## Abstract

We prove the following conjecture of Margulis. Let G be a higher rank simple Lie group, and let Λ ≤ G be a discrete subgroup of innite covolume. Then, the locally symmetric space Λ\G/K admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stuck-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.

Original language | English (US) |
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Pages (from-to) | 389-421 |

Number of pages | 33 |

Journal | Annals of Mathematics |

Volume | 197 |

Issue number | 1 |

DOIs | |

State | Published - 2023 |

### Funding

Keywords: locally symmetric spaces, discrete subgroups of Lie groups, stationary random subgroups, thin groups, lattices, normal subgroup theorem AMS Classification: Primary: 22E40, 53C24, 22E15, 22F30, 53C35, 32M15. T.G. Supported by ISF-Moked grant 2095/15. © 2023 Department of Mathematics, Princeton University. Acknowledgments. We thank Uri Bader for sharing with us some insights concerning stationary measures and Poisson boundaries. Our work was supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester. The second author was partially supported by the Israel Science Foundation grant No. 2919/19.

## Keywords

- discrete subgroups of Lie groups
- lattices
- locally symmetric spaces
- normal subgroup theorem
- stationary random subgroups
- thin groups

## ASJC Scopus subject areas

- Mathematics (miscellaneous)