Editing Building (mathematics) (section)

==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 [[group representation|representations]]. The results of Tits on determination of a group by its building have deep connections with [[Mostow rigidity theorem|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 algebra|Kac–Moody groups]] in algebra, and to nonpositively curved manifolds and [[hyperbolic group]]s in topology and [[geometric group theory]].