## Abstract

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 o: 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.

Original language | American English |
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Pages (from-to) | 445-454 |

Number of pages | 10 |

Journal | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

DOIs | |

State | Published - 2004 |

Event | Proceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States Duration: 13 Jun 2004 → 15 Jun 2004 |

## Keywords

- Cayley graphs
- Expanders
- Zig-zag product