Inducing Super-Approximation

Abstract

Let ${\Gamma }_2\subseteq {\Gamma }_1$ be finitely generated subgroups of ${\mathrm{GL}}_{n_0}(\mathbb{Z}[1/q_0])$. For $i=1$ or $2$, let ${\mathbb{G}}_i$ be the Zariski-closure of ${\Gamma }_i$ in $({\mathrm{GL}}_{n_0}{)}_{\mathbb{Q}}$, ${\mathbb{G}}_i^{\circ }$ be the Zariski-connected component of ${\mathbb{G}}_i$, and let $G_i$ be the closure of ${\Gamma }_i$ in ${\prod }_{p\nmid q_0}{\mathrm{GL}}_{n_0}({\mathbb{Z}}_p)$. 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.