Given a homogeneous space J\H, does there exist a discrete subgroup $\Gamma$ in $H$ such that J\H/$\Gamma$ is a compact manifold? These compact forms of homogeneous spaces turn out to be rare outside of a few natural cases. Their existence has been studied by a very wide range of techniques, one of which is via the action of the centralizer of $J$ in $H$. In this talk I'll show that no compact form exists when $H$ is a simple Lie group, $J$ is reductive and the acting group is higher-rank and semisimple. The proof uses cocycle superrigidity, Ratner's theorem and techniques from partially hyperbolic dynamics.