Inducing spectral gaps for the cohomological Laplacians of Sp 2 n (ℤ)

We show that the spectral gap of the first cohomological Laplacian Δ 1 for Sp 2n (ℤ) follows once a slightly stronger assumption holds for some Sp 2m (ℤ), where n ≥ m. As an application of this result, we provide explicit lower bounds for the maximal positive λ such that Δ 1 – λI is a sum of squares in the group ring of some quotients of Sp 2n (ℤ) for any n ≥ 2.