Lie type quotients of the maximal unipotent subgroup of Kac-Moody groups of type $\mathrm{HB}_{2}^{(2)}$

TL;DR

This paper constructs infinite families of finite simple groups of Lie type as quotients of a specific unipotent subgroup of Kac-Moody groups, leading to new examples of high-dimensional expanders with unbounded rank.

Contribution

It introduces a novel construction of finite simple groups as quotients of Kac-Moody unipotent subgroups, producing high-dimensional expanders with increasing rank.

Findings

Constructed infinite families of simple groups as quotients of unipotent subgroups.

Established these quotients as sources of high-dimensional expanders.

Provided the first examples of high-dimensional expanders from Lie type groups of unbounded rank.

Abstract

In this article, we construct infinite families $(G_n)_{n \in \mathbb{N}}$ of finite simple groups $G_n$ of Lie type, such that the rank of $G_n$ strictly increases as $n$ tends to infinity, and such that each $G_n$ is a quotient of the maximal unipotent subgroup $U^+$ of the (minimal) Kac-Moody group $\mathfrak{G}_{A}(\mathbb{K})$ of type $\mathrm{HB}_{2}^{(2)}$ over a finite field $\mathbb{K}$. Moreover, we show that the quotient maps lead to the construction of an infinite family of bounded degree spectral high-dimensional expanders. These provide the first class of examples of infinite families of high-dimensional expanders constructed from Lie type groups of unbounded rank.