  We prove that a large family of higher rank simple Lie groups (including $\rm
SL_n (\mathbb{R})$ for $n \geq 3$) and their lattices have Banach property (T)
with respect to all super-reflexive Banach spaces.
  Two consequences of this result are: First, we deduce Banach fixed point
properties with respect to all super-reflexive Banach spaces for a large family
of higher rank simple Lie groups. For example, we show that for every $n \geq
4$, the group $\rm SL_n (\mathbb{R})$ and all its lattices have the Banach
fixed point property with respect to all super-reflexive Banach spaces. Second,
we settle a long standing open problem and show that the Margulis expanders
(Cayley graphs of $\rm SL_{n} (\mathbb{Z} / m \mathbb{Z} )$ for a fixed $n \geq
3$ and $m$ tending to infinity) are super-expanders.
  All of our results stem from proving Banach property (T) for $\rm SL_3
(\mathbb{Z})$. Our method of proof for $\rm SL_3 (\mathbb{Z})$ relies on a
novel proof for relative Banach property (T) for the uni-triangular subgroup of
$\rm SL_3 (\mathbb{Z})$. This proof of relative property (T) is new even in the
classical Hilbert setting and is interesting in its own right.
