Global rigidity for some partially hyperbolic abelian actions with 1-dimensional center

Abstract

We obtain a global rigidity result for abelian partially hyperbolic higher rank actions on certain $2-$step nilmanifolds $X_{\Gamma}$. We show that, under certain natural assumptions, all such actions are $C^{\infty}-$conjugated to an affine model. As a consequence, we obtain a centralizer rigidity result, classifying all possible centralizers for any $C^{1}-$small perturbation of an irreducible, affine partially hyperbolic map on $X_{\Gamma}$. Along the way, we also prove two results of independent interest. We describe fibered partially hyperbolic diffeomorphisms on $X_{\Gamma}$ and we show that topological conjugacies between partially hyperbolic actions and higher rank affine actions are $C^{\infty}$.