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

The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for $\mathrm{SAut}(F_n)$ and ${\mathrm{SL}}_n(\mathbb{Z})$. 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 ${\mathrm{SL}}_n(\mathbb{Z})$ for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for ${\mathrm{SL}}_n(\mathbb{Z})$ whenever $n\ge 3$.