The extremal symmetry of arithmetic simplicial complexes

TL;DR

This paper investigates the highly symmetric properties of simplicial complexes derived from lattices in higher-rank semisimple Lie groups over nonarchimedean fields, revealing their extremal symmetry features.

Contribution

It proves that the simplicial complexes associated with these lattices have remarkable and extremal symmetry properties, surpassing other possible structures.

Findings

Simplicial complexes exhibit extremal symmetry properties.

Symmetry properties are compared to other simplicial structures.

Results apply to complexes from lattices in higher-rank semisimple Lie groups.

Abstract

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G={\rm PGL}(n,Q_P)$. To any lattice $L$ in $G$ there is an associated simplicial complex $B_L$, given by the quotient by $L$ of the Bruhat-Tits building associated to $G$. In this paper prove that the simplicial structure $B_L$ exhibits some remarkable and extremal symmetry properties, in particular when compared to any other simplicial structure on (any cover of) $B_L$.