Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

TL;DR

This paper investigates cocycles over higher-rank abelian group actions on quotients of semisimple Lie groups, demonstrating that smooth cocycles are cohomologous to constants via smooth transfer functions.

Contribution

It establishes the cohomology triviality of smooth cocycles for specific higher-rank abelian actions on semisimple Lie group quotients, extending understanding of dynamical rigidity.

Findings

Smooth cocycles are cohomologous to constants via smooth transfer functions.

Results apply to actions from flows of commuting elements and unipotent flows.

Provides new rigidity results for higher-rank abelian actions on Lie group quotients.

Abstract

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra--one nilpotent, and the other semisimple. Second, we consider actions from two commuting unipotent flows coming from two commuting embedded copies of SL(2,R). In both cases we show that any smooth real-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function.