High-Dimensional Expanders from Chevalley Groups

Abstract

Let $\Phi$ be an irreducible root system (other than $G_2$) of rank at least $2$, let $\mathbb{F}$ be a finite field with $p = \operatorname{char} \mathbb{F} > 3$, and let $\mathrm{G}(\Phi,\mathbb{F})$ be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension $\mathrm{rank}(\Phi)$, where $\mathrm{G}(\Phi,\mathbb{F})$ acts simply transitively on the top-dimensional faces; these are $\lambda$-spectral HDXs with $\lambda \to 0$ as $p \to \infty$. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case $\Phi = A_d$. Our work gives three new families of spectral HDXs of any dimension $\ge 2$, and four exceptional constructions of dimension $4$, $6$, $7$, and $8$.