The extremal symmetry of arithmetic simplicial complexes

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