On strong property (t) and fixed point properties for lie groups

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.