Logarithmic bounds for the diameters of some Cayley graphs

Lam Pham, Xin Zhang

Published 2019-10-13Version 1

Let $\mathcal S \subset{\text{SL}(d,\mathbb Z)\ltimes \mathbb Z^d}$ or $\mathcal S \subset\text{SL}(d,\mathbb Z)\times\cdots\times \text{SL}(d,\mathbb Z)$ be a finite symmetric set. We show that if $\Lambda=\langle\mathcal S\rangle$ is Zariski-dense, then the diameter of the Cayley graph $\mathcal Cay(\Lambda/\Lambda(q),\pi_q(\mathcal S))$ is $O(\log q)$, where $q$ is an arbitrary positive integer, $\pi_q: \Lambda\rightarrow\Lambda/\Lambda(q) $ is the canonical congruence projection, and the implied constant depends only on $\mathcal S$.