Editing Building (mathematics)

{{Short description|Mathematical structure}}
In [[mathematics]], a '''building''' (also '''Tits building''', named after [[Jacques Tits]]) is a combinatorial and geometric structure which simultaneously  generalizes certain aspects of [[flag manifold]]s, finite [[projective plane]]s, and [[Riemannian symmetric space]]s. Buildings were initially introduced by Jacques Tits as a means to understand the structure of [[Reductive group#Non-split reductive groups|isotropic]] [[Reductive group|reductive]] [[linear algebraic group]]s 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|{{mvar|p}}-adic Lie groups]] analogous to that of the theory of [[symmetric spaces]] in the theory of [[Lie group]]s.

==Overview==

[[File:Bruhat-Tits-tree-for-Q-2.png|thumb|The Bruhat&ndash;Tits tree for the 2-adic Lie group {{math|SL(2,''Q''<sub>2</sub>)}}.]]

The notion of a building was invented by [[Jacques Tits]] as a means of describing [[group of Lie type|simple algebraic groups]] over an arbitrary [[field (mathematics)|field]]. Tits demonstrated how to every such [[group (mathematics)|group]] {{mvar|G}} one can associate a [[simplicial complex]] {{math|Δ {{=}} Δ(''G'')}} with an [[Group action (mathematics)|action]] of {{mvar|G}}, called the '''spherical building''' of {{mvar|G}}. The group {{mvar|G}} imposes very strong combinatorial regularity conditions on the complexes {{math|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 {{math|Δ}} is a [[Coxeter group]] {{mvar|W}}, which determines a highly symmetrical simplicial complex {{math|Σ {{=}} Σ(''W'',''S'')}}, called the ''Coxeter complex''. A building {{math|Δ}} is glued together from multiple copies of {{math|Σ}}, called its ''apartments'', in a certain regular fashion. When {{mvar|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 {{mvar|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 {{math|''Ã''<sub>1</sub>}} is the same as an infinite [[tree (graph theory)|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 plane]]s and [[generalized quadrangle]]s 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 ({{harvnb|Tits|1974}}).

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 field|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 {{mvar|n}}-dimensional '''building''' {{mvar|X}} is an [[abstract simplicial complex]] which is a union of subcomplexes {{mvar|A}} called '''apartments''' such that

* every {{mvar|k}}-simplex of {{mvar|X}} is within at least three {{mvar|n}}-simplices if {{math|''k'' < ''n''}};
* any {{math|(''n'' – 1)}}-simplex in an apartment {{mvar|A}} lies in exactly two ''adjacent'' {{mvar|n}}-simplices of {{mvar|A}} and the [[graph theory|graph]] of adjacent {{mvar|n}}-simplices is connected;
* any two simplices in {{mvar|X}} lie in some common apartment {{mvar|A}};
* if two simplices both lie in apartments {{mvar|A}} and {{math|''A''′}}, then there is a simplicial isomorphism of {{mvar|A}} onto {{math|''A''′}} fixing the vertices of the two simplices.

An {{mvar|n}}-simplex in {{mvar|A}} is called a '''chamber''' (originally ''chambre'', i.e. ''room'' in [[French language|French]]).

The '''rank''' of the  building is defined to be {{math|''n'' + 1}}.

==Elementary properties==
Every apartment {{mvar|A}} in a building is a [[Coxeter complex]]. In fact, for every two {{mvar|n}}-simplices intersecting in an {{math|(''n'' – 1)}}-simplex or ''panel'', there is a unique period two simplicial automorphism of {{mvar|A}}, called a ''reflection'', carrying one {{mvar|n}}-simplex onto the other and fixing their common points. These reflections generate a [[Coxeter group]] {{mvar|W}}, called the [[Weyl group]] of {{mvar|A}}, and the simplicial complex {{mvar|A}} corresponds to the standard geometric realization of {{mvar|W}}. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in {{mvar|A}}. Since the apartment {{mvar|A}} is determined up to isomorphism by the building, the same is true of any two simplices in {{mvar|X}} lying in some common apartment {{mvar|A}}. When {{mvar|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 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 {{harvnb|Tits|1981}}).

Every building has a canonical [[intrinsic metric|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|{{math|CAT(0)}}]] comparison inequality of [[Aleksandr Danilovich Aleksandrov|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 {{harvnb|Bruhat|Tits|1972}}).

