Buekenhout geometries and the realization of groups as … · 2011-09-21 · Buekenhout geometries and the realization of groups as buildings. Part I: Buildings Nate Iverson Department

Buekenhout geometries and the realization of groups asbuildings.

Part I: Buildings

Nate Iverson

Department of Mathematics and StatisticsThe University of Toledo

Toledo, Ohio

September 20, 2011

Nate Iverson (UT) Buekenhout geometries and the realization of groups as buildings. September 20, 2011 1 / 21

Table of Contents1 Representations and Realizations2 Buildings: A History3 Coxeter Groups

Irreducible Spherical Coxeter GroupsThe classification of the irreducible spherical Coxeter groups:The classification of the irreducible spherical Coxeter groups:The classification of the irreducible spherical Coxeter groups:

4 Chamber Systems5 Coxeter chamber system6 Homotopy7 Distance8 Buildings

ApartmentsReflectionsRootsSome basic results on BuildingsBuilding Alternative Definition

9 BN-pairs10 Next Week...11 References


Representations and Realizations

DefinitionA representation is a group homomorphism

ρ : G → H

Where H is a subgroup of the group of invertible linear transformations GL(V ).

The main idea:Allows us to translate group theoretic questions into linear algebra questions

DefinitionA realization is a group homomorphism

ρ : G → H

Where H is a subgroup of the automorphism group Aut(Γ) for some object Γ .

Thus we can gain information about groups by realizing them into interestingobjects.




ρ : G → H


The main idea:

Allows us to translate group theoretic questions into linear algebra questions


ρ : G → H






ρ : G → H




ρ : G → H






ρ : G → H




ρ : G → H






ρ : G → H




ρ : G → H




Buildings: A History1934 H. S. M. Coxeter introduces a generalization of reflection groups thatbecome known as Coxeter groups[3].

1935 H. S. M. Coxeter classifies all finite Coxeter groups[4].

1962 J. Tits introduces the concept of a generalized polygon[10].

1964 J. Tits introduces BN-pairs for groups of Lie type and discovers hisfinite simple group[7].

1964 W. Feit and G. Higman show that generalized n-gons can only haven ∈ {2, 3, 4, 8, 6, 12} and some further restrictions on reflection groups usingBN-pairs[6].

1974 J. Tits shows introduces Moufang Polygons (rank 2 buildings) andclassifies all thick buildings of rank at least 3[11].

1981 J. Tits gives a second equivalent definition of a building[9].

1985 Stellmacher classifies weak BN-pairs[5].

2002 J. Tits and R. Weiss finish the classification of Moufang Polygons[12].

2004 M. Aschbacher and S. Smith finish the last step in the classification ofthe finite simple groups[1, 2]. Buildings are used to both describe thegeometry of the groups of Lie type and many sporadic groups. Aschbacherand Smith use theory of weak BN-pairs.
















































































































Coxeter Groups

Definition

Let I be a set with |I | = n.

A Coxeter matrix is an n × n symmetric matrix [mij ] with either positiveinteger or ∞ values and the property that mij = 1 if and only if i = j .

A Coxeter diagram Π = (I ,E ) is an undirected labeled graph withE = {(i , j) | i , j ∈ I with mij ≥ 3} and the label of each edge is mij .

A Coxeter group W of type Π is the group defined by generators {ri | i ∈ I}with relations {(ri rj)

mij = 1 | i , j ∈ I}. The rank of the Coxeter group is |I |.(W , r) is called a Coxeter System of type Π if r is the homomorphism fromthe free monoid MI to W that extends the injective map i → ri .

Notes:Since mii = 1 for each i ∈ I we have r 2

i = 1 for all i ∈ I .If i 6= j do not have an edge between them in Π then mij = 2. Thus,(ri rj)

2 = 1. Since r 2i = r 2

j = 1, (ri rj)2 is a commutator so ri and rj commute.

DefinitionA Coxeter group W of type Π is irreducible if and only if Π is a connected graph.


Coxeter Groups

Definition








2 = 1. Since r 2i = r 2




Coxeter Groups

Definition





mij = 1 | i , j ∈ I}. The rank of the Coxeter group is |I |.

(W , r) is called a Coxeter System of type Π if r is the homomorphism fromthe free monoid MI to W that extends the injective map i → ri .



2 = 1. Since r 2i = r 2




Coxeter Groups

Definition








2 = 1. Since r 2i = r 2




Coxeter Groups

Definition







i = 1 for all i ∈ I .

