Inducing Super-Approximation

Let $\Gamma_2\subseteq \Gamma_1$ be finitely generated subgroups of ${\rm GL}_{n_0}(\mathbb{Z}[1/q_0])$. For $i=1$ or $2$, let $\mathbb{G}_i$ be the Zariski-closure of $\Gamma_i$ in $({\rm GL}_{n_0})_{\mathbb{Q}}$, $\mathbb{G}_i^{\circ}$ be the Zariski-connected component of $\mathbb{G}_i$, and let $G_i$ be the closure of $\Gamma_i$ in $\prod_{p\nmid q_0}{\rm GL}_{n_0}(\mathbb{Z}_p)$. In this article we prove that, if $\mathbb{G}_1^{\circ}$ is the smallest closed normal subgroup of $\mathbb{G}_1^{\circ}$ which contains $\mathbb{G}_2^{\circ}$ and $\Gamma_2\curvearrowright G_2$ has spectral gap, then $\Gamma_1\curvearrowright G_1$ has spectral gap.