Rigidity of Measure-Preserving Actions of Higher Rank Lattices on Homogeneous Spaces

This monograph explores the rigidity phenomena inherent in the measure-preserving actions of lattices in higher-rank semisimple Lie groups, situated within the broader context of Zimmer's Program. We analyze the obstruction to the existence of non-trivial smooth actions of such lattices on low-dimensional manifolds, generalizing Margulis' superrigidity theorem to the non-linear setting. Specifically, we discuss the recent breakthroughs concerning the Zimmer Conjecture, establishing that any volume-preserving action of a lattice SL(n, R) (n 3) on a compact manifold M of dimension strictly less than n-1 must factor through a finite group. The exposition rigorously develops the machinery of cocycle superrigidity, the measurable reduction of principal bundles, and the role of property (T) in suppressing dynamical complexity. Furthermore, we examine the implications of subexponential growth of derivatives and the existence of invariant geometric structures, providing a comprehensive synthesis of the ergodic and geometric methods defining the current state of the field.