So that s definitely a possible fundamental group of a 3 manifold But if you consider all the other conditions of the fibering conjecture it s not allowed H 1 infinite means there s a non separating surface irrreduciblity means it can t be a sphere Compressing the surface gives a pi 1 injective surface so the 3 manifold group contains a surface subgroup which can t be the case if it s the integers since the sphere is excluded Actually the conclusion that the manifold fibers implies that the group contains a surface subgroup The surface can t be a sphere because of irreducibility so we have a more complicated surface whose fundamental group injects into the fundamental group of the 3 manifold