Building (mathematics)
Template: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|Template:Mvar-adic Lie groups]] analogous to that of the theory of symmetric spaces in the theory of Lie groups.
OverviewEdit
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 Template:Mvar one can associate a simplicial complex Template:Math with an action of Template:Mvar, called the spherical building of Template:Mvar. The group Template:Mvar imposes very strong combinatorial regularity conditions on the complexes Template: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 Template:Math is a Coxeter group Template:Mvar, which determines a highly symmetrical simplicial complex Template:Math, called the Coxeter complex. A building Template:Math is glued together from multiple copies of Template:Math, called its apartments, in a certain regular fashion. When Template: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 Template: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 Template: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 (Template: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.
DefinitionEdit
An Template:Mvar-dimensional building Template:Mvar is an abstract simplicial complex which is a union of subcomplexes Template:Mvar called apartments such that
- every Template:Mvar-simplex of Template:Mvar is within at least three Template:Mvar-simplices if Template:Math;
- any Template:Math-simplex in an apartment Template:Mvar lies in exactly two adjacent Template:Mvar-simplices of Template:Mvar and the graph of adjacent Template:Mvar-simplices is connected;
- any two simplices in Template:Mvar lie in some common apartment Template:Mvar;
- if two simplices both lie in apartments Template:Mvar and Template:Math, then there is a simplicial isomorphism of Template:Mvar onto Template:Math fixing the vertices of the two simplices.
An Template:Mvar-simplex in Template:Mvar is called a chamber (originally chambre, i.e. room in French).
The rank of the building is defined to be Template:Math.
Elementary propertiesEdit
Every apartment Template:Mvar in a building is a Coxeter complex. In fact, for every two Template:Mvar-simplices intersecting in an Template:Math-simplex or panel, there is a unique period two simplicial automorphism of Template:Mvar, called a reflection, carrying one Template:Mvar-simplex onto the other and fixing their common points. These reflections generate a Coxeter group Template:Mvar, called the Weyl group of Template:Mvar, and the simplicial complex Template:Mvar corresponds to the standard geometric realization of Template:Mvar. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in Template:Mvar. Since the apartment Template:Mvar is determined up to isomorphism by the building, the same is true of any two simplices in Template:Mvar lying in some common apartment Template:Mvar. When Template: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 Template: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|Template: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 Template:Harvnb).
Connection with Template:Math pairsEdit
If a group Template:Mvar acts simplicially on a building Template:Mvar, transitively on pairs Template:Math of chambers Template:Mvar and apartments Template:Mvar containing them, then the stabilisers of such a pair define a [[BN pair|Template:Math pair]] or Tits system. In fact the pair of subgroups
satisfies the axioms of a Template:Math pair and the Weyl group can be identified with Template:Math.
Conversely the building can be recovered from the Template:Math pair, so that every Template:Math pair canonically defines a building. In fact, using the terminology of Template:Math pairs and calling any conjugate of Template:Mvar a Borel subgroup and any group containing a Borel subgroup a parabolic subgroup,
- the vertices of the building Template:Mvar correspond to maximal parabolic subgroups;
- Template:Math vertices form a Template:Mvar-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic;
- apartments are conjugates under Template:Mvar of the simplicial subcomplex with vertices given by conjugates under Template:Mvar of maximal parabolics containing Template:Mvar.
The same building can often be described by different Template:Math pairs. Moreover, not every building comes from a Template:Math pair: this corresponds to the failure of classification results in low rank and dimension (see below).
The Solomon-Tits theorem is a result which states the homotopy type of a building of a group of Lie type is the same as that of a bouquet of spheres.<ref>https://www.ams.org/journals/proc/1998-126-07/S0002-9939-98-04453-0/S0002-9939-98-04453-0.pdf</ref>
Spherical and affine buildings for Template:MathEdit
The simplicial structure of the affine and spherical buildings associated to Template:Math, as well as their interconnections, are easy to explain directly using only concepts from elementary algebra and geometry (see Template: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 Template:Math by Template:Math-dimensional simplices; while for a spherical building it is the finite simplicial complex formed by all Template:Math simplices with a given common vertex in the analogous tessellation in Template:Math.
Each building is a simplicial complex Template:Mvar which has to satisfy the following axioms:
- Template:Mvar is a union of apartments.
- Any two simplices in Template: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 buildingEdit
Let Template:Mvar be a field and let Template:Mvar be the simplicial complex with vertices the non-trivial vector subspaces of Template:Math. Two subspaces Template:Math and Template:Math are connected if one of them is a subset of the other. The Template:Mvar-simplices of Template:Mvar are formed by sets of Template:Math mutually connected subspaces. Maximal connectivity is obtained by taking Template:Math proper non-trivial subspaces and the corresponding Template:Math-simplex corresponds to a complete flag
Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces Template:Math.
