Spectral gap for the cohomological Laplacian of $SL_3(ℤ)$

Abstract

We show that the cohomological Laplacian in degree 1 in the group cohomology of $\text{SL}_3(\mathbb{Z})$ is a sum of hermitian squares in the algebra $\mathbb{M}_n(\mathbb{R}G)$. We provide an estimate of the spectral gap for this Laplacian for every unitary representation.