A new family of Cayley expanders (?)

We assume that for some <i>fixed</i> large enough integer d, the symmetric group S<inf>d</inf> can be generated as an expander using d<sup>1/30</sup> generators. Under this assumption, we explicitly construct an infinite family of groups G<inf>n</inf>, and explicit sets of generators Y<inf>n</inf> ⊂ G<inf>n</inf>, such that all generating sets have bounded size (at most d<sup>1/7</sup>), and the associated Cayley graphs are all expanders. The groups G<inf>n</inf> above are very simple, and completely different from previous known examples of expanding groups. Indeed, G<inf>n</inf> 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<inf>n</inf> (iterated wreath products of the alternating group A<inf>d</inf>) 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<inf>n</inf> 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 <i>weak</i> 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.

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