Properly 3-realizable groups

A finitely presented group $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $\pi_1(K)\cong G$ and whose universal cover has the proper homotopy type of a 3-manifold (with boundary). We discuss the behavior of this property with respect to amalgamated products, HNN-extensions, and direct products, as well as the independence with respect to the chosen 2-polyhedron. We also exhibit certain classes of groups satisfying this property: finitely generated Abelian groups, (classical) hyperbolic groups, and one-relator groups.