We consider reduced group $C^{ \ast}$-algebras of finitely generated discrete groups metrized by seminorms obtained from word length functions. We study the metric dimensions of such $C^{ \ast}$-algebras as defined by David Kerr. We also study the entropy of the automorphisms of group $C^{ \ast}$-algebras induced by the automorphisms of the underlying groups. Both the metric dimension and entropy are related to the growth of the groups. We exhibit a class of examples of virtually abelian finitely generated discrete groups $\Gamma$ and automorphisms $\psi$ of $\Gamma$ such that $\psi$ has non-zero finite product entropy in the sense of David Kerr.
Categories: math.OA math.GR