### Abstract

Let X= G/ K be a symmetric space of noncompact type and rank k≥ 2. We prove that horospheres in X are Lipschitz (k- 2) -connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible Q-rank-1 lattice Γ in a linear, semisimple Lie group G of R-rank k is k- 1. That is, given m< k- 1 , a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) Γ , and a (m+ 1) -ball B in X (or G) filling S, there is a (m+ 1) -ball B^{′} in Γ filling S such that vol B^{′}∼ vol B. In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension k- 1.

Original language | English (US) |
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Pages (from-to) | 69-87 |

Number of pages | 19 |

Journal | Geometriae Dedicata |

Volume | 187 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1 2017 |

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### Keywords

- Arithmetic groups
- Dehn functions
- Horospheres
- Lipschitz connectivity
- Subgroup distortion
- Symmetric spaces

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Geometriae Dedicata*,

*187*(1), 69-87. https://doi.org/10.1007/s10711-016-0189-6