## Abstract

We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if is a cocompact lattice in SL(n;R), M is a compact manifold, and ! a volume form on M, we show that any homomorphism (Formula Presented)) has finite image if the dimension of M is less than n - 1 and that any homo-morphism (Formula Presented) has finite image if the dimension of M is less than n. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Laorgue's strong property (T) to establish the existence of an invariant Riemannian metric.

Original language | English (US) |
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Pages (from-to) | 891-940 |

Number of pages | 50 |

Journal | Annals of Mathematics |

Volume | 196 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2022 |

## Keywords

- Actions of abelian groups
- Actions of lattices
- Lattices in semisimple lie groups
- Lyapunov exponents
- Measure rigidity
- Property (t)
- Ratner theory
- Zimmer program

## ASJC Scopus subject areas

- Mathematics (miscellaneous)