In mathematics, a building (also Tits building, Bruhat–Tits building) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics Initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type, the theory has also been used to study the geometry and topology of homogeneous spaces of p-adic Lie groups and their discrete subgroups of symmetries, in the same way that trees have been used to study free groups. Jacques Tits (born August 12, 1930 in Uccle) is a French (and formerly Belgian) Mathematician. In Mathematics, a group of Lie type G(k is a (not necessarily finite group of rational points of a reductive Linear algebraic group G with In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Graph theory, a tree is a graph in which any two vertices are connected by exactly one path. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be

Contents 1 Overview 2 Definition 3 Elementary properties 4 Connection with BN pairs 5 Spherical and affine buildings for SLn 5.1 Spherical building 5.2 Affine building 5.3 Automorphisms 5.4 Geometric relations 6 Classification 7 Applications 8 See also 9 References

Overview

The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field. Jacques Tits (born August 12, 1930 in Uccle) is a French (and formerly Belgian) Mathematician. In Mathematics, a group of Lie type G(k is a (not necessarily finite group of rational points of a reductive Linear algebraic group G with In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Tits demonstrated how to every such group G one can associate a simplicial complex Δ = Δ(G) with an action of G, called the spherical building of G. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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 Σ = Σ(W,S), called the Coxeter complex. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries 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. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries An affine building of type is the same as an infinite tree without terminal vertices. In Graph theory, a tree is a graph in which any two vertices are connected by exactly one path.

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. A generalized quadrangle is an Incidence structure. A generalized quadrangle is by definition a Polar space of rank two An incidence geometry is a Mathematical structure composed of objects of various types and an Incidence relation between them 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. In Mathematics, a local field is a special type of field that is a Locally compact Topological field with respect to a non-discrete topology 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, any building of affine type and rank at least four arises from a group.

Definition

An n-dimensional building X is an abstract simplicial complex which is a union of subcomplexes A called apartments such that

• every k-simplex of X is contained in an at least three n-simplices if k < n;
• any (n – 1 )-simplex in an apartment A lies in exactly two adjacent n-simplices of A and the graph of adjacent n-simplices is connected;
• any two simplices in X lie in some common apartment A;
• if two simplices both lie in apartments A and A ', then there is a simplicial isomorphism of A onto A ' fixing the vertices of the two simplices. In Mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of a Simplicial complex, consisting of a family of In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

An n-simplex in A is called a chamber (originally chambre, i. e. room in French). French ( français,) is a Romance language spoken around the world by 118 million people as a native language and by about 180 to 260 million people

The rank of the building is defined to be n + 1.

Elementary properties

Every apartment A in a building is a Coxeter complex. 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. These reflections generate a Coxeter group W, called the Weyl group of A, and the simplicial complex A corresponds to the standard geometric realization of W. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries In Mathematics, in particular the theory of Lie algebras the Weyl group of a Root system &Phi is a Subgroup of the Isometry group Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in A. Since the apartment A is determined up to isomorphism by the building, the same is true of any two simplices in X lie in some common apartment A. When W 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. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries

The chamber system is given by 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 Tits 1981).

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. In the mathematical study of Metric spaces, one can consider the Arclength of paths in the space In Mathematics, an orthonormal basis of an Inner product space V (i This article assumes some familiarity with Analytic geometry and the concept of a limit. 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). In Mathematics, a CAT( k) space is a specific type of Metric space. Aleksandr Danilovich Aleksandrov ( Алекса́ндр Дани́лович Алекса́ндров, alternative Transliterations François Georges René Bruhat (8 April 1929 - July 2007 was a French mathematician who worked on algebraic groups

Connection with BN pairs

If a group G acts simplicially on a building X, transitively on pairs of chambers C and apartments A containing them, then the stabilisers of such a pair define a BN pair or Tits system. In fact the pair of subgroups

