Expansion of coset graphs of PSL_2(F_p)

Let $G$ be a finite group and let $H_1,H_2<G$ be two subgroups. In this paper, we are concerned with the bipartite graph whose vertices are $G/H_1\cup G/H_2$andacoset$g_1H_1$isconnectedwithanothercoset$g_2H_2$ if and only if $g_1{H}_1\cap g_2{H}_2\mathrm{\ne }\mathrm{\varnothing }$. The main result of the paper establishes the existence of such graphs with large girth and large spectral gap. Lubotzky, Manning and Wilton use such graphs to construct certain infinite groups of interest in geometric group theory.