A new family of Cayley expanders (?)

Eyal Rozenman*, Aner Shalev, Avi Wigderson

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

11 Scopus citations

Abstract

We assume that for some fixed large enough integer d, the symmetric group Sd can be generated as an expander using d1/30 generators. Under this assumption, we explicitly construct an infinite family of groups Gn, and explicit sets of generators Yn ⊂ 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 Gn above are very simple, and completely different from previous known examples of expanding groups. Indeed, Gn 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 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 [38]. The explicit construction of the generating sets Yn 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 languageAmerican English
Pages (from-to)445-454
Number of pages10
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs
StatePublished - 2004
EventProceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States
Duration: 13 Jun 200415 Jun 2004

Keywords

  • Cayley graphs
  • Expanders
  • Zig-zag product

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