A cohomological obstruction to the existence of compact Clifford-Klein forms

In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. $\mathrm{SO}_0(p+r, q)/(\mathrm{SO}_0(p,q) \times \mathrm{SO}(r))$ $(p,q,r \geq 1, \ q:\text{odd})$ and $\mathrm{SL}(p+q, \mathbb{C})/\mathrm{SU}(p,q)$ $(p,q \geq 1)$, of a homogeneous space that does not admit a compact Clifford-Klein form. To construct these examples, we apply H. Cartan's theorem on relative Lie algebra cohomology of reductive pairs and the theory of $\epsilon$-families of semisimple symmetric pairs.