Homotopy dominations by polyhedra with polycyclic-by-finite fundamental groups

In the previous papers, in connection with a question of K. Borsuk, we proved that there exist polyhedra with polycyclic fundamental groups homotopy dominating infinitely many different homotopy types. Here we consider a few problems of K. Borsuk concerning infinite chains of polyhedra or FANRʹs ordered by the relation of domination (in homotopy or shape category) and obtain that for polyhedra with polycyclic-by-finite fundamental groups, there are no pathology similar to the above.