Closed geodesics on compact Lorentzian solvmanifolds

The aim of this work is the study of geodesics on Lorentzian homogeneous spaces of the form $M=G/\Lambda$, where $G$ is a solvable Lie group endowed with a bi-invariant Lorentzian metric and $\Lambda < G$ is a cocompact lattice. Conditions to assert closedness of light, time or spacelike geodesics on the compact quotient spaces are given. This study implicitly requires additional information about the lattices in each case. We found conditions for which every lightlight geodesic on the quotient space is closed. And more important, this situation depends on the lattice. Moreover, even in dimension four, there are examples of compact solvmanifolds for which not every lightlike geodesic is closed. For time and spacelike geodesics, the conclusion are different. Finally, we study isometry groups of those compact spaces and show some computations in dimension six.