Cocompact discontinuous groups for tangential homogeneous spaces

Determine for which pairs (G,H) of real reductive Lie groups the tangential homogeneous space G_\theta/H_\theta (associated to the Cartan motion group of G) admits a cocompact discontinuous group.

Background

The tangential analogue replaces G/H with the simpler Cartan motion group quotient G_\theta/H_\theta to approximate the original classification problem for cocompact discontinuous groups.

Even in symmetric cases, the tangential problem remains unresolved, though it is expected to be more tractable; certain special cases (e.g., pseudo-Riemannian space forms) are settled via RadonHurwitz numbers.