Universal lattices and unbounded rank expanders

We study the representations of non-commutative universal lattices and use them to compute lower bounds of the τ-constant for the commutative universal lattices G _d,k =SL _d (ℤ[x ₁,...,x _k ]), for d≥3 with respect to several generating sets. As an application we show that the Cayley graphs of the finite groups can be made expanders with a suitable choice of generators. This provides the first example of expander families of groups of Lie type, where the rank is not bounded and provides counter examples to two conjectures of A. Lubotzky and B. Weiss.