New Cases of Differential Rigidity for Non-Generic Partially Hyperbolic Actions

TL;DR

This paper establishes new local differentiable rigidity results for certain high-rank partially hyperbolic actions on homogeneous spaces, expanding understanding of rigidity phenomena in dynamical systems.

Contribution

It proves the rigidity of generic partially hyperbolic abelian actions and provides a non-generic example, advancing the geometric approach to group relations.

Findings

Proved local differentiable rigidity for generic actions.

Constructed a non-generic rigidity example.

Progressed in computing relations in symplectic Lie groups.

Abstract

We prove the locally differentiable rigidity of generic partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from split symplectic Lie groups. We also gave a non-generic action rigidity example on compact homogeneous spaces obtained from SL(2n,R) or SL(2n,C). The conclusions are based on geometric Katok-Damjanovic way and progress towards computations of the generating relations in these groups.