  This article contains two rigidity type results for
$\mathrm{SL}(n,\mathbb{Z})$ for large $n$ that share the same proof. Firstly,
we prove that for every $p \in [1,\infty]$ different from $2$, the
noncommutative $L^p$-space associated with $\mathrm{SL}(n,\mathbb{Z})$ does not
have the completely bounded approximation property for sufficiently large $n$
depending on $p$.
  The second result concerns the coarse embeddability of expander families
constructed from $\mathrm{SL}(n,\mathbb{Z})$. Let $X$ be a Banach space and
suppose that there exist $\beta < \frac{1}{2}$ and $C>0$ such that the
Banach-Mazur distance to a Hilbert space of all $k$-dimensional subspaces of
$X$ is bounded above by $C k^\beta$. Then the expander family constructed from
$\mathrm{SL}(n,\mathbb{Z})$ does not coarsely embed into $X$ for sufficiently
large $n$ depending on $X$.
  More generally, we prove that both results hold for lattices in connected
simple real Lie groups with sufficiently high real rank.
