2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66758We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theory of general metric spaces of nonpositive curvature.26 pagesGroup TheoryDifferential GeometryPrimary 53C20, 20F67; Secondary 20F65text