Здания Тица: Математические структуры и обобщения.

Mathematical structure
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of isotropic reductive linear algebraic groups over arbitrary fields. The more specialized theory of Bruhat–Tits buildings (named also after François Bruhat) plays a role in the study of p adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.

The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field. Tits demonstrated how to every such group G one can associate a simplicial complex with an action of G, called the spherical building of G. The group G imposes very strong combinatorial regularity conditions on the complexes Δ that can arise in this fashion. By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at his first definition of a building. A part of the data defining a building Δ is a Coxeter group W, which determines a highly symmetrical simplicial complex , called the Coxeter complex. A building Δ is glued together from multiple copies of Σ, called its apartments, in a certain regular fashion. When W is a finite Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of spherical type. When W is an affine Weyl group, the Coxeter complex is a subdivision of the affine plane and one speaks of affine, or Euclidean, buildings. An affine building of type Ã1 is the same as an infinite tree without terminal vertices. Although the theory of semisimple algebraic groups provided the initial motivation for the notion of a building, not all buildings arise from a group. 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. This phenomenon turns out to be related to the low rank of the corresponding Coxeter system (namely, two). Tits proved a remarkable theorem: all spherical buildings of rank at least three are connected with a group; moreover, if a building of rank at least two is connected with a group then the group is essentially determined by the building
Iwahori–Matsumoto, Borel–Tits and Bruhat–Tits demonstrated that in analogy with Tits' construction of spherical buildings, affine buildings can also be constructed from certain groups, namely, reductive algebraic groups over a local non Archimedean field. Furthermore, if the split rank of the group is at least three, it is essentially determined by its building. Tits later reworked the foundational aspects of the theory of buildings using the notion of a chamber system, encoding the building solely in terms of adjacency properties of simplices of maximal dimension; this leads to simplifications in both spherical and affine cases. He proved that, in analogy with the spherical case, every building of affine type and rank at least four arises from a group.

Tits proved that any label preserving automorphism of the affine building arises from an element of SLn('Q'p). Since automorphisms of the building permute the labels, there is a natural homomorphism

Aut X → Sn. The action of GLn('Q'p) gives rise to an n cycle τ. Other automorphisms of the building arise from outer automorphisms of SLn('Q'p) associated with automorphisms of the Dynkin diagram. Taking the standard symmetric bilinear form with orthonormal basis vi, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the permutation σ that sends each label to its negative modulo n. The image of the above homomorphism is generated by σ and τ and is isomorphic to the dihedral group Dn of order 2n; when , it gives the whole of S3. If E is a finite Galois extension of 'Q'p and the building is constructed from SLn(E) instead of SLn('Q'p), the Galois group Gal(E / 'Q'p) will also act by automorphisms on the building.

When L is an archimedean local field then on the building for the group SL2(L) an additional structure can be imposed of a building with complex multiplication. These were first introduced by Martin L. Brown These buildings arise when a quadratic extension of L acts on the vector space L^(2). These building with complex multiplication can be extended to any global field. They describe the action of the Hecke operators on Heegner points on the classical modular curve X0(N) as well as on the Drinfeld modular curve These buildings with complex multiplication are completely classified for the case of SL2(L) in

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 2 (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 ); 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 (B, N) pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see ).

The theory of buildings has important applications in several rather disparate fields. Besides the already mentioned connections with the structure of reductive algebraic groups over general and local fields, buildings are used to study their representations. The results of Tits on determination of a group by its building have deep connections with rigidity theorems of George Mostow and Grigory Margulis, and with Margulis arithmeticity. Special types of buildings are studied in discrete mathematics, and the idea of a geometric approach to characterizing simple groups proved very fruitful in the classification of finite simple groups. The theory of buildings of type more general than spherical or affine is still relatively undeveloped, but these generalized buildings have already found applications to construction of Kac–Moody groups in algebra, and to nonpositively curved manifolds and hyperbolic groups in topology and geometric group theory.