On strong property (T) and fixed point properties for Lie groups - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Annales de l'Institut Fourier Année : 2016

On strong property (T) and fixed point properties for Lie groups

Résumé

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional subspaces is bounded above by a power of k strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank depending on X has strong property (T) of Lafforgue with respect to X. As a consequence, we obtain that every continuous affine isometric action of such a high rank group (or a lattice in such a group) on X has a fixed point. This result corroborates a conjecture of Bader, Furman, Gelander and Monod. 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. Without appealing to strong property (T), we prove that given a Banach space X as above, 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.
Fichier principal
Vignette du fichier
dLMdlS%20-%20AIF.pdf (760.45 Ko) Télécharger le fichier
Origine : Fichiers éditeurs autorisés sur une archive ouverte

Dates et versions

hal-01219531 , version 1 (06-02-2024)

Identifiants

Citer

Tim de Laat, Masato Mimura, Mikael de La Salle. On strong property (T) and fixed point properties for Lie groups. Annales de l'Institut Fourier, 2016, 66 (5), pp.1859 - 1893. ⟨10.5802/aif.3051⟩. ⟨hal-01219531⟩
44 Consultations
2 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More