Super-approximation, I: p-adic semisimple case

Abstract

Let $k$ be a number field, $\Omega$ be a finite symmetric subset of ${\mathbb{GL}}_{n_0}(k)$, and $\Gamma =⟨\Omega ⟩$. Let \[ C(Γ):=\{\mathfrak{p}\in V_f(k)|\hspace{1mm} Γ\text{is a bounded subgroup of} \mathbb{GL}_{n_0}(k_{\mathfrak{p}})\}, \] 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.

Revised and added explanations based on referee reports

Cited by in corpus (1)

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