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

Connes' metric for states in group algebras