Work

Non-Vanishing Unitary Cohomology of Low-Rank Integral Special Linear Groups

Year: 2025

Type: article

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

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf230

Authors: Benjamin Brück, Sam Hughes, Dawid Kielak, Piotr Mizerka