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

TL;DR

The paper constructs specific finite-dimensional representations of low-rank special linear groups over integers that have non-trivial cohomology, showing these groups lack certain higher Kazhdan properties.

Contribution

It explicitly constructs orthogonal representations of SL_N(Z) for N=3,4 with trivial invariant vectors but non-trivial cohomology, demonstrating the groups do not have certain higher property T variants.

Findings

Constructed explicit orthogonal representations with trivial invariants.

Proved non-trivial cohomology in degree N-1 for these groups.

Showed these groups lack higher Kazhdan properties.

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 $T$.