==Connection with {{math|(''B'', ''N'')}} pairs==
If a group {{mvar|G}} acts simplicially on a building {{mvar|X}}, transitively on pairs {{math|(''C'',''A'')}} of chambers {{mvar|C}} and apartments {{mvar|A}} containing them, then the stabilisers of such a pair define a [[BN pair|{{math|(''B'', ''N'')}} pair]] or [[Tits system]]. In fact the pair of subgroups
:{{math|''B'' {{=}} ''G''<sub>''C''</sub>}} and {{mvar|''N'' {{=}} ''G''<sub>''A''</sub>}}
satisfies the axioms of a {{math|(''B'', ''N'')}} pair and the Weyl group can be identified with {{math|''N'' / ''N'' ∩ ''B''}}.

Conversely the building can be recovered from the {{math|(''B'', ''N'')}} pair, so that every {{math|(''B'', ''N'')}} pair canonically defines a building. In fact, using the terminology of {{math|(''B'', ''N'')}} pairs and calling any conjugate of {{mvar|B}} a [[Borel subgroup]] and any group containing a Borel subgroup a parabolic subgroup,
* the vertices of the building {{mvar|X}} correspond to maximal parabolic subgroups;
* {{math|''k'' + 1}} vertices form a {{mvar|k}}-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic;
* apartments are conjugates under {{mvar|G}} of the simplicial subcomplex with vertices given by conjugates under {{mvar|N}} of maximal parabolics containing {{mvar|B}}.

The same building can often be described by different {{math|(''B'', ''N'')}} pairs. Moreover, not every building comes from a {{math|(''B'', ''N'')}} 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 {{math|SL<sub>''n''</sub>}}==
The simplicial structure of the affine and spherical buildings associated to {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}, as well as their interconnections, are easy to explain directly using only concepts from elementary [[algebra]] and [[geometry]] (see {{harvnb|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 building, an apartment is a simplicial complex [[tessellation|tessellating]] Euclidean space {{math|'''E'''<sup>''n''−1</sup>}} by {{math|(''n'' − 1)}}-dimensional simplices; while for a spherical building it is the finite simplicial complex formed by all {{math|(''n'' − 1)!}} simplices with a given common vertex in the analogous tessellation in {{math|'''E'''<sup>''n''−2</sup>}}.

Each building is a simplicial complex {{mvar|X}} which has to satisfy the following axioms:

* {{mvar|X}} is a union of apartments.
* Any two simplices in {{mvar|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 {{mvar|F}} be a [[field (mathematics)|field]] and let {{mvar|X}} be the simplicial complex with vertices the non-trivial vector subspaces of {{math|''V'' {{=}} ''F''<sup>''n''</sup>}}. Two subspaces {{math|''U''<sub>1</sub>}} and {{math|''U''<sub>2</sub>}} are connected if one of them is a subset of the other. The {{mvar|k}}-simplices of {{mvar|X}} are formed by sets of {{math|''k'' + 1}} mutually connected subspaces. Maximal connectivity is obtained by taking {{math|''n'' − 1}} proper non-trivial subspaces and the corresponding {{math|(''n'' − 1)}}-simplex corresponds to a ''[[Flag (linear algebra)|complete flag]]''

: {{math|(0) ⊂ ''U''<sub>1</sub> ⊂ ··· ⊂ ''U''<sub>''n'' – 1 </sub> ⊂ ''V''}}

Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces {{math|''U''<sub>''i''</sub>}}.

To define the apartments in {{mvar|X}}, it is convenient to define a ''frame'' in {{mvar|V}} as a basis ({{math|''v''<sub>''i''</sub>}}) determined up to scalar multiplication of each of its vectors {{math|''v''<sub>''i''</sub>}}; in other words a frame is a set of one-dimensional subspaces {{math|''L''<sub>''i''</sub> {{=}} ''F''·''v''<sub>''i''</sub>}} such that any {{mvar|k}} of them generate a {{mvar|k}}-dimensional subspace. Now an ordered frame {{math|''L''<sub>1</sub>, ..., ''L''<sub>''n''</sub>}} defines a complete flag via

: {{math|''U''<sub>''i''</sub> {{=}} ''L''<sub>1</sub> ⊕ ··· ⊕ ''L''<sub>''i''</sub>}}

Since reorderings of the various {{math|''L''<sub>''i''</sub>}} also give a frame, it is straightforward to see that the subspaces, obtained as sums of the {{math|''L''<sub>''i''</sub>}}, 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 theorem|Schreier refinement argument]] used to prove the uniqueness of the [[Jordan–Hölder decomposition]].

