Universal lattices and unbounded rank expanders

We study the representations of non-commutative universal lattices and use them to compute lower bounds for the τ -constant for the commutative universal lattices Gd,k = SLd(Z[x1, . . . , xk]) with respect to several generating sets. As an application of the above result we show that the Cayley graphs of the finite groups SL3k(Fp) can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and Benjamin Weiss.