Building (mathematics) - Classification

Classification

Tits proved that all irreducible spherical buildings (i.e. with finite Weyl group) of rank greater than 2 are associated to simple algebraic or classical groups. A similar result holds for irreducible affine buildings of dimension greater than two (their buildings "at infinity" are spherical of rank greater than two). In lower rank or dimension, there is no such classification. Indeed each incidence structure gives a spherical building of rank 2 (see Pott 1995); and Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the flag complex of a finite projective plane has the structure of a building, not necessarily classical. Many 2-dimensional affine buildings have been constructed using hyperbolic reflection groups or other more exotic constructions connected with orbifolds.

Tits also proved that every time a building is described by a BN pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see Tits 1974).