Editing Building (mathematics) (section)

==Classification==
Tits proved that all irreducible spherical buildings (i.e. with finite [[Weyl group]]) of rank greater than 2 are associated to simple algebraic groups, to classical groups (possibly infinite-dimensional), or to a special class of groups called "of mixed type" that only exist in characteristic 2 or 3.
A similar result holds for irreducible affine buildings of dimension greater than 2 (their buildings "at infinity" are spherical of rank greater than two).

In lower rank or dimension, there is no such classification.
Indeed, the spherical  buildings of rank 2 are precisely the [[generalized polygon]]s, and a plethora of examples exist. (There are free constructions of infinite generalized {{nowrap|''n''-gons}} for every <math>n \geq 3</math>.) 
Many 2-dimensional affine buildings have been constructed using hyperbolic [[reflection group]]s or other more exotic constructions connected with [[orbifold]]s.

Tits also proved that every time a building is described by a {{math|(''B'', ''N'')}} pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see {{harvnb|Tits|1974}}).