To define the apartments in Template:Mvar, it is convenient to define a frame in Template:Mvar as a basis (Template:Math) determined up to scalar multiplication of each of its vectors Template:Math; in other words a frame is a set of one-dimensional subspaces Template:Math such that any Template:Mvar of them generate a Template:Mvar-dimensional subspace. Now an ordered frame Template:Math defines a complete flag via
Since reorderings of the various Template:Math also give a frame, it is straightforward to see that the subspaces, obtained as sums of the Template: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 buildingEdit
Let Template:Mvar be a field lying between Template:Math and its [[p-adic number|Template:Mvar-adic completion]] Template:Math with respect to the usual non-Archimedean [[p-adic norm|Template:Mvar-adic norm]] Template:Math on Template:Math for some prime Template:Mvar. Let Template:Mvar be the subring of Template:Mvar defined by
When Template:Math, Template:Mvar is the localization of Template:Math at Template:Mvar and, when Template:Math, Template:Math, the [[p-adic integer|Template:Mvar-adic integers]], i.e. the closure of Template:Math in Template:Math.
The vertices of the building Template:Mvar are the Template:Mvar-lattices in Template:Math, i.e. Template:Mvar-submodules of the form
where Template:Math is a basis of Template:Mvar over Template:Mvar. Two lattices are said to be equivalent if one is a scalar multiple of the other by an element of the multiplicative group Template:Math of Template:Mvar (in fact only integer powers of Template:Mvar need be used). Two lattices Template:Math and Template:Math are said to be adjacent if some lattice equivalent to Template:Math lies between Template:Math and its sublattice Template:Math: this relation is symmetric. The Template:Mvar-simplices of Template:Mvar are equivalence classes of Template:Math mutually adjacent lattices, The Template:Math-simplices correspond, after relabelling, to chains
where each successive quotient has order Template:Mvar. Apartments are defined by fixing a basis Template:Math of Template:Mvar and taking all lattices with basis Template:Math where Template:Math lies in Template: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 Template: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 Template:Mvar is in fact independent of the choice of Template:Mvar. In particular taking Template:Math, it follows that Template:Mvar is countable. On the other hand, taking Template:Math, the definition shows that Template:Math admits a natural simplicial action on the building.
The building comes equipped with a labelling of its vertices with values in Template:Math. Indeed, fixing a reference lattice Template:Mvar, the label of Template:Mvar is given by
for Template:Mvar sufficiently large. The vertices of any Template:Math-simplex in Template:Mvar has distinct labels, running through the whole of Template:Math. Any simplicial automorphism Template:Mvar of Template:Mvar defines a permutation Template:Mvar of Template:Math such that Template:Math. In particular for Template:Mvar in Template:Math,
Thus Template:Mvar preserves labels if Template:Mvar lies in Template:Math.
AutomorphismsEdit
Tits proved that any label-preserving automorphism of the affine building arises from an element of Template:Math. Since automorphisms of the building permute the labels, there is a natural homomorphism
The action of Template:Math gives rise to an [[cyclic permutation|Template:Mvar-cycle]] Template:Mvar. Other automorphisms of the building arise from outer automorphisms of Template:Math associated with automorphisms of the Dynkin diagram. Taking the standard symmetric bilinear form with orthonormal basis Template:Math, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the permutation Template:Mvar that sends each label to its negative modulo Template:Mvar. The image of the above homomorphism is generated by Template:Mvar and Template:Mvar and is isomorphic to the dihedral group Template:Math of order Template:Math; when Template:Math, it gives the whole of Template:Math.
If Template:Mvar is a finite Galois extension of Template:Math and the building is constructed from Template:Math instead of Template:Math, the Galois group Template:Math will also act by automorphisms on the building.
Geometric relationsEdit
Spherical buildings arise in two quite different ways in connection with the affine building Template:Mvar for Template:Math:
- The link of each vertex Template:Mvar in the affine building corresponds to submodules of Template:Math under the finite field Template:Math. This is just the spherical building for Template:Math.
- The building Template:Mvar can be compactified by adding the spherical building for Template:Math as boundary "at infinity" (see Template:Harvnb or Template:Harvnb).
Bruhat–Tits trees with complex multiplicationEdit
When Template:Mvar is an archimedean local field then on the building for the group Template:Math an additional structure can be imposed of a building with complex multiplication. These were first introduced by Martin L. Brown (Template:Harvnb). These buildings arise when a quadratic extension of Template:Mvar acts on the vector space Template: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 Template:Math as well as on the Drinfeld modular curve Template:Math. These buildings with complex multiplication are completely classified for the case of Template:Math in Template:Harvnb
ClassificationEdit
Tits proved that all irreducible spherical buildings (i.e. with finite Weyl group) of rank greater than 2 are associated to simple algebraic groups, to classical groups (possibly infinite-dimensional), or to a special class of groups called "of mixed type" that only exist in characteristic 2 or 3. 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, the spherical buildings of rank 2 are precisely the generalized polygons, and a plethora of examples exist. (There are free constructions of infinite generalized Template:Nowrap for every <math>n \geq 3</math>.) 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 Template:Math pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see Template:Harvnb).
ApplicationsEdit
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 alsoEdit
- Buekenhout geometry
- Coxeter group
- [[(B, N) pair|Template:Math pair]]
- Affine Hecke algebra
- Bruhat decomposition
- Generalized polygon
- Mostow rigidity
- Coxeter complex
- Weyl distance function
ReferencesEdit
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External linksEdit
- Rousseau: Euclidean Buildings