Fundamental group of compact decent spacetimes with lightlike hypersurface curvature

In this paper, we study the fundamental group of compact de cent 4-dimensional spacetimes with lightlike hypersurface curvature. Such manifolds are foliated by totally geodesic null hypersurfaces. We prove that when the leaves of the foliation are compact then the fun damental group of the spacetime is polycyclic. We also prove that a non totally vicious 4-dimensional compact pp-wave spacetime has polycyclic fundamental group.