TY - JOUR

T1 - Global smooth and topological rigidity of hyperbolic lattice actions

AU - Brown, Aaron

AU - Hertz, Federico Rodriguez

AU - Wang, Zhiren

N1 - Funding Information:
Acknowledgments. We are grateful to Anatole Katok, David Fisher, and Ralf Spatzier for suggesting the problem and for inspiring discussions. We also thank Amir Mohammadi for helpful discussion regarding superrigidity. A. B. was partially supported by an NSF postdoctoral research fellowship DMS-1104013. F. R. H. was supported by NSF grants DMS-1201326 and DMS-1500947. Z. W. was supported by NSF grants DMS-1201453 and DMS-1501295, as well as a research membership at the Institute for Advanced Study.
Publisher Copyright:
© 2017 Department of Mathematics, Princeton University.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose Γ is a lattice in a semisimple Lie group, all of whose factors have rank 2 or higher. Let α be a smooth Γ-action on a compact nilmanifold M that lifts to an action on the universal cover. If the linear data ρ of α contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of α and ρ on a finite-index subgroup of Γ. If α is a C∞ action and contains an Anosov element, then the semiconjugacy is a C∞ conjugacy. As a corollary, we obtain C∞ global rigidity for Anosov actions by co- compact lattices in semisimple Lie groups with all factors rank 2 or higher. We also obtain global rigidity of Anosov actions of SL(n; Z) on Tn for n ≥ 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

AB - In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose Γ is a lattice in a semisimple Lie group, all of whose factors have rank 2 or higher. Let α be a smooth Γ-action on a compact nilmanifold M that lifts to an action on the universal cover. If the linear data ρ of α contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of α and ρ on a finite-index subgroup of Γ. If α is a C∞ action and contains an Anosov element, then the semiconjugacy is a C∞ conjugacy. As a corollary, we obtain C∞ global rigidity for Anosov actions by co- compact lattices in semisimple Lie groups with all factors rank 2 or higher. We also obtain global rigidity of Anosov actions of SL(n; Z) on Tn for n ≥ 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

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U2 - 10.4007/annals.2017.186.3.3

DO - 10.4007/annals.2017.186.3.3

M3 - Article

AN - SCOPUS:85039725048

SN - 0003-486X

VL - 186

SP - 913

EP - 972

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -