Tits alternative for closed real analytic 4-manifolds of nonpositive curvature

We 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.