Spacelike cocompact actions on pseudo-riemannian symmetric spaces

A reductive symmetric space $G/H$ has a natural pseudo-riemannian geometry. We call a subgroup $\Gamma$ of $G$ \emph{spacelike cocompact} if it acts properly and cocompactly on a spacelike submanifold of $G/H$ of maximal dimension. In this talk I will advocate for a theory of spacelike cocompact actions, present a general result and discuss various examples and open questions.