Building (mathematics)

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. 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.

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 "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. 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". 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 $\{scriptstyle\; ilde\{A\}\_1\}$ 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, 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.

An "n"-simplex in "A" is called a chamber (originally "chambre", i.e. "room" in French).

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". Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in "A". Sincethe 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.

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 standardgenerators of the Coxeter group (see ).

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 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 harvnb|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. 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 BN pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see harvnb|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. 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

* Coxeter group
* BN pair
* Affine Hecke algebra
* Bruhat decomposition
* Generalized polygon
* Tits geometry
* Twin building
* Hyperbolic building
* Tits simplicity theorem
* Mostow rigidity

References

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*citation|first=Sylvain|last=Barré|title=Polyèdres finis de dimension 2 à courbure ≤ 0 et de rang 2|journal=Ann. Inst. Fourier|volume=45|year=1995|pages=1037-1059|url=http://www.numdam.org/numdam-bin/fitem?id=AIF_1995__45_4_1037_0
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last=Brown|first= Kenneth S.|title= Buildings|publisher= Springer-Verlag|year= 1989|id= ISBN 0-387-96876-8
*citation|first=François|last= Bruhat|first2= Jacques|last2= Tits|title=Groupes réductifs sur un corps local, I. Données radicielles valuées|journal= Publ. Math. IHES |volume= 41 |year=1972|pages= 5–251
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*springer|id=T/t092900|title=Tits building|first=William M.|last= Kantor
* citation|first=William M.|last= Kantor|chapter=Generalized polygons, SCABs and GABs|editor-first= L.A. |editor-last=Rosati |title= Buildings and the Geometry of Diagrams (CIME Session, Como 1984) |series= Lect. notes in math. |volume= 1181 |publisher= Springer |year=1986|pages= 79–158
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