Expansion properties of Cayley graphs of the alternating groups

Let C be a conjugacy class in the alternating group A_n, and let supp(C) be the number of nonfixed digits under the action of a permutation in C. For every 1 > δ > 0 and n ≥5 there exists a constant c = c(δ) > 0 such that if supp(C) ≥ δn then the undirected Cayley graph X(A_n, C) is a c expander. A family of such Cayley graphs with supp(C) = o(√n) is not a family of c-expanders. For every δ > 0, if supp(C) ≥ √3n then sets of vertices of order at most (1/2 - δ)(n - (n/supp(C)))! in X(A_n, C) expand. The proof of the last result combines spectral and representation theory techniques with direct combinatorial arguments.