Chevalley群的高维扩张子
High-Dimensional Expanders from Chevalley Groups
摘要：
令$Φ$是秩至少为$2$的不可约根系（不同于$G_2$），令$\mathbb{F}$是具有$p = \operatorname{char} \mathbb{F} > 3$的有限域，令$\mathrm{G}(Φ,\mathbb{F})$是相应的Chevalley群。我们描述了一个强显式的高维扩展（HDX）家庭的尺寸$\mathrm{rank}(Φ)$，其中$\mathrm{G}(Φ,\mathbb{F})$的行为只是传递的顶维面，这些是$λ$ -光谱HDX与$λ\to 0$为$p \to \infty$。这概括了考夫曼和奥本海姆（STOC 2018）的结构，对应于$Φ= A_d$的情况。我们的工作给出了三个新的家庭的频谱HDX的任何尺寸$\ge 2$，和四个特殊的结构尺寸$4$，$6$，$7$，和$8$。
英文摘要：
Let $Φ$ 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 $\mathrm{G}(Φ,\mathbb{F})$ be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension $\mathrm{rank}(Φ)$, where $\mathrm{G}(Φ,\mathbb{F})$ acts simply transitively on the top-dimensional faces; these are $λ$-spectral HDXs with $λ\to 0$ as $p \to \infty$. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case $Φ= A_d$. Our work gives three new families of spectral HDXs of any dimension $\ge 2$, and four exceptional constructions of dimension $4$, $6$, $7$, and $8$.
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