Compact forms of homogeneous spaces and higher-rank semisimple group actions

Abstract

This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, techniques from partially hyperbolic dynamics, and the geometry and pseudo-Riemannian structure of the homogeneous space.