math.GR

# Title: 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 $\Gamma=\langle \Omega\rangle$. 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.
 Comments: Revised and added explanations based on referee reports Subjects: Group Theory (math.GR); Number Theory (math.NT) Journal reference: International Mathematics Research Notices 2017, Issue 23, (2017) 7190-7263 DOI: 10.1093/imrn/rnw208 Cite as: arXiv:1602.00403 [math.GR] (or arXiv:1602.00403v2 [math.GR] for this version)

## Submission history

From: Alireza Salehi Golsefidy [view email]
[v1] Mon, 1 Feb 2016 06:51:00 GMT (52kb)
[v2] Sat, 10 Feb 2018 09:15:52 GMT (108kb,D)

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