Examples of Groups: Coxeter Groups · Coxeter systems The cell complex Variation for Artin groups...

Geometric reflection groupsAbstract reflection groups

Euler characteristicsExercises

Examples of Groups:Coxeter Groups

Mike Davis

OSU

May 31, 2008http://www.math.ohio-state.edu/ mdavis/

Mike Davis Examples of Groups: Coxeter Groups



1 Geometric reflection groupsSome historyProperties

2 Abstract reflection groupsCoxeter systemsThe cell complex ΣVariation for Artin groups

3 Euler characteristics

4 Exercises




Some historyProperties

Dihedral groupsA dihedral gp is any gp which is generated by 2 involutions, callthem s, t . It is determined up to isomorphism by the order m ofst (m is an integer ≥ 2 or the symbol∞). Let Dm denote thedihedral gp corresponding to m.

For m 6=∞, Dm can berepresented as the subgp ofO(2) which is generated byreflections across lines L, L′,making an angle of π/m.

rL

rL´

π/m





- In 1852 Möbius determined the finite subgroups of O(3)generated by isometric reflections on the 2-sphere.

- The fundamental domain for such a group on the 2-spherewas a spherical triangle with angles πp ,

πq ,

πr , with p, q, r

integers ≥ 2.- Since the sum of the angles is > π, we have 1p +

1q +

1r > 1.

- For p ≥ q ≥ r , the only possibilities are: (p,2,2) for anyp ≥ 2 and (p,3,2) with p = 3, 4 or 5. The last three casesare the symmetry groups of the Platonic solids.

- Later work by Riemann and Schwarz showed there werediscrete gps of isometries of E2 or H2 generated byreflections across the edges of triangles with anglesintegral submultiples of π. Poincaré and Klein: similarly forpolygons in H2.





In 2nd half of the 19th century work began on finite reflectiongps on Sn, n > 2, generalizing Möbius’ results for n = 2. Itdeveloped along two lines.

- Around 1850, Schläfli classified regular polytopes in Rn+1,n > 2. The symmetry group of such a polytope was a finitegp generated by reflections and as in Möbius’ case, theprojection of a fundamental domain to Sn was a sphericalsimplex with dihedral angles integral submultiples of π.

- Around 1890, Killing and E. Cartan classified complexsemisimple Lie algebras in terms of their root systems. In1925, Weyl showed the symmetry gp of such a root systemwas a finite reflection gp.

- These two lines were united by Coxeter in the 1930’s. Heclassified discrete groups reflection gps on Sn or En.





Let K be a fundamental polytope for a geometric reflection gp.For Sn, K is a simplex. For En, K is a product of simplices. ForHn there are other possibilities, eg, a right-angled pentagon inH2 or a right-angled dodecahedron in H3.





Conversely, given a convex polytope K in Sn, En or Hn st alldihedral angles have form π/integer, there is a discrete gpW generated by isometric reflections across the codim 1faces of K .Let S be the set of reflections across the codim 1 faces ofK . For s, t ∈ S, let m(s, t) be the order of st . Then Sgenerates W , the faces corresponding to s and t intersectin a codim 2 face iff m(s, t) 6=∞, and for s 6= t , the dihedralangle along that face is π/m(s, t). Moreover,

〈S | (st)m(s,t), where (s, t) ∈ S × S〉

is a presentation for W .





Coxeter diagrams

Associated to (W ,S), there is a labeled graph Γ called its“Coxeter diagram.”

Vert(Γ) := S.

Connect distinct elements s, t by an edge iff m(s, t) 6= 2. Labelthe edge by m(s, t) if this is > 3 or =∞ and leave it unlabeled ifit is = 3. (W ,S) is irreducible if Γ is connected. (Thecomponents of Γ give the irreducible factors of W .)The next slide shows Coxeter’s classification of irreduciblespherical and cocompact Euclidean reflection gps.





Spherical Diagrams Euclidean Diagrams

I (p) p2

H 53

H 54

F 44

ωA1∼

4 4B 2∼

6G2

∼

4F4

∼

E6

∼

E7

∼

E8

∼

E6

E7

E8

An

B 4n

Dn

nA∼

4B n∼

4 4Cn∼

Dn∼




Coxeter systemsThe cell complex ΣVariation for Artin groups

QuestionGiven a gp W and a set S of involutions which generates it,when should (W ,S) be called an “abstract reflection gp”?

