Английская Википедия:Building (mathematics)
Шаблон:Short description 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 group|Шаблон:Mvar-adic Lie groups]] analogous to that of the theory of symmetric spaces in the theory of Lie groups.
Overview
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 Шаблон:Mvar one can associate a simplicial complex Шаблон:Math with an action of Шаблон:Mvar, called the spherical building of Шаблон:Mvar. The group Шаблон:Mvar imposes very strong combinatorial regularity conditions on the complexes Шаблон:Math 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 Шаблон:Math is a Coxeter group Шаблон:Mvar, which determines a highly symmetrical simplicial complex Шаблон:Math, called the Coxeter complex. A building Шаблон:Math is glued together from multiple copies of Шаблон:Math, called its apartments, in a certain regular fashion. When Шаблон:Mvar is a finite Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of spherical type. When Шаблон:Mvar 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 Шаблон:Math 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 (Шаблон:Harvnb).
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.
Definition
An Шаблон:Mvar-dimensional building Шаблон:Mvar is an abstract simplicial complex which is a union of subcomplexes Шаблон:Mvar called apartments such that
- every Шаблон:Mvar-simplex of Шаблон:Mvar is within at least three Шаблон:Mvar-simplices if Шаблон:Math;
- any Шаблон:Math-simplex in an apartment Шаблон:Mvar lies in exactly two adjacent Шаблон:Mvar-simplices of Шаблон:Mvar and the graph of adjacent Шаблон:Mvar-simplices is connected;
- any two simplices in Шаблон:Mvar lie in some common apartment Шаблон:Mvar;
- if two simplices both lie in apartments Шаблон:Mvar and Шаблон:Math, then there is a simplicial isomorphism of Шаблон:Mvar onto Шаблон:Math fixing the vertices of the two simplices.
An Шаблон:Mvar-simplex in Шаблон:Mvar is called a chamber (originally chambre, i.e. room in French).
The rank of the building is defined to be Шаблон:Math.
Elementary properties
Every apartment Шаблон:Mvar in a building is a Coxeter complex. In fact, for every two Шаблон:Mvar-simplices intersecting in an Шаблон:Math-simplex or panel, there is a unique period two simplicial automorphism of Шаблон:Mvar, called a reflection, carrying one Шаблон:Mvar-simplex onto the other and fixing their common points. These reflections generate a Coxeter group Шаблон:Mvar, called the Weyl group of Шаблон:Mvar, and the simplicial complex Шаблон:Mvar corresponds to the standard geometric realization of Шаблон:Mvar. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in Шаблон:Mvar. Since the apartment Шаблон:Mvar is determined up to isomorphism by the building, the same is true of any two simplices in Шаблон:Mvar lying in some common apartment Шаблон:Mvar. When Шаблон:Mvar is finite, the building is said to be spherical. When it is an affine Weyl group, the building is said to be affine or Euclidean.
The chamber system is the adjacency graph formed by the chambers; each pair of adjacent chambers can in addition be labelled by one of the standard generators of the Coxeter group (see Шаблон:Harvnb).
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(k) space|Шаблон:Math]] 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 Шаблон:Harvnb).
Connection with Шаблон:Math pairs
If a group Шаблон:Mvar acts simplicially on a building Шаблон:Mvar, transitively on pairs Шаблон:Math of chambers Шаблон:Mvar and apartments Шаблон:Mvar containing them, then the stabilisers of such a pair define a [[BN pair|Шаблон:Math pair]] or Tits system. In fact the pair of subgroups
satisfies the axioms of a Шаблон:Math pair and the Weyl group can be identified with Шаблон:Math.
Conversely the building can be recovered from the Шаблон:Math pair, so that every Шаблон:Math pair canonically defines a building. In fact, using the terminology of Шаблон:Math pairs and calling any conjugate of Шаблон:Mvar a Borel subgroup and any group containing a Borel subgroup a parabolic subgroup,
- the vertices of the building Шаблон:Mvar correspond to maximal parabolic subgroups;
- Шаблон:Math vertices form a Шаблон:Mvar-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic;
- apartments are conjugates under Шаблон:Mvar of the simplicial subcomplex with vertices given by conjugates under Шаблон:Mvar of maximal parabolics containing Шаблон:Mvar.
