Group actions and compact Clifford-Klein forms of homogeneous spaces

A compact Clifford-Klein form of the homogeneous space J\H is a compact manifold J\H/\Gamma constructed using a discrete subgroup \Gamma of H. I will briefly survey the existence problem for compact forms, and will show how the existence of an action by a large (higher-rank) semisimple group proves that many spaces do not have compact forms. I will also make some remarks on a conjecture of Kobayashi on the scarcity of compact forms.