Button observed that finitely generated matrix groups containing no nontrivial unipotent matrices behave much like groups admitting proper actions by semisimple isometries on complete CAT(0) spaces. It turns out that any finitely generated matrix group possesses an action on such a space whose restrictions to unipotent-free subgroups are in some sense tame. We discuss this phenomenon and its relevance to the following interesting characterization of nonpositive curvature for 3-manifolds: a closed aspherical 3-manifold admits a nonpositively curved Riemannian metric if and only if its fundamental group embeds in a compact Lie group.