## Abstract

The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When n is less than the rank of the associated symmetric space, we show that the n-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when n is equal to the rank, we show that the n-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky{Mozes{Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux{Wortman.

Original language | English (US) |
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Pages (from-to) | 733-792 |

Number of pages | 60 |

Journal | Annals of Mathematics |

Volume | 193 |

Issue number | 3 |

DOIs | |

State | Published - May 2021 |

## Keywords

- Arithmetic groups
- Dehn function
- Filling invariants
- Probabilistic methods
- Random ats

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty