On Diameters of Cayley Graphs over Matrix Groups

TL;DR

This paper proves that for the special linear group over a finite field, large symmetric generating sets have small covering numbers, revealing sharp bounds on the diameters of corresponding Cayley graphs.

Contribution

It establishes sharp bounds on the diameters of Cayley graphs over matrix groups, specifically for SL_n(F_p), with explicit constants and conditions.

Findings

Covering number is at most Cn^2 for large generating sets.

Results are sharp up to the constant C.

Provides bounds on diameters of Cayley graphs over SL_n(F_p).

Abstract

We establish for the matrix group $G=\mathrm{SL}_{n}\left(\mathbb{F}_{p}\right)$ that there exist absolute constants $c\in\left(0,1\right)$ and $C>0$ such that any symmetric generating set $A$, with $\left|A\right|\geq\left|G\right|^{1-c}$ has a covering number $\leq Cn^{2}.$ This result is sharp up to the value of the constant $C>0$.