III.5 Buildings by Mark Ronan, Apartments and Chambers: Page 162.

Invertible linear transformations on a vector space form a group called the general linear group, GLn(K), where n is the dimension of the vector space and K is the field of scalars.  Given a basis for the vector space then each group element can be written as an n x n matrix whose determinant is non-zero.

The subgroups of this group is studied.  The subgroups such as all lines, or two dimensional subspaces, are studied as projective spaces.  The study of the subspaces removes the special role of zero in the original space.

If one restricts the space GLn(K) to those linear transformations with determinant equal to one then we have the space SLn(K).

Another example is the group O(n) which consists of all linear transformations α of an n-dimensional real inner product space such that <αv,αw> = <v,w>, for any two vectors v and w (or in matrix terms for all real matrices A such that AAT = I).

Many groups may be created from GLn(K) and these subgroups are referred top as the classical groups.  The classical groups are either simple or close to simple.  Close to simple may mean for example that we can make them simple by quotienting out by the subgroup of scalar matrices.  When K is the field of real or complex numbers then the classical groups become the Lie groups. 

The simple Lie groups fall into one of four families called: An, Bn, Cn and Dn.  Then there are other types such 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 of n+1 dimensions.

Lie groups have analogues over any field K where they are called of Lie type.  K can be a finite group in which case the the groups are finite.  Almost all finite simple groups are of Lie type.

Jacques Tits found a geometric theory that would embrace groups called the theory of buildings.

The building associated with the groups GLn(K) and SLn(K) are of type An-1.  The building is an abstract simplical complex.   It is a high dimensional analogue of a graph.  It has vertices, some pairs of vertices form edges, triples of vertices form edges and k vertices form k-1 dimensional simplexes.

A simplex is a convex hull of a finite set of points.  A three dimensional simplex is a tetrahedron. 

To form a building of type An-1 we take all 1-spaces, in projective space and treat them as vertices.   The simplexes are formed as nested sequences of proper subspaces.  For example, a 2-space inside a 3-space inside a 7-space form a triangle whose vertices are these three subspaces.

A chamber is formed as a simplex of maximal dimension n-1.  This would be a 1-space inside a 2-space inside a 3-space and so on. 

Buildings have important subgeometries called apartments.  In the An-1 case one takes a basis for the vector space and then generates subspaces formed by subsets of the basis.  Certain subgeometries, as mentioned above, of apartments are called chambers. 

In the A3 case the vector space is 4 dimensional.  A basis has four elements.  There are 4 subspaces formed by of one basis vector,  six formed by two basis vectors and four formed by three basis vectors.  One may view the four one spaces as vertices of a tetrahedron, the six two spaces as the midpoints of the edges and the four 3 spaces as the midpoints of the faces.  The apartment has 24 chambers.  There are six chambers for each face.  The triangular chambers form a triangular tiling of the tetrahedron.  The tailing surface is topologically equivalent to s sphere.  All apartments of a building that are topologically equivalent to a sphere are called spherical buildings. 

The buildings for groups of Lie type are all spherical.  The apartments of other Lie types are related to the regular and semi regular polyhedra in n dimensions.

Any two chambers of a building lie in a common apartment.  All apartments of a building are isomorphic.  Apartments have a nice intersection property. 

The theory of spherical buildings gives a geometric basis for the Lie groups and the other non Lie groups mentioned above.  A theorem of Tits on the existence of automorphisms shows that the group’s themselves must also exist.

If the group K is a p-adic field then one may construct an affine, as well as an associated spherical, building.  The affine building carries more information and yields the spherical one as a structure at infinity.  The apartments are tilings of Euclidean space.

One may further generalize to hyperbolic buildings, whose apartments are tilings of hyperbolic space, that arise in the study of hyperbolic Kac-Moody groups.

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