Non-vanishing unitary cohomology of low-rank integral special linear groups

Abstract: We construct explicit finite-dimensional orthogonal representations $\pi_N$ of $\operatorname{SL}{N}(\mathbb{Z})$ for $N \in {3,4}$ 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 $T$.