Suzuki groups as expanders. (Q664233)

Let \(\varepsilon>0\) be a real number. A graph \(X\) is called an \(\varepsilon\)-expander if, for all sets \(A\) consisting of at most half the vertices of \(X\), we have \(|\partial A|\geq\varepsilon|A|\) where \(\partial A\) is the set of vertices of \(X\setminus A\) that are joined to a vertex in \(A\). An important particular case is a Cayley graph. Given a finite group \(G\) with a symmetric generating set \(S\), one defines the Cayley graph \(\text{Cay}(G,S)\) to be the graph with vertex set \(G\) in which vertices \(x\) and \(y\) are joined to an edge if and only if \(x=ys\) for some \(s\in S\). \textit{M. Kassabov, A. Lubotzky} and \textit{N. Nikolov} [Proc. Natl. Acad. Sci. USA 103, No. 16, 6116-6119 (2006; Zbl 1161.20010)] announced the following result: There exist \(k\in\mathbb N\) and \(\varepsilon>0\) such that, for every non-Abelian finite simple group \(G\) with the possible exception of the Suzuki groups \(Sz(q)\), one may select a symmetric generating set \(S\) of \(k\) generators for which \(\text{Cay}(G,S)\) is an \(\varepsilon\)-expander. The main aim of this paper is to remove the lacuna in this result. It is proved that there exists \(\varepsilon>0\) such that, for every Suzuki group \(G=Sz(q)\), one may select a generating pair \(a,b\) in \(G\) such that, setting \(S:=\{a^{\pm 1},b^{\pm 1}\}\), the Cayley graph \(\text{Cay}(G,S)\) is an \(\varepsilon\)-expander. In fact, the authors prove that a random pair of elements \(a,b\) will generate \(Sz(q)\) and have the above expansion property with probability going to 1 as \(q\to\infty\).