## Abstract

We prove that a large family of higher rank simple Lie groups (including SL _{n}(R) for n≥ 3) and their lattices have Banach property (T) with respect to all super-reflexive Banach spaces. Two consequences of this result are: First, we deduce Banach fixed point properties with respect to all super-reflexive Banach spaces for a large family of higher rank simple Lie groups. For example, we show that for every n≥ 4 , the group SL _{n}(R) and all its lattices have the Banach fixed point property with respect to all super-reflexive Banach spaces. Second, we settle a long standing open problem and show that the Margulis expanders (Cayley graphs of SL _{n}(Z/ mZ) for a fixed n≥ 3 and m tending to infinity) are super-expanders. All of our results stem from proving Banach property (T) for SL _{3}(Z) . Our method of proof for SL _{3}(Z) relies on a novel proof for relative Banach property (T) for the uni-triangular subgroup of SL _{3}(Z) . This proof of relative property (T) is new even in the classical Hilbert setting and is interesting in its own right.

Original language | English |
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Pages (from-to) | 893-930 |

Number of pages | 38 |

Journal | Inventiones Mathematicae |

Volume | 234 |

Issue number | 2 |

DOIs | |

State | Published - 1 Nov 2023 |

## ASJC Scopus subject areas

- General Mathematics

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