Building (mathematics)

A building, also called a Tits building after Jacques Tits, is a combinatorial and geometric structure that simultaneously generalizes certain aspects of flag manifolds, finite projective planes and Riemannian symmetric spaces. Tits introduced buildings as a way to understand the structure of isotropic reductive linear algebraic groups over arbitrary fields: to each such group he attached a simplicial complex on which the group acts, called its spherical building. A building is glued together from copies of a Coxeter complex, called its apartments, in a regular fashion. When the underlying Coxeter group is finite the building is of spherical type, and when it is an affine Weyl group the building is affine, also called Euclidean; an affine building of the simplest type is an infinite tree without terminal vertices. Not every building comes from a group, since projective planes and generalized quadrangles satisfy the axioms without necessarily being connected to one. Tits proved that every spherical building of rank at least three arises from a group, and later that the same holds for affine buildings of rank at least four. The more specialized Bruhat-Tits buildings, named also for François Bruhat, play a role in the study of p-adic Lie groups similar to that of symmetric spaces in the theory of Lie groups.