Logarithmic bounds for the diameters of some Cayley graphs

2019/10/13 by Pham, Lam, Zhang, Xin
#FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1910.05718

Abstract

Let S\subsetGL_n(\mathbb Z) be a finite symmetric set. We show that if the Zariski closure of Γ=⟨ S⟩ is a product of SL_d or a special affine linear group, then the diameter of the Cayley graph Cay(Γ/Γ(q),π_q(S)) is O(log q), where q is an arbitrary positive integer, π_q:Γ→ Γ/Γ(q) is the canonical projection induced by the reduction modulo q, and the implied constant depends only on S.

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