2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/166172We study some basic properties of the variety of characters in PSL(2,C) of a finitely generated group. In particular we give an interpretation of its points as characters of representations. We construct 3-manifolds whose variety of characters has arbitrarily many components that do not lift to SL(2,C). We also study the singular locus of the variety of characters of a free group.Final version published by Sociedad Matematica Mexicana. BoletinGeometric Topology57M50, 57M05, 20C15text