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

Diffeomorphism group valued cocycles over higher rank abelian Anosov actions