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.
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