On strong property (T) and fixed point properties for Lie groups
Annales de l'Institut Fourier, Volume 66 (2016) no. 5, p. 1859-1893

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.

Nous considérons certains renforcements de la propriété (T) Banachique. Soit $X$ un espace de Banach pour lequel la distance de Banach–Mazur à un espace euclidien de tout sous-espace de dimension $k$ croît comme une puissance de $k$ strictement inférieure à un demi. Nous prouvons que tout groupe de Lie simple connexe et de rang réel suffisament grand a la propriété (T) renforcée de Lafforgue relativement à $X$. Par conséquent toute action continue par isométries affines d’un tel groupe (ou d’un réseau dans un tel groupe) sur $X$ a un point fixe. Pour les groupes spéciaux linéaires, nous présentons aussi une approche plus directe aux propriétés de point fixe. Plus précisément nous prouvons que tout groupe spécial linéaire de rang suffisament grand a la propriété suivante : tous ses quasi-$1$-cocycles à valeurs dans une représentations isométrique sur $X$ sont bornés.

Revised : 2015-11-20
Accepted : 2016-01-21
Published online : 2016-07-28
DOI : https://doi.org/10.5802/aif.3051
Classification:  20J06,  22D12,  22E45,  46B20
Keywords: Strong property (T), Banach space representations, geometry of Banach spaces, bounded cohomology
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author = {de Laat, Tim and Mimura, Masato and de la Salle, Mikael},
title = {On strong property (T) and fixed point properties for Lie groups},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {66},
number = {5},
year = {2016},
pages = {1859-1893},
doi = {10.5802/aif.3051},
language = {en},
url = {http://www.numdam.org/item/AIF_2016__66_5_1859_0}
}

de Laat, Tim; Mimura, Masato; de la Salle, Mikael. On strong property (T) and fixed point properties for Lie groups. Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 1859-1893. doi : 10.5802/aif.3051. http://www.numdam.org/item/AIF_2016__66_5_1859_0/

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