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

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... ach 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 {Ck^{\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.