On properly 3-realizable groups

We study the class of those finitely presented groups which are “properly 3-realizable”, i.e., those groups G for which there exists a compact 2-polyhedron having G as fundamental group and whose universal cover is proper homotopy equivalent to a 3-manifold (with boundary). We show that certain amalgamated free product of groups G1∗FG2 (HNN-extensions G∗F), over a cyclic group F, are properly 3-realizable. As a consequence, we note that there are properly 3-realizable groups which cannot be realized by a fake surface without points of type III. The question of whether or not every finitely presented group is properly 3-realizable still remains open.