Super-approximation, I: p-adic semisimple case.

Let k be a number field, Ω be a finite symmetric subset of GL_n0 (k), and Γ = Ω. Let C(Γ):= {p ∈ V_f (k) ∣ Γ is a bounded subgroup of GL_n0 (k_p)}, and Γ_p be the closure of Γ in GL_n0 (k_p). Assuming that the Zariski-closure of Γ is semisimple, we prove that the family of left translation actions {Γ → Γ_p}_p∈C(Γ) has uniform spectral gap. As a corollary we get that the left translation action Γ → 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 ℚ_p-basis of Lie(G). This can be viewed as a (stronger) p-adic version of [10, Theorem A], which enables us to give applications to the Banach-Ruziewicz problem and orbit equivalence rigidity. [ABSTRACT FROM AUTHOR]