B = G_C and N = G_A

satisfies the axioms of a BN pair and the Weyl group can identified with N / N B. Conversely the building can be recovered from the BN pair, so that every BN pair canonically defines a building. In fact, using the terminology of BN pairs and calling any conjugate of B a Borel subgroup and any group containing a Borel subgroup a parabolic subgroup,

• the vertices of the building X correspond to maximal parabolic subgroups;
• k + 1 vertices form a k-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic;
• apartments are conjugates under G of the simplicial subcomplex with vertices given by conjugates under N of maximal parabolics containing B. In the theory of Algebraic groups, a Borel subgroup of an Algebraic group G is a maximal Zariski closed and connected solvable In the theory of Algebraic groups, a Borel subgroup of an Algebraic group G is a maximal Zariski closed and connected solvable

The same building can often be described by different BN pairs. Moreover not every building comes from a BN pair: this corresponds to the failure of classification results in low rank and dimension (see below).

Spherical and affine buildings for SL_n

The simplicial structure of the affine and spherical buildings associated to SL_n(Q_p), as well as their interconnections, are easy to explain directly using only concepts from elementary algebra and geometry (see Garrett 1997). Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position 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 group, an apartment is just the simplicial complex obtained from the standard tessellation of Euclidean space E^n-1 by equilateral (n-1)-simplices; while for a spherical building it is the finite simplicial complex formed by all (n-1)! simplices with a given common vertex in the analogous tessellation in E^n-2. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps

Each building is a simplicial complex X which has to satisfy the following axioms:

• X is a union of apartments.
• Any two simplices in X 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 F be a field and let X be the simplicial complex with vertices the non-trivial vector subspaces of V=Fⁿ. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Two subspaces U₁ and U₂ are connected if one of them is a subset of the other. The k-simplices of X are formed by sets of k + 1 mutually connected subspaces. Maximal connectivity is obtained by taking n - 1 subspaces and the corresponding (n-2)-simplex corresponds to a complete flag

(0) U₁ ··· U_{n – 1} V

Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces U_i.

To define the apartments in X, it is convenient to define a frame in V as a basis (v_i) determined up to scalar multiplication of each of its vectors v_i; in other words a frame is a set of one-dimensional subspaces L_i = F·v_i such that any k of them generate a k-dimensional subspace. Now an ordered frame L₁, . . . , L_n defines a complete flag via

U_i = L₁ ··· L_i

Since reorderings of the L_i's also give a frame, it is straightforward to see that the subspaces, obtained as sums of the L_i's, 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. In Mathematics, the Schreier refinement theorem of Group theory states that any two Normal series of Subgroups ending with the Trivial group In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces

Affine building

Let K be a field lying between Q and its p-adic completion Q_p with respect to the usual non-archimedean p-adic norm ||x||_p on Q for some prime p. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 Let R be the subring of K defined by

When K = Q, R is the localisation of Z at p and, when K = Q_p, R = Z_p, the p-adic integers, i. In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 e. the closure of Z in Q_p.

The vertices of the building X are the R-lattices in V = Kⁿ, i. e. R-submodules of the form

L = R·v₁ ··· R·v_n

where (v_i) is a basis of V over K. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Two lattices are said to be equivalent if one is a scalar multiple of the other by an element of the multiplicative group K* of K (in fact only integer powers of p need be used). Two lattice L₁ and L₂ are said to be adjacent if some lattice equivalent to L₂ lies between L₁ and its sublattice p·L₁: this relation is symmetric. The k-simplices of X are equivalence classes of k + 1 mutually adjacent lattices, The (n - 1)- simplices correspond, after relabelling, to chains

p·L_n L₁ L₂ ··· L_{n – 1} L_n

where each successive quotient has order p. Apartments are defined by fixing a basis (v_i) of V and taking all lattices with basis (p^a_i v_i) where (a_i) lies in Zⁿ 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 X. 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

L + p^k ·L_i / p^k ·L_i .

A standard compactness argument shows that X is in fact independent of the choice of K. In particular taking K = Q, it follows that X is countable. On the other hand taking K = Q_p, the definition shows that GL_n(Q_p) admits a natural simplicial action on the building.

