Iterative Construction of Cayley Expander Graphs

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 (τ) with respect to the family of subgroups ker(fn). The proof is elementary, using only simple combinatorics and linear algebra. The recursive 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 [4] of the zig-zag graph product of [42]. The basis of the inductive proof is a recent result by Kassabov [22] on expanding generating sets for the group Ad. Essential use is made of the fact that our groups have the commutator property: every element is a commutator. We prove that direct products of such groups are expanding even with highly correlated tuples of generators. Equivalently, highly dependent random walks on several copies of these groups converge to stationarity on all of them essentially as quickly as independent random walks. Moreover, our explicit construction of the generating sets Yn above uses an efficient algorithm for solving certain equations over these groups, which relies on the work of [37] on the commutator width of perfect groups. The Hebrew university, Jerusalem. E-mail: [email protected]. Part of this research was performed while visiting the Institute for Advanced Study, Princeton, NJ. The Hebrew university, Jerusalem. E-mail: [email protected]. Partially supported by BSF grant 2000-53 and a grant from the Israel Science Foundation Institute for Advanced Study, Princeton. E-mail: [email protected] Partially supported by NSF grant CCR0324906