Building (mathematics)
In mathematics, a building (also Tits building, Bruhat–Tits building, named after François Bruhat and Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type, the theory has also been used to study the geometry and topology of homogeneous spaces of p-adic Lie groups and their discrete subgroups of symmetries, in the same way that trees have been used to study free groups.
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 G one can associate a simplicial complex Δ = Δ(G) with an action of G, called the spherical building of G. The group G imposes very strong combinatorial regularity conditions on the complexes Δ 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 Δ is a Coxeter group W, which determines a highly symmetrical simplicial complex Σ = Σ(W,S), called the Coxeter complex. A building Δ is glued together from multiple copies of Σ, 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 is the same as an infinite tree without terminal vertices.
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