===Affine building===
Let {{mvar|K}} be a field lying between {{math|'''Q'''}} and its [[p-adic number|{{mvar|p}}-adic completion]] {{math|'''Q'''<sub>''p''</sub>}} with respect to the usual [[Archimedean property|non-Archimedean]] [[p-adic norm|{{mvar|p}}-adic norm]] {{math|{{norm|''x''}}<sub>''p''</sub>}} on {{math|'''Q'''}} for some prime {{mvar|p}}. Let {{mvar|R}} be the [[subring]] of {{mvar|K}} defined by

:{{math|''R'' {{=}} { ''x'' : {{norm|''x''}}<sub>''p''</sub> ≤ 1 } }}

When {{math|''K'' {{=}} '''Q'''}}, {{mvar|R}} is the [[Localization of a ring|localization]] of {{math|'''Z'''}} at {{mvar|p}} and, when {{math|''K'' {{=}} '''Q'''<sub>''p''</sub>}}, {{math|''R'' {{=}} '''Z'''<sub>''p''</sub>}}, the [[p-adic integer|{{mvar|p}}-adic integers]], i.e. the closure of {{math|'''Z'''}} in {{math|'''Q'''<sub>''p''</sub>}}.

The vertices of the building {{mvar|X}} are the {{mvar|R}}-lattices in {{math|''V'' {{=}} ''K''<sup>''n''</sup>}}, i.e. {{mvar|R}}-[[submodules]] of the form

:{{math|''L'' {{=}} ''R''·''v''<sub>1</sub> ⊕ ··· ⊕  ''R''·''v''<sub>''n''</sub>}}

where {{math|(''v''<sub>''i''</sub>)}} is a basis of {{mvar|V}} over {{mvar|K}}. Two lattices are said to be ''equivalent'' if one is a scalar multiple of the other by an element of the multiplicative group {{math|''K''*}} of {{mvar|K}} (in fact only integer powers of {{mvar|p}} need be used). Two lattices {{math|''L''<sub>1</sub>}} and {{math|''L''<sub>2</sub>}} are said to be ''adjacent'' if some lattice equivalent to {{math|''L''<sub>2</sub>}} lies between {{math|''L''<sub>1</sub>}} and its sublattice {{math|''p''·''L''<sub>1</sub>}}: this relation is symmetric. The {{mvar|k}}-simplices of {{mvar|X}} are equivalence classes of {{math|''k'' + 1}} mutually adjacent lattices, The {{math|(''n'' − 1)}}-simplices correspond, after relabelling, to chains

:{{math|''p''·''L''<sub>''n''</sub> ⊂ ''L''<sub>1</sub> ⊂ ''L''<sub>2</sub> ⊂ ··· ⊂ ''L''<sub>''n'' – 1 </sub> ⊂ ''L''<sub>''n''</sub>}}

where each successive quotient has order {{mvar|p}}. Apartments are defined by fixing a basis {{math|(''v''<sub>''i''</sub>)}} of {{mvar|V}} and taking all lattices with basis {{math|(''p''<sup>''a''<sub>''i''</sub></sup> ''v''<sub>''i''</sub>)}} where {{math|(''a''<sub>''i''</sub>)}} lies in {{math|'''Z'''<sup>''n''</sup>}} 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 {{mvar|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

:{{math|''L'' + ''p''<sup>''k''</sup> ·''L''<sub>''i''</sub> / ''p''<sup>''k''</sup> ·''L''<sub>''i''</sub>}}

A standard compactness argument shows that {{mvar|X}} is in fact independent of the choice of {{mvar|K}}. In particular taking {{math|''K'' {{=}} '''Q'''}}, it follows that {{mvar|X}} is countable. On the other hand, taking {{math|''K'' {{=}} '''Q'''<sub>''p''</sub>}}, the definition shows that {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} admits a natural simplicial action on the building.

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

:{{math|label(''M'') {{=}} log<sub>''p''</sub> {{abs|''M'' / ''p''<sup>''k''</sup> ''L''}} modulo ''n''}}

for {{mvar|k}} sufficiently large. The vertices of any {{math|(''n'' – 1)}}-simplex in {{mvar|X}} has distinct labels, running through the whole of {{math|'''Z''' / ''n'''''Z'''}}. Any simplicial automorphism {{mvar|φ}} of {{mvar|X}} defines a permutation {{mvar|π}} of {{math|'''Z''' / ''n'''''Z'''}} such that {{math|label(''φ''(''M'')) {{=}} ''π''(label(''M''))}}. In particular for {{mvar|g}} in {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}},

:{{math|label(''g''·''M'') {{=}} label(''M'') + log<sub>''p''</sub> {{norm|det ''g''}}<sub>''p''</sub> modulo ''n''}}.

Thus {{mvar|g}} preserves labels if {{mvar|g}} lies in {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}.

