  Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an
integer q, denote by Ga_q the subgroup of Ga consisting of the elements that
project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q
with respect to the generating set S form a family of expanders when q ranges
over square-free integers with large prime divisors if and only if the
connected component of the Zariski-closure of Ga is perfect.
