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 et al. [On property [Formula: see text] for [Formula: see text] and [Formula: see text], Ann. of Math. (2) 193 (2021) 539–562] to prove property [Formula: see text] for [Formula: see text] and [Formula: see text]. 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 [Formula: see text] for every unitary representation. In particular, one gets in that way an alternative proof of property [Formula: see text] for [Formula: see text] whenever [Formula: see text].