===Automorphisms===
Tits proved that any label-preserving [[automorphism]] of the affine building arises from an element of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}. Since automorphisms of the building permute the labels, there is a natural homomorphism

:{{math|Aut ''X'' → ''S''<sub>''n''</sub>}}.

The action of {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} gives rise to an [[cyclic permutation|{{mvar|n}}-cycle]]&nbsp;{{mvar|τ}}. Other automorphisms of the building arise from [[outer automorphism]]s of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} associated with automorphisms of the [[Dynkin diagram]]. Taking the standard symmetric bilinear form with orthonormal basis {{math|''v''<sub>''i''</sub>}}, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the permutation {{mvar|σ}} that sends each label to its negative modulo {{mvar|n}}. The image of the above homomorphism is generated by {{mvar|σ}} and {{mvar|τ}} and is isomorphic to the [[dihedral group]] {{math|''D''<sub>''n''</sub>}} of order {{math|2''n''}}; when {{math|''n'' {{=}} 3}}, it gives the whole of {{math|''S''<sub>3</sub>}}.

If {{mvar|E}} is a finite [[Galois extension]] of {{math|'''Q'''<sub>''p''</sub>}} and the building is constructed from {{math|SL<sub>''n''</sub>(''E'')}} instead of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}, the [[Galois group]] {{math|Gal(''E'' / '''Q'''<sub>''p''</sub>)}} will also act by automorphisms on the building.

===Geometric relations===
Spherical buildings arise in two quite different ways in connection with the affine building {{mvar|X}} for {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}:

* The [[link (geometry)|link]] of each vertex {{mvar|L}} in the affine building corresponds to submodules of {{math|''L'' / ''p''·''L''}} under the finite field {{math|''F'' {{=}} ''R'' / ''p''·''R'' {{=}} '''Z''' / (''p'')}}. This is just the spherical building for {{math|SL<sub>''n''</sub>(''F'')}}.
* The building {{mvar|X}} can be ''[[compactification (mathematics)|compactified]]'' by adding the spherical building for {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} as boundary "at infinity" (see {{harvnb|Garrett|1997}} or {{harvnb|Brown|1989}}).

===Bruhat–Tits trees with complex multiplication===

When {{mvar|L}} is an archimedean local field then on the building for the group {{math|SL<sub>2</sub>(''L'')}} an additional structure can be imposed of a building with complex multiplication. These were first introduced by [[Martin L. Brown]]  ({{harvnb|Brown|2004}}). These buildings arise when a quadratic extension of {{mvar|L}} acts on the vector space {{math|''L''<sup>2</sup>}}. 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 {{math|''X''<sub>0</sub>(''N'')}} as well as on the Drinfeld modular curve {{math|''X''{{su|b=0|p=Drin}}(''I'')}}. These buildings with complex multiplication are completely classified for the case of {{math|SL<sub>2</sub>(''L'')}} in {{harvnb|Brown|2004}}

==Classification==
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 polygon]]s, and a plethora of examples exist. (There are free constructions of infinite generalized {{nowrap|''n''-gons}} for every <math>n \geq 3</math>.) 
Many 2-dimensional affine buildings have been constructed using hyperbolic [[reflection group]]s or other more exotic constructions connected with [[orbifold]]s.

Tits also proved that every time a building is described by a {{math|(''B'', ''N'')}} 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 [[group representation|representations]]. The results of Tits on determination of a group by its building have deep connections with [[Mostow rigidity theorem|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 algebra|Kac–Moody groups]] in algebra, and to nonpositively curved manifolds and [[hyperbolic group]]s in topology and [[geometric group theory]].

== See also ==
{{colbegin|colwidth=25em}}
* [[Buekenhout geometry]]
* [[Coxeter group]]
* [[(B, N) pair|{{math|(''B'', ''N'')}} pair]]
* [[Affine Hecke algebra]]
* [[Bruhat decomposition]]
* [[Generalized polygon]]
* [[Mostow rigidity]]
* [[Coxeter complex]]
* [[Weyl distance function]]
{{colend}}

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*{{citation|last=Weiss|first =Richard M.|title=The structure of spherical buildings|publisher=Princeton University Press|year= 2003|isbn= 978-0-691-11733-1}}

==External links==
* Rousseau: [http://hal.inria.fr/docs/00/09/43/63/PDF/_04a5_Euclidean_buildings_Grenoble_.pdf Euclidean Buildings]

[[Category:Group theory]]
[[Category:Algebraic combinatorics]]
[[Category:Geometric group theory]]
[[Category:Mathematical structures]]