Random Schreier graphs and expanders

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.