Expansion Properties of Cayley Graphs of the Alternating Groups

LetCbe a conjugacy class in the alternating groupAn, and let supp(C) be the number of nonfixed digits under the action of a permutation inC. For every 1>δ>0 andn⩾5 there exists a constantc=c(δ)>0 such that if supp(C)⩾δnthen the undirected Cayley graphX(An, C) is acexpander. A family of such Cayley graphs withsupp(C)=o(n)is not a family ofc-expanders. For everyδ>0, ifsupp(C)⩾nthen sets of vertices of order at most(12−δ)(n−(n/supp(C)))!inX(An, C) expand. The proof of the last result combines spectral and representation theory techniques with direct combinatorial arguments.