The invertible linear transformations on a vector space form a group, called the general linear group. If n is the dimension of the vector space and K is the field of scalars, then it is denoted by GLn (K), and if we pick a basis for the vector space, then each group element can be written as an n × n matrix whose DETERMINANT [III.15] is nonzero. This group and its subgroups are of great interest in mathematics, and can be studied “geometrically” in the following way. Instead of looking at the vector space V, where of course the origin plays a unique role and is fixed by the group, we use the PROJECTIVE SPACE [I.3 §6.7] associated with V: the points of projective space are the one-dimensional subspaces of V, the lines are the two-dimensional subspaces, the planes are the three-dimensional subspaces, and so on.
Several important subgroups of GLn(K) can be obtained by imposing constraints on the linear maps (or matrices). For example, SLn (K) consists of all linear transformations of determinant 1. The group O(n) consists of all linear transformations α of an n-dimensional real inner-product space such that (αvαw,) = (v, w) for any two vectors v and w (or in matrix terms all real matrices A such that AAT = I); more generally, one can define many similar subgroups of GLn(K) by taking all linear maps that preserve certain forms, such as bilinear or sesquilinear forms. These subgroups are called classical groups. The classical groups are either simple or close to simple (for example, we can often make them simple by quotienting out by the subgroup of scalar matrices). When K is the field of real or complex numbers, the classical groups are Lie groups.
Lie groups and their classification are discussed in LIE THEORY [III.48]: the simple Lie groups comprise the classical groups, which fall into one of four families, known as An, Bn, Cn, and Dn (where n is a natural number), along with other types known as E6, E7, E8, F4, and G2. The subscripts are related to the dimensions of the groups. For example, the groups of type An are the groups of invertible linear transformations in n+ 1 dimensions.
These simple Lie groups have analogues over any field, where they are often referred to as groups of Lie type. For example, K can be a finite field, in which case the groups are finite. It turns out that almost all finite simple groups are of Lie type: see THE CLASSIFICATION OF FINITE SIMPLE GROUPS [V.7]. A geometric theory underlying the classical groups had been developed by the first half of the twentieth century. It used projective space and various subgeometries of projective space, which made it possible to provide analogues for the classical groups, but it did not provide analogues for the groups of types E6, E7, E8, F4, and G2. For this reason, Jacques Tits looked for a geometric theory that would embrace all families, and ended up creating the theory of buildings.
The full abstract definition of a building is somewhat complicated, so instead we shall try to give some idea of the concept by looking at the building associated with the groups GLn(K) and SLn (K), which are of type An-l. This building is an abstract simplicial complex, which can be thought of as a higher-dimensional analogue of a GRAPH [III.34]. It consists of a collection of points called vertices; as in a graph, some pairs of vertices form edges; however, it is then possible for triples of vertices to form two-dimensional faces, and for sets of k vertices to form (k - 1)-dimensional “simplexes.” (The geometrical meaning of the word “simplex” is a convex hull of a finite set of points in general position: for instance, a three-dimensional simplex is a tetrahedron.) All faces of simplexes must also be included, so for example three vertices cannot form a two-dimensional face unless each pair is joined by an edge.
To form the building of type An-1, we start by taking all the 1-spaces, 2-spaces, 3-spaces, and so on (corresponding to points, lines, planes, and so on, in projective space), and treat them as “vertices.” The simplexes are formed by all nested sequences of proper subspaces: for example, a 2-space inside a 4-space inside a 5-space will form a “triangle” whose vertices are these three subspaces. The simplexes of maximal dimension have n - 1 vertices: a 1-space inside a 2-space inside a 3-space, and so on. These simplexes are called chambers.
There are many subspaces, so a building is a huge object. However, buildings have important subgeometries called apartments, which in the An-1 case are obtained by taking a basis for the vector space, and then taking all subspaces generated by subsets of this basis. For example, in the A3 case our vector space is four dimensional, so a basis has four elements; its subsets span four 1-spaces, six 2-spaces, and four 3-spaces. To visualize this apartment it helps to view the four 1-spaces as the vertices of a tetrahedron, the six 2-spaces as the midpoints of its edges, and the four 3-spaces as the midpoints of its faces. The apartment has twenty-four chambers, six for each face of the original tetrahedron, and they form a triangular tiling of the surface of the tetrahedron. This surface is topologically equivalent to a sphere, as are all apartments of this building: such buildings are called spherical. The buildings for the groups of Lie type are all spherical, and, just as A3 is related to the tetrahedron, their apartments are related to the regular and semiregular polyhedra in n dimensions, where n is the subscript in the Lie notation given earlier.
Buildings have the following two noteworthy features. First, any two chambers lie in a common apartment: this is not obvious in the example above but it can be proved using linear algebra. Second, in any building all apartments are isomorphic and any two apartments intersect nicely: more precisely, if A and A' are apartments, then A ∩ A' is convex and there is an isomorphism from A to A' that fixes A ∩ A'. These two features were originally used by Tits in defining buildings.
The theory of spherical buildings does not just give a pleasing geometric basis for the groups of Lie type: it can also be used to construct the ones of types E6, E7, E8, and F4, for an arbitrary field K, without the need for sophisticated machinery such as Lie algebras. Once the building has been constructed (and a construction can be given in a surprisingly simple manner), a theorem of Tits on the existence of automorphisms shows that the groups themselves must exist.
In a spherical building the apartments are tilings of a sphere, but other types of buildings also play significant roles. Of particular importance are affine buildings, in which the apartments are tilings of Euclidean space; such buildings arise in a natural way from groups, such as GLn (K), where K is a p-ADIC FIELD [III.51]. For such fields there are two buildings, one spherical and one affine, but the affine one carries more information and yields the spherical building as a structure “at infinity.” Going beyond affine buildings, there are hyperbolic buildings, whose apartments are tilings of hyperbolic space; they arise naturally in the study of hyperbolic Kac-Moody groups.