  We study the representations of non-commutative universal lattices and use
them to compute lower bounds for the \TauC for the commutative universal
lattices $G_{d,k}= \SL_d(\Z[x_1,...,x_k])$ with respect to several generating
sets.
  As an application of the above result we show that the Cayley graphs of the
finite groups $\SL_{3k}(\F_p)$ 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.
