Building (mathematics)
Buildings were initially introduced by Jacques Tits as a means to understand the structure of isotropic reductive linear algebraic groups over arbitrary fields.
The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field.
By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at his first definition of a building.
A building Δ is glued together from multiple copies of Σ, called its apartments, in a certain regular fashion.
In particular, projective planes and generalized quadrangles form two classes of graphs studied in incidence geometry which satisfy the axioms of a building, but may not be connected with any group.
He proved that, in analogy with the spherical case, every building of affine type and rank at least four arises from a group.
In fact, for every two n-simplices intersecting in an (n – 1)-simplex or panel, there is a unique period two simplicial automorphism of A, called a reflection, carrying one n-simplex onto the other and fixing their common points.
Every building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space.
For affine buildings, this metric satisfies the CAT(0) comparison inequality of Alexandrov, known in this setting as the Bruhat–Tits non-positive curvature condition for geodesic triangles: the distance from a vertex to the midpoint of the opposite side is no greater than the distance in the corresponding Euclidean triangle with the same side-lengths (see Bruhat & Tits 1972).
The simplicial structure of the affine and spherical buildings associated to SLn(Qp), as well as their interconnections, are easy to explain directly using only concepts from elementary algebra and geometry (see Garrett 1997).
The axioms for a building can easily be verified using the classical Schreier refinement argument used to prove the uniqueness of the Jordan–Hölder decomposition.
The k-simplices of X are equivalence classes of k + 1 mutually adjacent lattices, The (n − 1)-simplices correspond, after relabelling, to chains where each successive quotient has order p. Apartments are defined by fixing a basis (vi) of V and taking all lattices with basis (pai vi) where (ai) lies in Zn and is uniquely determined up to addition of the same integer to each entry.
Tits proved that any label-preserving automorphism of the affine building arises from an element of SLn(Qp).
Since automorphisms of the building permute the labels, there is a natural homomorphism The action of GLn(Qp) gives rise to an n-cycle τ.
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.
Besides the already mentioned connections with the structure of reductive algebraic groups over general and local fields, buildings are used to study their representations.