The building comes equipped with a labelling of its vertices with values in Z / n Z. Indeed, fixing a reference lattice L, the label of M is given by

label (M) = log_p |M/ p^k L| modulo n

for k sufficiently large. The vertices of any (n – 1)-simplex in X have distinct labels, running through the whole of Z / n Z. Any simplicial automorphism φ of X defines a permutation π of Z / n Z such that label (φ(M)) = π(label (M)). In particular for g in GL_n (Q_p),

label (g·M) = label (M) + log_p || det g ||_p modulo n.

Thus g preserves labels if g lies in SL_n(Q_p).

Automorphisms

Tits proved that any label-preserving automorphism of the affine building arises from an element of SL_n(Q_p). In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Since automorphisms of the building permute the labels, there is a natural homomorphism

Aut X S_n.

The action of GL_n(Q_p) gives rise to an n-cycle τ. Let S be a set A cycle is a Permutation ( bijective function of a set onto itself such that there exist distinct elements a_1 a_2\ldotsa_k Other automorphisms of the building arise from outer automorphisms of SL_n(Q_p) associated with automorphisms of the Dynkin diagram. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner This article discusses root systems in mathematics For root systems of Plants see Root. Taking the standard symmetric bilinear form with orthonormal basis v_i, the map sending a lattice to its dual lattice gives an automorphism with square 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 D_n of order 2n; when n = 3, it gives the whole of S₃. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections

If E is a finite Galois extension of Q_p and the building is constructed from SL_n(E) instead of SL_n(Q_p), the Galois group Gal (E/Q_p) will also act by automorphisms on the building. In Mathematics, a Galois extension is an algebraic field extension E / F satisfying certain conditions (described below one also says that the In Mathematics, a Galois group is a group associated with a certain type of Field extension.

Geometric relations

Spherical buildings arise in two quite different ways in connection with the affine building X for SL_n(Q_p):

• The link of each vertex L in the affine building corresponds to submodules of L/p·L under the finite field F = R/p·R = Z/(p). In Geometry, the link of a vertex of a 2- Dimensional Simplicial complex is a graph that encodes information about the local structure This is just the spherical building for SL_n(F).
• The building X can be compactified by adding the spherical building for SL_n(Q_p) as boundary "at infinity" (see Garrett 1997 or Brown 1989).

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. In Mathematics, in particular the theory of Lie algebras the Weyl group of a Root system &Phi is a Subgroup of the Isometry group A similar result holds for irreducible affine buildings of dimension greater than two (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 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. In combinatorial Mathematics, an incidence structure is a triple C=(PLI Many 2-dimensional affine buildings have been constructed using hyperbolic reflection groups or other more exotic constructions connected with orbifolds. A reflection group is a Group action, acting on a finite dimensional Vector space, which is generated by reflections elements that fix a Hyperplane in In the mathematical disciplines of Topology and Geometric group theory, an orbifold (for "orbit-manifold" is a generalization of a Manifold.

Tits also proved that every time a building is described by a BN pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see Tits 1974).

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. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of 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. In Mathematics, Mostow's rigidity theorem, sometimes called the strong rigidity theorem, essentially states that the geometry of a finite volume Hyperbolic manifold George Mostow (full name George Daniel Mostow is an American Mathematician, a member of the National Academy of Sciences, Professor emeritus of Yale Gregori Aleksandrovich Margulis (Григорий Александрович Маргулис first name often given as Gregory, Grigori or Grigory) (born

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 classification of the finite simple groups, also called the enormous theorem is believed to classify all 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. In Mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional that can be defined by generators and relations through a Generalized Cartan In Group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated Geometric group theory is an area in Mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and

