Abstract
We construct explicit finite-dimensional orthogonal representations $\pi_N$ of $\operatorname{SL}_n(\mathbb{Z})$ for $N=3,4$ all of whose invariant vectors are trivial, and such that $H^{N-1}(\operatorname{SL}_N(\mathbb{Z}),\pi_N)$ is non-trivial. This implies that for $N$ as above, the group $\operatorname{SL}_N(\mathbb{Z})$ does not have property $(T_N$-$_1)$ of Bader–Sauer and therefore is not $(N-1)$-Kazhdan in the sense of De Chiffre–Glebsky–Lubotzky–Thom, both being higher versions of Kazhdan’s property $T$.
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