Iterative Construction of Cayley Expander Graphs

Abstract

We construct a sequence of groups Gn, and explicit sets of generators Yn Gn, such that all generating sets have bounded size, and the associated Cayley graphs are all expanders. The group G1 is the alternating group Ad, the set of even permutations on the elements {1, 2,..., d}. The group Gn is the group of all even symmetries of the rooted d-regular tree of depth n. Our results hold for any large enough d. We also describe a finitely generated infinite group G∞ with generating set Y∞, given with a mapping fn from G∞ to Gn for every n, which sends Y∞ to Yn. In particular, under the assumption described above, G∞ has property (t) with respect to the family of subgroups ker( fn). The proof is elementary, using only simple combinatorics and linear algebra. The re- cursive structure of the groups Gn (iterated wreath products of the alternating group Ad) allows for an inductive proof of expansion, using the group theoretic analogue (of Alon et