Strong property (T) for higher-rank lattices

Mikael de la Salle École Normale Supérieure de Lyon

Functional Analysis mathscidoc:1911.43044

Acta Mathematica, 223, (1), 151 – 193, 2019
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 SLn(Z) for n⩾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. Full Text (PDF format)
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@inproceedings{mikael2019strong,
  title={Strong property (T) for higher-rank lattices},
  author={Mikael de la Salle},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191128155412839007555},
  booktitle={Acta Mathematica},
  volume={223},
  number={1},
  pages={151 – 193},
  year={2019},
}
Mikael de la Salle. Strong property (T) for higher-rank lattices. 2019. Vol. 223. In Acta Mathematica. pp.151 – 193. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191128155412839007555.
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