EXPANSION OF COSET GRAPHS OF PSL2(Fp)

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 boolean OR G/H-2 and a coset g(1)H(1) is connected with another coset g(2)H(2) if and only if g(1)H(1) boolean AND g(2)H(2) not equal empty set. 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.