Inducing Super-Approximation

Let ${\mathrm{\Gamma }}_2\subseteq {\mathrm{\Gamma }}_1$ be finitely generated subgroups of ${\rm GL}{n_0}($\mathbb{Z}$[1/q_0])$\mathrm{.}For$i=1$or$2$,let$$\mathbb{G}$i$ be the Zariski-closure of ${\mathrm{\Gamma }}_i$ in $({\mathrm{G}\mathrm{L}}_{n_0}{)}_{\mathbb{Q}}$, ${\mathbb{G}}_i^{\circ }$ be the Zariski-connected component of ${\mathbb{G}}_i$, and let $G_i$ be the closure of ${\mathrm{\Gamma }}_i$ 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 ${\mathrm{\Gamma }}_2\curvearrowright G_2$ has spectral gap, then ${\mathrm{\Gamma }}_1\curvearrowright G_1$ has spectral gap.