a remarkable constructioon of expander graphs needed in the Spiser-Spielman code. In addition, their works introduce a probabilistic viewpoint of expansion. In previous liter­ ature, the spectral techniques are used to analyze the expans...

We study the representations of non-commutative universal lattices and use them to compute lower bounds for the τ -constant for the commutative universal lattices Gd,k = SLd(Z[x1, . . . , xk]) with respect to several generating sets. As an application of the above result we show that the Cayley graphs of the finite groups SL3k(Fp) can be made expanders using suitable choice of the generators. T...

We present a novel error correcting code and decoding algorithm which have construction similar to expander codes. The code is based on a bipartite graph derived from the subsumption relations of finite projective geometry, and ReedSolomon codes as component codes. We use a modified version of well-known Zemor’s decoding algorithm for expander codes, for decoding our codes. By derivation of geo...