Inducing spectral gaps for the cohomological Laplacians of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$

The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for $\operatorname{SAut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$. In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for $\operatorname{SL}_n(\mathbb{Z})$ for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for $\operatorname{SL}_n(\mathbb{Z})$ whenever $n\geq 3$.