# Cocycle rigidity of abelian partially hyperbolic actions

## TL;DR

This paper investigates cocycle rigidity for abelian higher-rank partially hyperbolic actions, characterizing obstructions, constructing smooth solutions, and establishing Sobolev estimates, thus advancing understanding of rigidity in non-hyperbolic directions.

## Contribution

It introduces the first analysis of cocycle rigidity where hyperbolic directions do not span the entire tangent space, extending rigidity results to partially hyperbolic actions.

## Key findings

- Characterization of obstructions to solving twisted cohomological equations
- Construction of smooth solutions with tame Sobolev estimates
- Proof of cocycle rigidity for abelian higher-rank partially hyperbolic algebraic actions

## Abstract

Suppose $G$ is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let $\mathbb{G}=G$ or $\mathbb{G}=G\ltimes V$, where $V$ is a finite dimensional vector space $V$. For any unitary representation $(\pi,\mathcal{H})$ of $\GG$, we study the twisted cohomological equation $\pi(a)f-\lambda f=g$ for partially hyperbolic element $a\in \mathbb{G}$ and $\lambda\in U(1)$, as well as the twisted cocycle equation $\pi(a_1)f-\lambda_1f=\pi(a_2)g-\lambda_2 g$ for commuting partially hyperbolic elements $a_1,\,a_2\in \mathbb{G}$. We characterize the obstructions to solving these equations, construct smooth solutions and obtain tame Sobolev estimates for the solutions. These results can be extended to partially hyperbolic flows parallelly. As an application, we prove cocycle rigidity for any abelian higher-rank partially hyperbolic algebraic actions. This is the first paper exploring rigidity properties of partially hyperbolic that the hyperbolic directions don't generate the whole tangent space. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for these actions in future works.