Logarithmic diameter bounds for some Cayley graphs

Abstract

Abstract Let S ⊂ GL n ⁢ ( 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 ⁡ ( Γ / Γ ⁢ ( q ) , π q ⁢ ( S ) ) \operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O ⁢ ( log ⁡ q ) O(\log q) , where 𝑞 is an arbitrary positive integer, π q : Γ → Γ / Γ ⁢ ( 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.