Inducing spectral gaps for the cohomological Laplacians of SLₙ(Z) and SAut(Fₙ)

The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for SAut(Fₙ) and SLₙ(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 SLₙ(Z) for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for SLₙ(Z) whenever n≥ 3.