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

Abstract

We construct explicit finite-dimensional orthogonal representations ${\pi }_N$ of ${\mathrm{SL}}_N(\mathbb{Z})$ for $N\in \{3,4\}$ all of whose invariant vectors are trivial, and such that $H^{N-1}({\mathrm{SL}}_N(\mathbb{Z}),{\pi }_N)$ is non-trivial. This implies that for $N$ as above, the group ${\mathrm{SL}}_N(\mathbb{Z})$ 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 $T$.

21 pages, code available under https://zenodo.org/records/14008647 ; v2: minor corrections and updated references

References in corpus (2)

• Improved homological stability for certain general linear groups
• Higher Kazhdan property and unitary cohomology of arithmetic groups