Random Schreier graphs and expanders

Abstract

Let the group G act transitively on the finite set $$\Omega $$ , and let $$S \subseteq G$$ be closed under taking inverses. The Schreier graph $$Sch(G \circlearrowleft \Omega ,S)$$ is the graph with vertex set $$\Omega $$ and edge set $$\{ (\omega ,\omega ^s) : \omega \in \Omega , s \in S \}$$ . In this paper, we show that random Schreier graphs on $$C \log |\Omega |$$ elements exhibit a (two-sided) spectral gap with high probability, magnifying a well-known theorem of Alon and Roichman for Cayley graphs. On the other hand, depending on the particular action of G on $$\Omega $$ , we give a lower bound on the number of elements which are necessary to provide a spectral gap. We use this method to estimate the spectral gap when G is nilpotent.