  In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a
classification of non-compact Lie groups which can act isometrically and
locally effectively on compact Lorentzian manifolds. In the case that the
corresponding Lie algebra contains a direct summand isomorphic to the
two-dimensional special linear algebra or to a twisted Heisenberg algebra,
Zeghib also described the geometric structure of the manifolds. Using these
results, we investigate the local geometry of compact homogeneous Lorentz
spaces whose isometry groups have non-compact connected components. It turns
out that they all are reductive. We investigate the isotropy representation and
curvatures. In particular, we obtain that any Ricci-flat compact homogeneous
Lorentz space is flat or has compact isometry group.
