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arXiv · 2401.00654

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

Sven Sandfeldt
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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 $C^{\infty}$.

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Sven Sandfeldt. 2024-01-01. Global rigidity for some partially hyperbolic abelian actions with 1-dimensional center. https://arxiv.org/abs/2401.00654

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