The extremal symmetry of arithmetic simplicial complexes

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G=\mathrm{P}\mathrm{G}\mathrm{L}(n,Q_P)$. 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) $B_L$.