Super-approximation, I: p-adic semisimple case

Let $k$ be a number field, $\Omega$ be a finite symmetric subset of $\mathbb{GL}_{n_0}(k)$, and $\Gamma=\langle \Omega\rangle$. Let $$C(\Gamma):=\{\mathfrak{p}\in V_f(k)|\hspace{1mm} \Gamma \text{is a bounded subgroup of} \mathbb{GL}_{n_0}(k_{\mathfrak{p}})\},$$ and $\Gamma_{\mathfrak{p}}$ be the closure of $\Gamma$ in $\mathbb{GL}_{n_0}(k_{\mathfrak{p}})$. Assuming that the Zariski-closure of $\Gamma$ is semisimple, we prove that the family of left translation actions $\{\Gamma\curvearrowright \Gamma_{\mathfrak{p}}\}_{\mathfrak{p}\in C(\Gamma)}$ has {\em uniform spectral gap}. As a corollary we get that the left translation action $\Gamma\curvearrowright G$ has {\em local spectral gap} if $\Gamma$ is a countable dense subgroup of a semisimple $p$-adic analytic group $G$ and Ad$(\Gamma)$ 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.