If i 6= j do not have an edge between them in Π then mij = 2. Thus,(ri rj)

2 = 1. Since r 2i = r 2




Coxeter Groups

Definition








2 = 1. Since r 2i = r 2




Coxeter Groups

Definition








2 = 1. Since r 2i = r 2




Irreducible Spherical Coxeter Groups

DefinitionCoxeter group W of type Π is spherical if it is a group of finite order.

The classification of the irreducible spherical Coxeter groups[4]:

r r r r. . .1 2 n

An r r r r. . .1 2 n

BCn 4 r r r rr. . . ��

@@1 2

n

n − 1

Dn

r r rr r rE6 r r rr r r rE7 r r rr r r r rE8

r r r rF4 4 r r rH3 5 r r r rH4 5 r rI2(m) m

Theorem

If a Coxeter system (W , r) of type Π has a diagram with connected components{A1,A2, . . . ,Am}, then the Coxeter group W of type Π is the direct sum ofgroups W (Ak) = 〈r(k) | k ∈ Ak〉. Therefore, all spherical Coxeter groups aredirect sums of irreducible sperical Coxeter groups.





r r r r. . .1 2 n

An r r r r. . .1 2 n

BCn 4 r r r rr. . . ��

@@1 2

n

n − 1

Dn



Theorem






r r r r. . .1 2 n

An r r r r. . .1 2 n

BCn 4 r r r rr. . . ��

@@1 2

n

n − 1

Dn



Theorem



Chamber SystemLet I be a set and ∆ = (V ,E ) an I -colored graph(edges). When a graph iscolored such as this we call the verticies chambers and we call a path a gallery.Notions of distance, diameter, connectedness and convexity all follow from graphtheory. For chambers x , y we call dist(x , y) the distance from x to y in the graph.If x and y are on an i-edge then we say x ∼i y . For J ⊂ I a J-residue of ∆ is aconnected component with colors only in J. An i -panel is an {i}-residue.

DefinitionA chamber system ∆ is an I -colored graph as above where every i-panel is acomplete graph.

A chamber system is thin if every panel contains at exactly two chambers.A chamber system is thick if every panel contains at least three chambers.

Definition

Two chamber systems ∆ and ∆′ with index sets I and I ′ are called isomorphic ifthere are bijections σ : I → I ′ and φ : ∆→ ∆′ such that

x ∼i y if and only if φ(x) ∼σ(i) φ(y)

. If σ = 1 it is called a special isomorphism.





Definition







A chamber system is thin if every panel contains at exactly two chambers.

A chamber system is thick if every panel contains at least three chambers.

Definition








Definition








Definition



.

If σ = 1 it is called a special isomorphism.





Definition



. If σ = 1 it is called a special isomorphism.Nate Iverson (UT) Buekenhout geometries and the realization of groups as buildings. September 20, 2011 7 / 21

Coxeter chamber system

Definition

Let (W , r) be a Coxeter system of type Π. The Coxeter chamber system ΣΠ is(up to special isomorphism) the following: Let the chambers be the elements ofW , then the edges are defined

x ∼i y if and only if x−1y = ri

Note:

To show ΣΠ is well defined, note |ri | = 2 for all i ∈ I .ΣΠ is infact just the Cayley graph of W w.r.t. the generators {ri | i ∈ I}.ΣΠ is a thin Chamber system.

TheoremLet Σ be the Coxeter chamber system of type Π and i 6= j in I .

Every {i , j}-residue contains exactly 2mij chambers.

For every f ∈ MI and chamber x there is a unique gallery of type f startingat x and ending with xrf .

If R is a J-residue and x , y ∈ R with x ∼i y then i ∈ J.



Definition



Note:

To show ΣΠ is well defined, note |ri | = 2 for all i ∈ I .

ΣΠ is infact just the Cayley graph of W w.r.t. the generators {ri | i ∈ I}.ΣΠ is a thin Chamber system.







Definition



Note:

To show ΣΠ is well defined, note |ri | = 2 for all i ∈ I .ΣΠ is infact just the Cayley graph of W w.r.t. the generators {ri | i ∈ I}.

ΣΠ is a thin Chamber system.







Definition



Note:








Definition



Note:








Definition



Note:








Definition



Note:







Homotopy IDefine p : I × I → MI via

p(i , j) =

(ij)mij/2 if mij is even

j(ij)(mij−1)/2 if mij is odd

undefined if mij =∞

= · · · ijij︸︷︷︸mij terms

Definition

An elementary homotopy is transforms word f1p(i , j)f2 to f1p(j , i)f2

A contraction transforms word f1iif2 to f1f2.

