Local Rigidity for Cocycles

TL;DR

This paper demonstrates that continuous perturbations of constant cocycles for higher rank semi-simple group actions are actually conjugate to the original cocycle, with implications for local rigidity and superrigidity results.

Contribution

It establishes that continuous perturbations of constant cocycles are continuously conjugate to the original, extending Zimmer's superrigidity and aiding in local rigidity proofs.

Findings

Perturbed cocycles are continuously conjugate to original cocycles.

Results apply to standard actions of higher rank semisimple Lie groups.

Improves and extends Zimmer's cocycle superrigidity statements.

Abstract

In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued cocycle. The main point of this article is to see that a cocycle which is a continuous perturbation of a constant cocycle is actually continuously conjugate back to the original constant cocycle modulo a cocycle that is continuous and "small". We give some applications to perturbations of standard actions of higher rank semisimple Lie groups and their lattices. Some of the results proven here are used in our proof of local rigidity for affine and quasi-affine actions of these groups. We also improve and extend the statements and proofs of Zimmer's cocycle superrigidity.