Finite simple groups as expanders

Abstract

We prove that there exist k ʦ ‫ގ‬ and 0 < ʦ ‫ޒ‬ such that every non-abelian finite simple group G, which is not a Suzuki group, has a set of k generators for which the Cayley graph Cay(G; S) is an -expander.expander graphs ͉ Ramanujan complexes L et X be a finite graph and 0 Ͻ ʦ ‫.ޒ‬ Then X is called an -expander if for every subset A of the vertices of X with ͉A͉ Յ (1͞2)͉X͉ we have ͉ѨA͉ Ն ͉A͉, where ѨA denotes the boundary of A, i.e., the vertices of distance 1 from A.Expander graphs play an important role in computer science and combinatorics, and many efforts has been dedicated to their constructions (cf. ref. 1, the references therein, and ref.2). Many of these constructions are of Cayley graphs, and in particular various infinite families of finite simple groups have been shown to be expanding families.A finite group G is called an -expander with respect to a generating subset S if the Cayley graph Cay(G; S) is an -expander. We will say that an infinite family G of groups is a family of expanders if there exists k ʦ ‫ގ‬ and 0 Ͻ ʦ ‫ޒ‬ such that every group G ʦ G has a subset S of k generators with respect to (w.r.t.) which G is an -expander. In this situation we also say that the groups G ʦ G are expanders ''uniformly.'' Until recently all known such families consisting of simple groups were of bounded Lie rank (cf. refs. 1 and 3), a fact that has raised speculation that this is the only possibility (see refs. 4 and 5).It is easy to see that the diameter of each graph X i from an expander family {X i } is most c log͉X i ͉ (where c is a constant). In ref. 6 it was shown that every non-abelian finite simple group has a set S of seven generators for which the Cayley graph Cay(G; S) has a diameter bounded by C log(͉G͉) for an absolute constant C. It was conjectured there that one can even make all finite simple groups expanders uniformly, although, as observed by Y. Luz (see ref. 4), the generators used in ref. 6 do not give rise to expanders.The main goal of this note is to announce a proof of almost the whole of this conjecture. Theorem 1. There exist k ʦ ‫ގ‬ and 0 Ͻ ʦ ‫ޒ‬ such that every non-abelian finite simple group G, which is not a Suzuki group, has a set S of k generators for which Cay(G; S) is an -expander.In fact careful estimates using variations of some of the arguments below yield k Ͻ 1,000 and Ͼ 10 Ϫ10 .We believe that the above theorem holds also for the Suzuki groups, but our (diverse) methods do not apply to them. What makes them exceptional is the fact that they do not contain copies of SL 2 ‫ކ(‬ p ) or PSL 2 ‫ކ(‬ p ) like all the other finite simple groups (see below for more details).The proof of Theorem 1 is the accumulation of the works (refs. 7-10 and A.L., unpublished results) in the following chronological order.In ref. . This includes, in particular, the groups EL 3 (Mat n ‫ކ(‬ q )) Ӎ SL 3n ‫ކ(‬ q ), and thus SL 3n ‫ކ(‬ q ) are uniformly expanders for all n and for all prime powers q. For every d Ն 3, the group SL d ‫ކ(‬ q ) is a bounded product of copies of SL 3n ‫ކ(‬ q ) for n ϭ d͞3...

Symmetric groups and expander graphs

“…These results, together with some new ones in the rank one case, are combined in [18], which almost proves Conjecture 5.3.…”

Expander graphs and their applications

“…(We only know this for "most" simple groups, since the claim is not known to hold for the socalled simple groups of Suzuki type). All these exciting developments are explained in [KLN05].…”

On the number of universal sofic groups

“…(For example, see Lubotzky [8] and Kassabov-Lubotzky-Nikolov [6].) Here if G is a finite group and S ⊆ G 1 is a generating set, then the corresponding Cayley graph Cay(G, S) is the graph with vertex set G and edge set…”