An induction result reduces the spectral gap problem for the cohomological Laplacian of Sp_{2n}(Z) to a base case, and computer-assisted checks give lower bounds for quotients and small n.
Braid groups and symplectic S teinberg groups
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Inducing spectral gaps for the cohomological Laplacians of $\operatorname{Sp}_{2n}(\mathbb{Z})$
An induction result reduces the spectral gap problem for the cohomological Laplacian of Sp_{2n}(Z) to a base case, and computer-assisted checks give lower bounds for quotients and small n.