Metric dimension and product entropy of group $C^{\ast }$-algebras

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 product entropy of the automorphisms of group $C^{\ast }$-algebras induced by the automorphisms of the underlying groups. We get a lower bound and an upper bound of the product entropy of an automorphism in terms of the classical group theoretic algebraic and geometric entropy of the automorphisms, provided the group has polynomial growth property. For groups with exponential growth, we show that the metric dimension of the group $C^{\ast }$-algebras is generically $+\mathrm{\infty }$.