Inducing spectral gaps for the cohomological Laplacians of ${\mathrm{Sp}}_{2n}(\mathbb{Z})$

We show that the spectral gap of the first cohomological Laplacian ${\mathrm{\Delta }}_1$ for ${\mathrm{Sp}}_{2n}(\mathbb{Z})$ follows once a slightly stronger assumption holds for some ${\mathrm{Sp}}_{2m}(\mathbb{Z})$, where $n\ge m$. As an application of this result, we provide explicit lower bounds for some quotients of ${\mathrm{Sp}}_{2n}(\mathbb{Z})$ for any $n\ge 2$.