Suzuki groups as expanders

Abstract

Abstract. We show that pairs of generators for the family Sz.q/ of Suzuki groups may be selected so that the corresponding Cayley graphs are expanders. By combining this with several deep works of Kassabov, Lubotzky and Nikolov, this establishes that the family of all non-abelian finite simple groups can be made into expanders in a uniform fashion. Mathematics Subject Classification (2010). 20D60, 11B30.

Expansion in SL d (Z/q Z), q arbitrary

“…A partial result in this direction is given in [8]. Expanding properties of groups of Lie type with respect to random generators are studied in [10]- [11].…”

Expansion in Finite Simple Groups of Lie Type

“…The method used in the present paper follows the Bourgain-Gamburd strategy first introduced in [BG], as did the paper [BGT2] on Suzuki groups by three of the authors, and will be outlined in the next section. To verify the various steps of the BourgainGamburd argument, we will need a number of existing results in the literature, such as the quasirandomness properties and product theorems for finite simple groups of Lie type, as well as the existence of strongly dense free subgroups that was (mostly) established in a previous paper [BGGT] of the authors.…”

Non-expander Cayley Graphs of Simple Groups

“…The proof of the fact was completed by Kassabov, Lubotzky and Nikolov in [3] based on several earlier work (see [4], [5], [6], [8], [12]), that there exist k ∈ N and 0 < ǫ ∈ R such that every non-abelian finite simple group which is not a Suzuki group has a set of generators S of size at most k for which Cay(G, S) is an ǫ-expander. This work was extended by Breuillard, Green and Tao in [1] to the Suzuki groups. These results can also motivate the question which was asked by Lubotzky in [11] whether every family of Chevalley groups of bounded rank is a family of uniformly expanding group.…”