Inducing spectral gaps for the cohomological Laplacians of $\operatorname{Sp}_{2n}(\mathbb{Z})$
Piotr Mizerka, Jakub Szymański
TL;DR
The work develops an inductive, SOS-based framework to induce spectral gaps for the cohomological Laplacians of $\operatorname{Sp}_{2n}(\mathbb{Z})$ from lower-rank data. By decomposing the first cohomological Laplacian $\Delta_1$ into four compatible summands and leveraging Fox calculus alongside the Steinberg presentation, the authors show that a positive gap at level $m$ propagates to level $n\ge m$ as $\Delta_1-\frac{n-2}{m-2}\lambda I_n$ being a sum of squares. They prove the central theorem and apply it to obtain explicit lower bounds for $\Delta_1-\lambda I$ for $G_2$, $G_3$, and quotients $H_n$ for all $n\ge 2$, with concrete numerical bounds ($\lambda=0.82$ for $G_2$ and $\lambda=0.99$ for $G_3$) and comparison to known results. The paper couples rigorous algebraic decompositions with computer-assisted certifications (including Wedderburn-based acceleration and interval arithmetic) to produce reproducible, verifiable bounds on spectral gaps, contributing a scalable method for establishing property (T)-related gaps in families of arithmetic groups.
Abstract
We show that the spectral gap of the first cohomological Laplacian $Δ_1$ for $\operatorname{Sp}_{2n}(\mathbb{Z})$ follows once a slightly stronger assumption holds for some $\operatorname{Sp}_{2m}(\mathbb{Z})$, where $n\geq m$. As an application of this result, we provide explicit lower bounds for some quotients of $\operatorname{Sp}_{2n}(\mathbb{Z})$ for any $n\geq 2$.