The same building can often be described by different Шаблон:Math pairs. Moreover, not every building comes from a Шаблон:Math pair: this corresponds to the failure of classification results in low rank and dimension (see below).
Spherical and affine buildings for Шаблон:Math
The simplicial structure of the affine and spherical buildings associated to Шаблон:Math, as well as their interconnections, are easy to explain directly using only concepts from elementary algebra and geometry (see Шаблон:Harvnb). In this case there are three different buildings, two spherical and one affine. Each is a union of apartments, themselves simplicial complexes. For the affine building, an apartment is a simplicial complex tessellating Euclidean space Шаблон:Math by Шаблон:Math-dimensional simplices; while for a spherical building it is the finite simplicial complex formed by all Шаблон:Math simplices with a given common vertex in the analogous tessellation in Шаблон:Math.
Each building is a simplicial complex Шаблон:Mvar which has to satisfy the following axioms:
- Шаблон:Mvar is a union of apartments.
- Any two simplices in Шаблон:Mvar are contained in a common apartment.
- If a simplex is contained in two apartments, there is a simplicial isomorphism of one onto the other fixing all common points.
Spherical building
Let Шаблон:Mvar be a field and let Шаблон:Mvar be the simplicial complex with vertices the non-trivial vector subspaces of Шаблон:Math. Two subspaces Шаблон:Math and Шаблон:Math are connected if one of them is a subset of the other. The Шаблон:Mvar-simplices of Шаблон:Mvar are formed by sets of Шаблон:Math mutually connected subspaces. Maximal connectivity is obtained by taking Шаблон:Math proper non-trivial subspaces and the corresponding Шаблон:Math-simplex corresponds to a complete flag
Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces Шаблон:Math.
To define the apartments in Шаблон:Mvar, it is convenient to define a frame in Шаблон:Mvar as a basis (Шаблон:Math) determined up to scalar multiplication of each of its vectors Шаблон:Math; in other words a frame is a set of one-dimensional subspaces Шаблон:Math such that any Шаблон:Mvar of them generate a Шаблон:Mvar-dimensional subspace. Now an ordered frame Шаблон:Math defines a complete flag via
Since reorderings of the various Шаблон:Math also give a frame, it is straightforward to see that the subspaces, obtained as sums of the Шаблон:Math, form a simplicial complex of the type required for an apartment of a spherical building. 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.
Affine building
Let Шаблон:Mvar be a field lying between Шаблон:Math and its [[p-adic number|Шаблон:Mvar-adic completion]] Шаблон:Math with respect to the usual non-Archimedean [[p-adic norm|Шаблон:Mvar-adic norm]] Шаблон:Math on Шаблон:Math for some prime Шаблон:Mvar. Let Шаблон:Mvar be the subring of Шаблон:Mvar defined by
When Шаблон:Math, Шаблон:Mvar is the localization of Шаблон:Math at Шаблон:Mvar and, when Шаблон:Math, Шаблон:Math, the [[p-adic integer|Шаблон:Mvar-adic integers]], i.e. the closure of Шаблон:Math in Шаблон:Math.
The vertices of the building Шаблон:Mvar are the Шаблон:Mvar-lattices in Шаблон:Math, i.e. Шаблон:Mvar-submodules of the form
where Шаблон:Math is a basis of Шаблон:Mvar over Шаблон:Mvar. Two lattices are said to be equivalent if one is a scalar multiple of the other by an element of the multiplicative group Шаблон:Math of Шаблон:Mvar (in fact only integer powers of Шаблон:Mvar need be used). Two lattices Шаблон:Math and Шаблон:Math are said to be adjacent if some lattice equivalent to Шаблон:Math lies between Шаблон:Math and its sublattice Шаблон:Math: this relation is symmetric. The Шаблон:Mvar-simplices of Шаблон:Mvar are equivalence classes of Шаблон:Math mutually adjacent lattices, The Шаблон:Math-simplices correspond, after relabelling, to chains
where each successive quotient has order Шаблон:Mvar. Apartments are defined by fixing a basis Шаблон:Math of Шаблон:Mvar and taking all lattices with basis Шаблон:Math where Шаблон:Math lies in Шаблон:Math and is uniquely determined up to addition of the same integer to each entry.
