Random unitaries, amenable linear groups and Jordan's theorem

It is well known that a dense subgroup $G$ of the complex unitary group $U(d)$ cannot be amenable as a discrete group when $d>1$. When $d$ is large enough we give quantitative versions of this phenomenon in connection with certain estimates of random Fourier series on the compact group $\bar G$ that is the closure of $G$. Roughly, we show that if $\bar G$ covers a large enough part of $U(d)$ in the sense of metric entropy then $G$ cannot be amenable. The results are all based on a version of a classical theorem of Jordan that says that if $G$ is finite, or amenable as a discrete group, then $G$ contains an Abelian subgroup with index $e^{o(d^2)}$.