Super-approximation, I: $\mathfrak {p}$-adic semisimple case

Abstract

Abstract. Let k be a number field, Ω be a finite symmetric subset of GLn 0 (k), and Γ = Ω . Letand Γp be the closure of Γ in GLn 0 (kp). Assuming that the Zariski-closure of Γ is semisimple, we prove that the family of left translation actions {Γ Γp} 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 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.

Super-approximation, II: the p-adic and bounded power of square-free integers cases

“…Proof. A similar argument as in the proof of [SG,Lemma 18] works here. For the convenience of the reader, we present an outline of the proof.…”

“…Let us recall a couple of well-known results which give us a connection between having spectral gap and explicit construction of expanders (see [Lub94,Chapter 4.3], [LZ03, Chapter 1.4], [SG,Remark 15]).…”

Balanced Product Quantum Codes

“…It was proven in [29] (using results from [30]) that all hyperbolic regular tessellations {r, s} give families of sregular expander graphs. Unfortunately, we are not aware of any explicit bounds on λ 2 .…”

Balanced Product Quantum Codes

“…It was proven in [25] (using results from [26]) that all hyperbolic regular tessellations {r, s} give families of sregular expander graphs. Unfortunately, we are not aware of any explicit bounds on λ 2 .…”