Tits building

In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings (named also after François Bruhat) plays a role in the study of -adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups. Overview The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field. Tits demonstrated how to every such group one can associate a simplicial complex with an action of , called the spherical building of . The group imposes very strong combinatorial regularity conditions on the complexes that can arise in this fashion. By treating these 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 is a Coxeter group , which determines a highly symmetrical simplicial complex , called the Coxeter complex. A building is glued together from multiple copies of , called its apartments, in a certain regular fashion. When is a finite Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of spherical type. When 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 is the same as an infinite tree without terminal vertices. Although the theory of semisimple 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 which satisfy the axioms of a building, but may not be connected with any group. This 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.