Use this tag for questions about combinatorial and geometric structures that generalize simultaneously certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.
A building is a combinatorial and geometric structure that generalizes simultaneously certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.
Buildings were invented by Jacques Tits to describe simple algebraic groups over an arbitrary field. Tits demonstrated how to every such group $G$ one can associate a simplicial complex $\Delta = \Delta(G)$ with an action of $G$, called the spherical building of $G.$ The group $G$ imposes very strong combinatorial regularity conditions on the complexes $\Delta$ that can arise in that fashion. By treating those conditions as axioms for a class of simplicial complexes, Tits arrived at his first definition of a building. A part of the data defining a building $|Delta$ is a Coxeter group $W,$ which determines a highly symmetrical simplicial complex $\Sigma = \Sigma(W,S),$ called the Coxeter complex. A building $\Delta$ is glued together from multiple copies of $\Sigma,$ called its apartments, in a certain regular fashion. When $W$ is a finite Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of spherical type. When $W$ is an affine Weyl group, the Coxeter complex is a subdivision of the affine plane and one speaks of affine, or Euclidean, buildings. An affine building of type $\tilde A_1$ is the same as an infinite tree without terminal vertices.
Although the theory of semi-simple algebraic groups provided the initial motivation for the notion of a building, not all buildings arise from a group. In particular, projective planes and generalized quadrangles form two classes of graphs studied in incidence geometry that satisfy the axioms of a building but may not be connected with any group. That phenomenon turns out to be related to the low rank of the corresponding Coxeter system (namely, two). Tits proved a remarkable theorem: all spherical buildings of rank at least three are connected with a group; moreover, if a building of rank at least two is connected with a group then the group is essentially determined by the building.
Iwahori–Matsumoto, Borel–Tits and Bruhat–Tits demonstrated that in analogy with Tits's construction of spherical buildings, affine buildings can also be constructed from certain groups, namely, reductive algebraic groups over a local non-Archimedean field. Furthermore, if the split rank of the group is at least three, it is essentially determined by its building. Tits later reworked the foundational aspects of the theory of buildings using the notion of a chamber system, encoding the building solely in terms of adjacency properties of simplices of maximal dimension; that leads to simplifications in both spherical and affine cases. He proved that, in analogy with the spherical case, every building of affine type and rank at least four arises from a group.
