We prove that Cayley graphs of SL2 (Fp ) are expanders with respect to the projection of any fixed elements in SL(2, Z) generating a non-elementary subgroup, and with respect to generators chosen at random in SL2 (Fp ). 1. Introduction Expanders are highly-connected sparse graphs widely used in computer science, in areas ranging from parallel computation to complexity theory and cryptography; recently they also have found some remarkable applications in pure mathematics; see [5],[10], [15], [20], [21] and references therein. .
