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

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