Diffeomorphism group valued cocycles over higher rank abelian Anosov actions

TL;DR

This paper establishes new rigidity results for smooth cocycles valued in diffeomorphism groups over higher rank abelian Anosov actions, showing they are trivial or reducible under broad conditions, without smallness restrictions.

Contribution

It provides the first general rigidity results for such cocycles without smallness or group restrictions, extending the understanding of cocycle trivialization over higher rank actions.

Findings

Cocycles trivial at a fixed point are smooth coboundaries on a finite cover.

Non-trivial cocycles can be reduced to constant cocycles on the universal cover.

Results apply to maximal Cartan actions on any compact smooth manifold.

Abstract

We prove that every smooth diffeomorphism group valued cocycle over certain abelian Anosov actions on tori (and more generally on infranilmanifolds), is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialisation apply, via the existing global rigidity results, to maximal Cartan actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over higher rank abelian actions, with values in diffeomorphism groups, which does not require any restrictions on the smallness of the cocycle, nor on the diffeomorphism group.