  Banica and Vergnioux have shown that the dual discrete quantum group of a
compact simply connected Lie group has polynomial growth of order the real
manifold dimension. We extend this result to a general compact group and its
topological dimension, by connecting it with the Gelfand-Kirillov dimension of
an algebra. Furthermore, we show that polynomial growth for a compact quantum
group G of Kac type implies *-regularity of the Fourier algebra A(G), that is
every closed ideal of C(G) has a dense intersection with A(G). In particular,
A(G) has a unique C*-norm.
