Logarithmic bounds for the diameters of some Cayley graphs

TL;DR

This paper proves that for certain linear groups, the diameters of associated Cayley graphs grow logarithmically with the modulus, extending understanding of expansion properties in algebraic groups.

Contribution

It establishes logarithmic diameter bounds for Cayley graphs of specific linear groups over finite quotients, under Zariski closure conditions.

Findings

Diameter of Cayley graphs is O(log q) for groups with Zariski closure as a product of SL_d or affine groups.

Bounds depend only on the generating set, not on the modulus q.

Results apply to a broad class of linear groups, enhancing understanding of their expansion properties.

Abstract

Let $S\subset\text{GL}_n(\mathbb Z)$ be a finite symmetric set. We show that if the Zariski closure of $\Gamma=\langle S\rangle$ is a product of $\text{SL}_d$ or a special affine linear group, then the diameter of the Cayley graph $\text{Cay}(\Gamma/\Gamma(q),\pi_q(S))$ is $O(\log q)$, where $q$ is an arbitrary positive integer, $\pi_q:\Gamma\to \Gamma/\Gamma(q)$ is the canonical projection induced by the reduction modulo $q$, and the implied constant depends only on $S$.