2003/03/19 by David Fisher, G. A. Margulis, Fisher, David +1 · 1 citation
Mathematics · #37C85 #53C24 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.DG #math.DS #msc:37C85 #msc:53C24
paper · pdf · doi:10.48550/arxiv.math/0303236
58 pages. to appear Surveys in Differential Geometry Vol. 7
openalex publication_date 2003/03/19 · arxiv created 2003/11/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.