2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77637Let M^3 be a compact, oriented, irreducible, and boundary incompressible 3-manifold. Assume that its fundamental group is without rank two abelian subgroups and its boundary is non-empty. We will show that every homomorphism from pi_1(M) to PSL(2,C) which is not `boundary elementary' is induced by a possibly branched complex projective structure on the boundary of a hyperbolic manifold homeomorphic to M.6 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon1/paper16.abs.htmlGeometric Topology30F50, 30F45, 30F60, 30F99, 30C99text