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

We construct explicit finite-dimensional orthogonal representations $π_N$ of $\operatorname{SL}{N}(\mathbb{Z})$ for $N \in {3,4}$ all of whose invariant vectors are trivial, and such that $H^{N - 1}(\operatorname{SL}{N}(\mathbb{Z}),π_N)$ 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 $T$.

💡 Research Summary

The paper investigates higher‑dimensional analogues of Kazhdan’s property T for the integral special linear groups $\operatorname{SL}_N(\mathbb Z)$. Bader and Sauer introduced two generalisations: the weaker property $(T_n)$, which requires the $n$‑th reduced cohomology to vanish for unitary representations without invariant vectors, and the stronger property $