Inducing Super-Approximation

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