Spectral gap for the cohomological Laplacian of ${\mathrm{SL}}_3(\mathbb{Z})$

We show that the cohomological Laplacian in degree 1 in the group cohomology of ${\mathrm{SL}}_3(\mathbb{Z})$ 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.