Logarithmic diameter bounds for some Cayley graphs

Abstract

Abstract Let S⊂GLn\u2062(Z)S\\subset\\mathrm{GL}_{n}(\\mathbb{Z}) be a finite symmetric set. We show that if the Zariski closure of Γ=⟨S⟩\\Gamma=\\langle S\\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay\u2061(Γ/Γ\u2062(q),πq\u2062(S))\\operatorname{Cay}(\\Gamma/\\Gamma(q),\\pi_{q}(S)) is O\u2062(log\u2061q)O(\\log q), where 𝑞 is an arbitrary positive integer, πq:Γ→Γ/Γ\u2062(q)\\pi_{q}\\colon\\Gamma\\to\\Gamma/\\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.