### Abstract

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 T^{n} for n ≥ 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

Original language | English (US) |
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Pages (from-to) | 913-972 |

Number of pages | 60 |

Journal | Annals of Mathematics |

Volume | 186 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1 2017 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

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*Annals of Mathematics*, vol. 186, no. 3, pp. 913-972. https://doi.org/10.4007/annals.2017.186.3.3

**Global smooth and topological rigidity of hyperbolic lattice actions.** / Brown, Aaron; Rodriguez Hertz, Federico Juan; Wang, Zhiren.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Global smooth and topological rigidity of hyperbolic lattice actions

AU - Brown, Aaron

AU - Rodriguez Hertz, Federico Juan

AU - Wang, Zhiren

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

VL - 186

SP - 913

EP - 972

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 3

ER -