Symmetric Groups and Expanders

We construct an explicit generating sets Fn and F̃n of the alternating and the symmetric groups, which make the Cayley graphs C(Alt(n), Fn) and C(Sym(n), F̃n) a family of bounded degree expanders for all sufficiently large n. These expanders have many applications in the theory of random walks on groups and other areas of mathematics. A finite graph Γ is called an ǫ-expander for some ǫ ∈ (0, 1), if for any subset A ⊆ Γ of size at most |Γ|/2 we have |∂(A)| > ǫ|A| (where ∂(A) is the set of vertices of Γ\A of edge distance 1 to A). The largest such ǫ is called the expanding constant of Γ. Constructing families of ǫ-expanders with bounded valency is an important practical problem in computer science, because such graphs have many nice properties — for example they have a logarithmic diameter. For an excellent introduction to the subject we refer the reader to the book [14] by A. Lubotzky. Using counting arguments it can be shown that almost any 5 regular graph is 1/5-expander. However constructing an explicit examples of families expander graphs is a difficult problem. The first explicit construction of a family of expanders was done by G. Margulis in [19], using Kazhdan property T of SL3(Z). Currently there are several different construction of expanders. With the exception of a few recent ones based on the zig-zag products of graphs (see [2, 23, 24]), all constructions are based groups theory and use some variant of property T (property τ , Selberg property etc.). Kazhdan Property T is not very interesting for a given finite group G (all finite groups have property T ), but the related Kazhdan constant with respect to some generating F set is. Given an infinite collection of finite groups Gi, it is a challenge to prove the existence of uniform Kazhdan constants with respect to properly chosen generating sets. This problem is related to construction a family of expanders using the Cayley graphs of the groups Gi. The original definition of property T uses the Fell topology of the unitary dual, see [11]. Here we will use an equivalent definition (only for discrete groups) which also addresses the notion of the Kazhdan constants. 2000 Mathematics Subject Classification: Primary 20B30; Secondary 05C25, 05E15, 20C30, 20F69, 60C05, 68R05, 68R10.