A expansion transforms word f1f2 to f1iif2.

A word f is reduced if it is not homotopic to a contraction.

Two words are equivalent if they can be transformed into each other by afinite number of elementary homotopies, expansions and contractions.

For word f define f to be the reverse of f . Conjugation by f ∈ M1 is the mapx → fx f from MI to itself.

For word f define f −1 to be the inverse of f in the free group FI .Conjugation by f ∈ FI is the map x → fxf −1 from FI to itself.



p(i , j) =



undefined if mij =∞= · · · ijij︸︷︷︸

mij terms

Definition










p(i , j) =




mij terms

Definition










p(i , j) =




mij terms

Definition










p(i , j) =




mij terms

Definition










p(i , j) =




mij terms

Definition










p(i , j) =




mij terms

Definition










p(i , j) =




mij terms

Definition









Homotopy II

Theorem

f , g ∈ MI are equivalent (with respect to Π) if and only if rf = rg .

If rf = rg for two words f , g ∈ MI then |f | and |g | have the same parity.


Distance

DefinitionNote there is a unique chamber in ΣΠ that corresponds to the identity 1 ∈W .

‖rf ‖ = dist(1,w)

|rf | = length(f ) where r(f ) = rf and f is a reduced word with respect to thediagram Π (in free group FI ).

Theorem

‖rf ‖ = |rf |


Distance

DefinitionNote there is a unique chamber in ΣΠ that corresponds to the identity 1 ∈W .

‖rf ‖ = dist(1,w)

|rf | = length(f ) where r(f ) = rf and f is a reduced word with respect to thediagram Π (in free group FI ).

Theorem

‖rf ‖ = |rf |


Buildings

Definition

Let (W , r) be a Coxeter system of type Π with vertex set I . A building of type Πis a pair (∆, δ) where ∆ is a chamber system whose index set is I andδ : ∆×∆→W such that

δ(x , y) = rf if and only if there is a gallery in ∆ of type f from x to y

W is called the Weyl group and δ is called the Weyl distance function of ∆.

Theorem

Let (∆, δ) be a building and x , y chambers in ∆. Then,

δ is surjective and ∆ is connected.

δ(x , y) = δ(y , x)−1

dist(x , y) = ‖δ(x , y)‖

(ΣΠ, δ) is a building when δ(x , y) = x−1y and ΣΠ is the Coxeter chamber systemof type Π.


Buildings

Definition




Theorem



δ(x , y) = δ(y , x)−1

dist(x , y) = ‖δ(x , y)‖



Buildings

Definition




Theorem



δ(x , y) = δ(y , x)−1

dist(x , y) = ‖δ(x , y)‖



Buildings

Definition




Theorem



δ(x , y) = δ(y , x)−1

dist(x , y) = ‖δ(x , y)‖



Apartments

Definition

Let (∆, δ) be a building of type Π and ΣΠ the Coxeter chamber system of type Π.

A subgraph X of ∆ is called an apartment if there is an isometryπ : ΣΠ → ∆ with π(ΣΠ) = X .

Let X ⊂ ΣΠ. An isometry is a function π : X → ∆ sending chambers tochambers and edges to edges such that

δ (xπ, yπ) = x−1y for all chambers x , y in X

TheoremLet X ⊂ ΣΠ Every isometry f : X → ∆ extends to an isometry F : ΣΠ → ∆.

Every two chambers in a building are contained in a common apartment.

Let R be a J-residue of ∆ and Σ and apartment. Then R ∩Σ is either emptyor a J-residue of Σ.

Let x , y , z be chambers in apartment Σ. If y and z are adjacent, thendist(x , y) = dist(x , z)± 1


Apartments

Definition










Apartments

Definition










Apartments

Definition










Apartments

Definition










Reflections I

Note W acts on the special automorphisms Aut◦(Σ) by multiplication. Let x be achamber in ΣP i and rf ∈W the automorphism associated with rf is the mapx → rf x . Since Σ is connected and every chamber is i-adjacent to exactly onechamber for each i ∈ I , the identity is the only special automorphism of Σ whichfixes a chamber. So the map that sends w ∈W to “left multiplication by w” is anisomorphism.

Definition

A reflection is a non-trivial element of W that fixes edges of Σ. (Note edgesare panels).

The set of edges fixed by reflection s is called the wall of s and is denoted Ms .

The gallery γ = (c0, c1, . . . , ck) crosses the wall Ms at the panel {ci−1, ci} forsome i ∈ [1, k] if the panel {ci−1, ci} is contained in Ms .

