Building (mathematics)

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For the architectural term, see building.

In mathematics, a building (also Tits building) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. 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 Δ = Δ(G) 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. Not all buildings arise from a group. In particular, projective planes and generalized quadrangles form two classes of graphs studied in incidence geometry which fall under the definition of a building, but may not be connected with any group. Nevertheless, Tits proved a remarkable theorem: that all spherical buildings of rank at least three are connected with a group; and, moreover, the group is essentially determined by the building.

1 Formulation
2 Spherical and affine buildings for SL_n
3 Classification
4 Applications
5 See also
6 References

Formulation

A part of the data defining a building Δ is a certain Coxeter group W. If this group is finite, the corresponding building is of spherical type. If W is an affine Weyl group, the corresponding building is of affine (or Euclidean) type. In the simplest possible case <math>{\scriptstyle\tilde{A}_1,}</math> an affine building is the same as an infinite tree without terminal vertices. 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, which leads to simplifications in both spherical and affine cases. Tits proved that buildings of affine type and rank at least four arise from a group, just as in the spherical case.

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). 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 tesselation 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 tesselation in E^n-2. 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. 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) <math>\subset</math> U₁ <math>\subset</math> ··· <math>\subset</math> U_{n – 1} <math>\subset</math> 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 <math> \oplus</math> ··· <math> \oplus</math> 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.

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. 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.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 <math>\oplus</math> ···<math>\oplus</math> R·v_n

where (v_i) is a basis of V over K. 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 <math>\subset</math> L₁ <math>\subset</math> L₂ <math>\subset</math> ··· <math> \subset</math> L_{n – 1} <math> \subset</math> 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 sufficently 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). Since automorphisms of the building permute the labels, there is a natural homomorphism

Aut X <math>\rightarrow</math> S_n.

The action of GL_n(Q_p) gives rise to an n-cycle τ. Other automorphisms of the building arise from outer automorphisms of SL_n(Q_p) associated with automorphisms of the Dynkin diagram. 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. 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.

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). 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; a similar result holds for irreducible affine buildings of rank greater than two. In rank 2, there is no such classification. In fact Tits 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 rank 2 buildings have been constructed using hyperbolic reflection groups or other more exotic constructions connected with orbifolds.

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.

References

Tits, Jacques (1974), Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, ISBN 0-387-06757-4
Brown, Kenneth S. (1989), Buildings, Springer-Verlag, ISBN 0-387-96876-8
Ronan, Mark Lectures on buildings. Perspectives in Mathematics, 7. Academic Press, Inc., Boston, MA, 1989. xiv+201 pp. ISBN 0-12-594750-X
Garrett, Paul (1997), Buildings and Classical Groups, Chapman & Hall, ISBN 0-412-06331-X Contents, index, bibliography and errata
Weiss, Richard M. The structure of spherical buildings. Princeton University Press, Princeton, NJ, 2003. xiv+135 pp. ISBN 0-691-11733-0
Barré, Sylvain (1995), "Polyèdres finis de dimension 2 à courbure ≤ 0 et de rang 2", Ann. Inst. Fourier 45: 1037-1059 Article in djvu and pdf format.
Barré, Sylvain & Pichot, Mikaël (2007), "Sur les immeubles triangulaires et leurs automorphismes", Geom. Dedicata 130: 71-91 Preprint of article