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September 2019 Strong property $(T)$ for higher-rank lattices
Mikael de la Salle
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Acta Math. 223(1): 151-193 (September 2019). DOI: 10.4310/ACTA.2019.v223.n1.a3


We prove that every lattice in a product of higher-rank simple Lie groups or higher-rank simple algebraic groups over local fields has Vincent Lafforgue’s strong property $(T)$. Over non-Archimedean local fields, we also prove that they have strong Banach property $(T)$ with respect to all Banach spaces with non-trivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-co-compact lattices, such as $\mathop{\rm SL}_n(\mathbf{Z})$ for $n \geqslant 3$. To do so, we introduce a stronger form of strong property $(T)$ which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher-rank groups have this property and that this property passes to undistorted lattices.


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Mikael de la Salle. "Strong property $(T)$ for higher-rank lattices." Acta Math. 223 (1) 151 - 193, September 2019.


Received: 4 October 2018; Published: September 2019
First available in Project Euclid: 16 April 2020

zbMATH: 1429.22008
MathSciNet: MR4018265
Digital Object Identifier: 10.4310/ACTA.2019.v223.n1.a3

Rights: Copyright © 2019 Institut Mittag-Leffler


Vol.223 • No. 1 • September 2019
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