We say γ crosses Ms m times if m is the number of indices i ∈ [1, k] suchthat γ crosses Ms at {ci−1, ci}.


Reflections I


Definition






Reflections I


Definition






Reflections I


Definition






Reflections I


Definition






Roots IConsequences:

Let s be a reflection and {x , y} and edge fixed by s. Since s is non-trivial itcannot fix chambers x or y , thus it must exchange them and hence |s| = 2.

s is uniquely determined by {x , y}. If {x , y} is an edge, then y = xri forexactly one i ∈ I . In particular xrix

−1 is the reflection that fixes {x , y}.In Σ a minimal gallery cannot cross a wall more than once.

Let x , y ∈ Σ and s be a reflection in Σ. For each gallery γ from x to y letn(y) denote the number of times γ crosses Ms . The parity of n(γ) dependsonly on x , y and s not on γ.

DefinitionLet s be a reflection of Σ and x , y chambers of Σ

Define ∼=s on Σ by x ∼=s y if and only if there is a gallery in Σ from x to ythat crosses the wall Ms an even number of times (Note this is well definedby above).∼=s is an equivalence relation. A root of Σ is an equivalence class withrespect to ∼=s for some reflection s.





−1 is the reflection that fixes {x , y}.

In Σ a minimal gallery cannot cross a wall more than once.



























Define ∼=s on Σ by x ∼=s y if and only if there is a gallery in Σ from x to ythat crosses the wall Ms an even number of times (Note this is well definedby above).

∼=s is an equivalence relation. A root of Σ is an equivalence class withrespect to ∼=s for some reflection s.










Roots II

TheoremLet s be a reflection in Σ .

There are exactly two roots for each reflection s in Σ. For each panel{x , y} ∈ Ms the two roots are

{u | dist(u, x) < dist(u, y)} and {u | dist(u, x) ≥ dist(u, y)}

The two roots of s are interchanged by s.

A panel is contained in Ms if and only if it contains one chamber from eachof the two roots.

The subgraph spanned by a root is convex.

All residues of Σ are convex.

DefinitionA root of building ∆ is a root of some apartment Σ of ∆.


Roots II










Roots II










Roots II










Roots II










Roots II










Some basic results on Buildings

A residue of a building ∆ is convex

Let R and R ′ are J and J ′ residues respectively and S = R ∩ R ′. If S 6= ∅then S is a J ∩ J ′ residue.

If Σ is and apartment of ∆ such that Σ ∩ R 6= ∅ and α is a root of Σ thenΣ∩R is an apartment of R and either α∩R is a root of Σ∩R or Σ∩R ⊂ αor Σ ∩ R ⊂ Σ\α.

If Σ0 is an apartment of R, then Σ0 = Σ ∩ R for some apartment Σ of ∆,and for each apartment Σ and for each root α0 of Σ0, there is a unique rootα of Σ such that α0 = α ∩ R.

Let ∆ be a building, R a residue of ∆ and Σ0 and apartment of R. If Σ is anapartment of ∆ containing Σ0 then Σ0 = Σ ∩ R.

Let ∆ be a building, R a residue of ∆ and x a chamber of ∆. Then there isa unique chamber w ∈ R nearest to x . Furthermore, w is contained in everyapartment containing x and some chamber of R.

Suppose R and R ′ are two residues of building ∆ such that R ∩ R ′ 6= ∅ andlet P be a panel containing at least two chambers in R ∪ R ′. Then P ⊂ R orP ⊂ R ′
























































Building Alternative Definition

DefinitionAn n-dimensional building X is an abstract simplicial complex which is a union ofsubcomplexes A called apartments such that:

Every k-simplex of X is within at least three n-simplices if k < n;

any (n − 1)-simplex in an apartment A lies in exactly two adjacentn-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 simplicialisomorphism of A onto A′ fixing the vertices of the two simplices.

An n-simplex in A is called a chamber. The rank of the building is defined to ben + 1.

This is actually the original definition. To see that they are the same see [9].


Building Alternative Definition

DefinitionAn n-dimensional building X is an abstract simplicial complex which is a union ofsubcomplexes A called apartments such that:

Every k-simplex of X is within at least three n-simplices if k < n;

any (n − 1)-simplex in an apartment A lies in exactly two adjacentn-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 simplicialisomorphism of A onto A′ fixing the vertices of the two simplices.

An n-simplex in A is called a chamber. The rank of the building is defined to ben + 1.

This is actually the original definition. To see that they are the same see [9].


