  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.
