Connes' metric for states in group algebras

In this article we follow the main idea of A. Connes for the construction of a metric in the state space of a C*-algebra. We focus in the reduced algebra of a discrete group $\Gamma$, and prove some equivalences and relations between two central objects of this category: the word-length growth (connected with the degree of the extension of $\Gamma$ when the group is an extension of Z by a finite group), and the topological equivalence between the w*-topology and the one introduced with this metric in the state space of $C_r*(\Gamma)$.