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Tim de Laat; Masato Mimura; Mikael de la Salle
On strong property (T) and fixed point properties for Lie groups
(Sur la propriété (T) renforcée et la propriété de point fixe pour les groupes de Lie)
Annales de l'institut Fourier, 66 no. 5 (2016), p. 1859-1893, doi: 10.5802/aif.3051
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Class. Math.: 20J06, 22D12, 22E45, 46B20
Keywords: Strong property (T), Banach space representations, geometry of Banach spaces, bounded cohomology

Résumé - Abstract

We consider certain strengthenings of property (T) relative to Banach spaces. Let $X$ be a Banach space for which the Banach–Mazur distance to a Hilbert space of all $k$-dimensional subspaces grows as a power of $k$ strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank has strong property (T) of Lafforgue with respect to $X$. As a consequence, every continuous affine isometric action of such a high rank group (or a lattice in such a group) on $X$ has a fixed point. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. We prove that every special linear group of sufficiently large rank satisfies the following property: every quasi-$1$-cocycle with values in an isometric representation on $X$ is bounded.

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