See also

References

• Ballmann, Werner & Brin, Michael (1995), “Orbihedra of nonpositive curvature”, Publications Mathématiques de l'IHÉS 82: 169-209, <http://www.numdam.org/item?id=PMIHES_1995__82__169_0> 
• Barré, Sylvain (1995), “Polyèdres finis de dimension 2 à courbure ≤ 0 et de rang 2”, Ann. In Mathematics, a Coxeter group, named after HSM Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries Definition Let V be a Euclidean space of a finite dimension and \Sigma an Affine root system on V In mathematics the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a In Combinatorial theory, a generalized n -gon is an incidence structure introduced by Jacques Tits. In Mathematics, Mostow's rigidity theorem, sometimes called the strong rigidity theorem, essentially states that the geometry of a finite volume Hyperbolic manifold Inst. Fourier 45: 1037-1059, <http://www.numdam.org/numdam-bin/fitem?id=AIF_1995__45_4_1037_0> 
• Barré, Sylvain & Pichot, Mikaël (2007), “Sur les immeubles triangulaires et leurs automorphismes”, Geom. Dedicata 130: 71-91, <http://web.univ-ubs.fr/lmam/barre/henri.pdf> 
• Bourbaki, Nicolas (1968), Lie Groups and Lie Algebras: Chapters 4-6, Elements of Mathematics, Hermann, ISBN 3-540-42650-7 
• Brown, Kenneth S. (1989), Buildings, Springer-Verlag, ISBN 0-387-96876-8 
• Bruhat, François & Tits, Jacques (1972), “Groupes réductifs sur un corps local, I. Données radicielles valuées”, Publ. Math. IHES 41: 5–251, <http://www.numdam.org/item?id=PMIHES_1972__41__5_0> 
• Garrett, Paul (1997), Buildings and Classical Groups, Chapman & Hall, ISBN 0-412-06331-X, <http://www.math.umn.edu/~garrett/m/buildings> 
• Kantor, William M. (2001), “Tits building”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Kantor, William M. The Encyclopaedia of Mathematics is a large reference work in Mathematics. (1986), “Generalized polygons, SCABs and GABs”, in Rosati, L. A. , Buildings and the Geometry of Diagrams (CIME Session, Como 1984), vol. 1181, Lect. notes in math. , Springer, pp. 79–158, DOI 10. 1007/BFb0075513 
• Pott, Alexander (1995), Finite Geometry and Character Theory, vol. 1601, Lect. Notes in Math. , Springer-Verlag, ISBN 354059065X, DOI 10. 1007/BFb0094449 
• Ronan, Mark (1995), A construction of buildings with no rank 3 residues of spherical type, vol. 1181, Lect. Notes in Math. , Springer-Verlag, pp. 159-190, DOI 10. 1007/BFb0075518 
• Ronan, Mark (1992), “Buildings: main ideas and applications. II. Arithmetic groups, buildings and symmetric spaces”, Bull. London Math. Soc. 24 (2): 97--126, MR1148671 
• Ronan, Mark (1992), “Buildings: main ideas and applications. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many I. Main ideas. ”, Bull. London Math. Soc. 24 (1): 1--51, MR1139056 
• Ronan, Mark (1989), Lectures on buildings, Perspectives in Mathematics 7, Academic Press, ISBN 0-12-594750-X 
• Tits, Jacques (1974), Buildings of spherical type and finite BN-pairs, vol. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many 386, Lecture Notes in Mathematics, Springer-Verlag, ISBN 0-387-06757-4, DOI 10. 1007/BFb0057391 
• Tits, Jacques (1981), “A local approach to buildings”, The geometric vein: The Coxeter Festschrift, Springer-Verlag, pp. 519-547, ISBN 0387905871 
• Tits, Jacques (1986), “Immeubles de type affine”, in Rosati, L. A. , Buildings and the Geometry of Diagrams (CIME Session, Como 1984), vol. 1181, Lect. notes in math. , Springer, pp. 159-190, DOI 10. 1007/BFb0075514 
• Weiss, Richard M. (2003), The structure of spherical buildings, Princeton University Press, ISBN 0-691-11733-0 

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