By definition each apartment has the required form and their union is the whole of Шаблон:Mvar. The second axiom follows by a variant of the Schreier refinement argument. The last axiom follows by a simple counting argument based on the orders of finite Abelian groups of the form
A standard compactness argument shows that Шаблон:Mvar is in fact independent of the choice of Шаблон:Mvar. In particular taking Шаблон:Math, it follows that Шаблон:Mvar is countable. On the other hand, taking Шаблон:Math, the definition shows that Шаблон:Math admits a natural simplicial action on the building.
The building comes equipped with a labelling of its vertices with values in Шаблон:Math. Indeed, fixing a reference lattice Шаблон:Mvar, the label of Шаблон:Mvar is given by
for Шаблон:Mvar sufficiently large. The vertices of any Шаблон:Math-simplex in Шаблон:Mvar has distinct labels, running through the whole of Шаблон:Math. Any simplicial automorphism Шаблон:Mvar of Шаблон:Mvar defines a permutation Шаблон:Mvar of Шаблон:Math such that Шаблон:Math. In particular for Шаблон:Mvar in Шаблон:Math,
Thus Шаблон:Mvar preserves labels if Шаблон:Mvar lies in Шаблон:Math.
Automorphisms
Tits proved that any label-preserving automorphism of the affine building arises from an element of Шаблон:Math. Since automorphisms of the building permute the labels, there is a natural homomorphism
The action of Шаблон:Math gives rise to an [[cyclic permutation|Шаблон:Mvar-cycle]] Шаблон:Mvar. Other automorphisms of the building arise from outer automorphisms of Шаблон:Math associated with automorphisms of the Dynkin diagram. Taking the standard symmetric bilinear form with orthonormal basis Шаблон:Math, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the permutation Шаблон:Mvar that sends each label to its negative modulo Шаблон:Mvar. The image of the above homomorphism is generated by Шаблон:Mvar and Шаблон:Mvar and is isomorphic to the dihedral group Шаблон:Math of order Шаблон:Math; when Шаблон:Math, it gives the whole of Шаблон:Math.
If Шаблон:Mvar is a finite Galois extension of Шаблон:Math and the building is constructed from Шаблон:Math instead of Шаблон:Math, the Galois group Шаблон:Math will also act by automorphisms on the building.
Geometric relations
Spherical buildings arise in two quite different ways in connection with the affine building Шаблон:Mvar for Шаблон:Math:
- The link of each vertex Шаблон:Mvar in the affine building corresponds to submodules of Шаблон:Math under the finite field Шаблон:Math. This is just the spherical building for Шаблон:Math.
- The building Шаблон:Mvar can be compactified by adding the spherical building for Шаблон:Math as boundary "at infinity" (see Шаблон:Harvnb or Шаблон:Harvnb).
Bruhat–Tits trees with complex multiplication
When Шаблон:Mvar is an archimedean local field then on the building for the group Шаблон:Math an additional structure can be imposed of a building with complex multiplication. These were first introduced by Martin L. Brown (Шаблон:Harvnb). These buildings arise when a quadratic extension of Шаблон:Mvar acts on the vector space Шаблон:Math. 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 Шаблон:Math as well as on the Drinfeld modular curve Шаблон:Math. These buildings with complex multiplication are completely classified for the case of Шаблон:Math in Шаблон:Harvnb
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 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 Шаблон:Harvnb); 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 Шаблон:Math pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see Шаблон:Harvnb).
Applications
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.
See also
- Buekenhout geometry
- Coxeter group
- [[(B, N) pair|Шаблон:Math pair]]
- Affine Hecke algebra
- Bruhat decomposition
- Generalized polygon
- Mostow rigidity
- Coxeter complex
- Weyl distance function
References
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Springer
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
External links
- Rousseau: Euclidean Buildings
