A new family of Cayley expanders (?)

We assume that for some fixed large enough integer d, the symmetric group S_d can be generated as an expander using d^1/30 generators. Under this assumption, we explicitly construct an infinite family of groups G_n, and explicit sets of generators Y_n ⊂ G_n, such that all generating sets have bounded size (at most d^1/7), and the associated Cayley graphs are all expanders. The groups G_n above are very simple, and completely different from previous known examples of expanding groups. Indeed, G_n is (essentially) all symmetries of the d-regular tree of depth n. The proof is completely elementary, using only simple combinatorics and linear algebra. The recursive structure of the groups G_n (iterated wreath products of the alternating group A_d) allows for an inductive proof of expansion, using the group theoretic analogue [4] of the zig-zag graph product of [38]. The explicit construction of the generating sets Y_n uses an efficient algorithm for solving certain equations over these groups, which relies on the work of [33] on the commutator width of perfect groups.We stress that our assumption above on weak expansion in the symmetric group is an open problem. We conjecture that it holds for all d. We discuss known results related to its likelihood in the paper.