Super-approximation, I: p-adic semisimple case

Let $k$ be a number field, $\mathrm{\Omega }$ be a finite symmetric subset of ${\mathbb{G}\mathbb{L}}_{n_0}(k)$, and $\mathrm{\Gamma }=⟨\mathrm{\Omega }⟩$. Let $$C(\mathrm{\Gamma }):=\{\mathfrak{p}\in V_f(k)\mathrm{\mid }\mathrm{\Gamma }\text{is a bounded subgroup of}{\mathbb{G}\mathbb{L}}_{n_0}(k_{\mathfrak{p}})\},$$ and ${\mathrm{\Gamma }}_{\mathfrak{p}}$ be the closure of $\mathrm{\Gamma }$ in ${\mathbb{G}\mathbb{L}}_{n_0}(k_{\mathfrak{p}})$. Assuming that the Zariski-closure of $\mathrm{\Gamma }$ is semisimple, we prove that the family of left translation actions $\{\mathrm{\Gamma }\curvearrowright {\mathrm{\Gamma }}_{\mathfrak{p}}{\}}_{\mathfrak{p}\in C(\mathrm{\Gamma })}$ has {\em uniform spectral gap}. As a corollary we get that the left translation action $\mathrm{\Gamma }\curvearrowright G$ has {\em local spectral gap} if $\mathrm{\Gamma }$ is a countable dense subgroup of a semisimple $p$-adic analytic group $G$ and Ad$(\mathrm{\Gamma })$ consists of matrices with algebraic entries in some ${\mathbb{Q}}_p$-basis of Lie$(G)$. This can be viewed as a (stronger) $p$-adic version of \cite[Theorem A]{BISG}, which enables us to give applications to the Banach-Ruziewicz problem and orbit equivalence rigidity.