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In Mathematics , a building (also '''Tits building''') is a combinatorial and geometric structure which simultaneously generalizes certain aspects of Flag Manifold s, finite Projective Plane s, and Riemannian Symmetric Space s. 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. Not all buildings arise from a group. In particular, projective planes and Generalized Quadrangle s form two classes of graphs studied in Incidence Geometry which fall under the definition of a building, but may not be connected with any group. Nevertheless, Tits proved a remarkable theorem: that all spherical buildings of rank at least three are connected with a group; and, moreover, the group is essentially determined by the building. FORMULATION A part of the data defining a building Δ is a certain Coxeter Group ''W''. If this group is finite, the corresponding building is of spherical type. If ''W'' is an Affine Weyl Group , the corresponding building is of '''affine''' (or '''Euclidean''') type. In the simplest possible case an affine building is the same as an infinite Tree without terminal vertices. Iwahori-Matsumoto, Borel-Tits and Bruhat-Tits demonstrated that in analogy with Tits' 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, which leads to simplifications in both spherical and affine cases. Tits proved that buildings of affine type and rank at least four arise from a group, just as in the spherical case. APPLICATIONS The theory of buildings has important applications in several rather disparate fields. Besides the already mentioned connections with the structure of reductive algebraic groups over general and local fields, buildings are used to study their Representations . The results of Tits on determination of a group by its building have deep connections with rigidity theorems of George Mostow and Grigory Margulis , and with Margulis Arithmeticity . Special types of buildings are studied in discrete mathematics, and the idea of a geometric approach to characterizing simple groups proved very fruitful in the Classification Of Finite Simple Groups . The theory of buildings of type more general than spherical or affine is still relatively undeveloped, but these generalized buildings have already found applications to construction of Kac-Moody Group s in algebra, and to nonpositively curved manifolds and Hyperbolic Group s in topology and Geometric Group Theory . SEE ALSO
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