Building

Summary

A building is a simplicial complex glued together from copies of a Coxeter complex, called apartments, under strong compatibility rules. Jacques Tits introduced buildings to give a geometric picture of reductive algebraic groups and of the groups of Lie type. Each group with a BN-pair acts on a building, and the Bruhat decomposition becomes a statement about the relative position of two chambers. Buildings of $p$-adic groups are spaces of nonpositive curvature, used much like symmetric spaces for real Lie groups.