On properly 3-realizable groups (Q2577126)

A group \(G\) is called \textit{properly 3-realizable} if there exists a compact 2-polyhedron having \(G\) as fundamental group whose universal cover is properly homotopy equivalent to a 3-manifold with boundary. The authors show that certain amalgamated free products and HNN-extensions over cyclic groups are properly 3-realizable. As a consequence it can be seen that there are properly 3-realizable groups which can only be realized by a fake surface with points of type III.