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

TL;DR

This paper demonstrates the non-existence of compact forms for many homogeneous spaces with higher-rank semisimple Lie group actions, using advanced rigidity and dynamical techniques.

Contribution

It extends Zimmer's approach by proving non-existence results for compact forms of certain homogeneous spaces with higher-rank group actions.

Findings

No compact forms for a large class of homogeneous spaces

Application of cocycle superrigidity and measure rigidity techniques

Integration of geometric and dynamical methods

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.