SUPER-APPROXIMATION: p-ADIC SEMISIMPLE CASE

Let k be a number field, Ω be a finite symmetric subset of GLn0 (k), and Γ = 〈Ω〉. Let C(Γ) := {p ∈ Vf (k)| Γ is a bounded subgroup of GLn0 (kp)}, and Γp be the closure of Γ in GLn0 (kp). Assuming that the Zariski-closure of Γ is semisimple, we prove that the family of left translation actions {Γ y Γp}p∈C(Γ) has uniform spectral gap. As a corollary we get that the left translation action Γ y G has local spectral gap if Γ is a countable dense subgroup of a semisimple p-adic analytic group G and Ad(Γ) consists of matrices with algebraic entries in some Qp-basis of Lie(G). This can be viewed as a (stronger) p-adic version of [BISG, Theorem A], which enables us to give applications to the Banach-Ruziewicz problem and orbit equivalence rigidity.