BN-pairs

DefinitionA BN-pair is a pair of subgroups B and N of a group G such that the followingaxioms hold:

G is generated by B and N.H = B ∩ N is a normal subgroup of N.The group W = N/H is generated by a set of elements wi of order 2, for i insome non-empty set I .If wi is one of the generators of W and w is any element of W , then wiBw iscontained in BwiwB ∪ BwB.No generator wi normalizes B. The group B is called the Borel subgroup, His called the Cartan Subgroup and W is called the Weyl group.

Note:

G = GL(V ) then B is the invertible upper triangular matrices, H is thediagonal matrices, and N is the monomial matrices.If G acts transitively on the chambers and containing apartments. Let C be achamber and A an apartment containing C then B = GC , N = GA is aBN-pair.BN-pairs and buildings are interchangeable for rank ≥ 3 but in lower rank notevery BN-pair describes a building.


BN-pairs



Note:

G = GL(V ) then B is the invertible upper triangular matrices, H is thediagonal matrices, and N is the monomial matrices.

If G acts transitively on the chambers and containing apartments. Let C be achamber and A an apartment containing C then B = GC , N = GA is aBN-pair.BN-pairs and buildings are interchangeable for rank ≥ 3 but in lower rank notevery BN-pair describes a building.


BN-pairs



Note:

G = GL(V ) then B is the invertible upper triangular matrices, H is thediagonal matrices, and N is the monomial matrices.If G acts transitively on the chambers and containing apartments. Let C be achamber and A an apartment containing C then B = GC , N = GA is aBN-pair.

BN-pairs and buildings are interchangeable for rank ≥ 3 but in lower rank notevery BN-pair describes a building.


BN-pairs



Note:

G = GL(V ) then B is the invertible upper triangular matrices, H is thediagonal matrices, and N is the monomial matrices.If G acts transitively on the chambers and containing apartments. Let C be achamber and A an apartment containing C then B = GC , N = GA is aBN-pair.BN-pairs and buildings are interchangeable for rank ≥ 3 but in lower rank notevery BN-pair describes a building.


Next Week...

Define Buekenhout geometry and discuss under what conditions these arebuildings. This will allow us to realize a general group, with a geometric structureinside the structure of a building.


References[1] Michael Aschbacher and Stephen D. Smith. The classification of quasithin

groups. I, volume 111 of Mathematical Surveys and Monographs. AmericanMathematical Society, Providence, RI, 2004. Structure of strongly quasithinK -groups.

[2] Michael Aschbacher and Stephen D. Smith. The classification of quasithingroups. II, volume 112 of Mathematical Surveys and Monographs. AmericanMathematical Society, Providence, RI, 2004. Main theorems: theclassification of simple QTKE-groups.

[3] H. S. M. Coxeter. Discrete groups generated by reflections. Ann. of Math.(2), 35(3):588–621, 1934.

[4] H. S. M. Coxeter. The complete enumeration of finite groups of the formr 2i = (ri rj)

kij = 1. London Math. Soc. 10, pages 21–25, 1935.

[5] A. Delgado, D. Goldschmidt, and B. Stellmacher. Groups and graphs: newresults and methods, volume 6 of DMV Seminar. Birkhauser Verlag, Basel,1985. With a preface by the authors and Bernd Fischer.

[6] Walter Feit and Graham Higman. The nonexistence of certain generalizedpolygons. J. Algebra, 1:114–131, 1964.

[7] J. Tits. Algebraic and abstract simple groups. Ann. of Math. (2),80:313–329, 1964.

[8] J. Tits. Classification of buildings of spherical type and Moufang polygons: asurvey. In Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973),Tomo I, pages 229–246. Atti dei Convegni Lincei, No. 17. Accad. Naz.Lincei, Rome, 1976.

[9] J. Tits. A local approach to buildings. In The geometric vein, pages 519–547.Springer, New York, 1981.

[10] Jacques Tits. Theoreme de Bruhat et sous-groupes paraboliques. C. R. Acad.Sci. Paris, 254:2910–2912, 1962.

[11] Jacques Tits. Buildings of spherical type and finite BN-pairs. Lecture Notesin Mathematics, Vol. 386. Springer-Verlag, Berlin, 1974.

[12] Jacques Tits and Richard M. Weiss. Moufang polygons. SpringerMonographs in Mathematics. Springer-Verlag, Berlin, 2002.


Documents

Buekenhout geometries and the realization of groups as … · 2011-09-21 · Buekenhout geometries and the realization of groups as buildings. Part I: Buildings Nate Iverson Department