2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161250In 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 $Γ$, 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 $Γ$ 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 $C_r*(Γ)$.Operator AlgebrasMetric Geometry46L30 (Primary) 46L05, 46L85 (Secondary)text