Local Rigidity of Higher Rank Homogeneous Abelian Actions: a Complete Solution via the Geometric Method

TL;DR

This paper proves local and cocycle rigidity for a broad class of partially hyperbolic translation actions on homogeneous spaces, including new twisted symmetric space examples, using a novel combination of geometric methods and central extension theory.

Contribution

It provides a complete solution to local rigidity for higher rank homogeneous abelian actions, extending previous results to more complex geometric cases and new examples.

Findings

Established local and cocycle rigidity for a wide class of actions

First treatment of partially hyperbolic twisted symmetric space examples

Introduced a new proof technique combining geometric methods and central extensions

Abstract

We show local and cocycle rigidity for $\R^k \times \Z^l$ partially hyperbolic translation actions on homogeneous spaces $\mc G/ \Lambda$. We consider a large class of actions whose geometric properties are more complicated than previously treated cases. It is also the first time that partially hyperbolic twisted symmetric space examples have been treated in the literature. The main new ingredient in the proof is a combination of geometric method and the theory of central extensions.