Two answersLet Cay(W ,S) be the Cayley graph (ie, its vertex set is Wand {w , v} spans an edge iff v = ws for some s ∈ S).First answer: for each s ∈ S, the fixed set of s separatesCay(W ,S).Second answer: W has a presentation of the form:

〈S | (st)m(s,t), where (s, t) ∈ S × S〉.

These two answers are equivalent!





If either answer holds, (W ,S) is a Coxeter system and W aCoxeter group.

Terminology

Given a subset T ⊂ S, put WT := 〈T 〉 (called a special subgp).T is spherical if WT is finite. Let S denote the poset ofspherical subsets of S. (N.B. ∅ ∈ S.)

DefinitionThe nerve of (W ,S) is the simplicial complex L (= L(W ,S))with Vert(L) := S and with T ⊂ S a simplex iff T is spherical.(So, S(L) = S.)





ExampleIn previous lecture, given simplicial cx L, we defined aright-angled Coxeter system (WL,S) (meaning that ∀s 6= t ,m(s, t) is 2 or∞). L is the nerve of (WL,S) iff L is a flag cx.(Pf: WT is finite ⇐⇒ any two vertices of T span an edge.)

QuestionCan every Coxeter system be geometrized?

AnswerYes. In fact, there are two different answers.





First answer: Tits showed ∃ a linear action of W on RSgenerated by (not necessarily orthogonal) reflections acrossthe faces of the standard simplicial cone C ⊂ RS giving afaithful linear representation W ↪→ GL(RS).

Moreover, WC (=⋃

w∈W wC) is a convex cone and if Idenotes the interior of the cone, thenI = U(W ,Cf ), where Cf is the complement of thenonspherical faces of C and U(W , ) is defined as before,ie, U(W ,Cf ) = (W × Cf )/ ∼, where (w , x) ∼ (w ′, x ′) ⇐⇒x = x ′ and w−1w ′ ∈WS(x).

AdvantagesI is contractible (since it is convex) and W acts properly (ie,with finite stabilizers) on it.





One consequenceW is virtually torsion-free. (This is true for any finitely generatedlinear gp.)

Disadvantage

The action is not cocompact (since Cf is not compact).

The second answer is to construct of contractible cell cx Σ onwhich W acts properly and cocompactly as a gp generated byreflections. When W is right-angled, Σ = P̃L, the complex fromthe previous lecture.





There are two constructions of Σ.

Geometric realization of a poset

Given a poset P, let Flag(P) denote the abstract simplicialcomplex with vertex set P and with simplices all finite, totallyordered subsets of P. The geometric realization of Flag(P) isdenoted |P|.

First construction of ΣRecall S is the poset of spherical subsets of S. Thefundamental chamber K is defined by K := |S|. (K is thecone on the barycentric subdivision of L.)Mirror structure: Ks := |S≥{s}|.Σ := U(W ,K ).





Second constructionLet WS denote the disjoint union of all spherical cosets(partially ordered by inclusion:

WS :=∐T∈S

W/WT and Σ := |WS|.

Coxeter polytopes

Suppose WT is finite reflection gp on RT . Choose a point x inthe interior of fundamental simplicial cone and let PT be convexhull of WT x . PT is determined up to isometry once we specifythe distance of x from each bounding hyperplane.





t

s

1

t

s

s

s

t

t

st

sts= tst

ts

There is a cell structure on Σ with {cells} = WS.

This follows from fact that poset of cells in PT is ∼= WTS≤T .The cells of Σ are defined as follows: the geometric realizationof subposet of cosets ≤ wWT is ∼= barycentric subdiv of PT .





Properties of this cell structure on Σ- W acts cellularly on Σ.- Σ has one W -orbit of cells for each spherical subset T ∈ S

and dim(cell) = Card(T ).- The 0-skeleton of Σ is W- The 1-skeleton of Σ is Cay(W ,S).- The 2-skeleton of Σ is the Cayley 2 complex of the

presentation.- If W is right-angled, then each Coxeter cell is a cube and

we have the cubical cell structure on P̃L discussed in thelast lecture.

- Moussong: the induced piecewise Euclidean metric on Σ isCAT(0).





More properties

- Σ is contractible. (This follows from the fact it is CAT(0).- The W -action is proper (by construction each isotropy

subgp is conjugate to some finite WT ).- Σ/W = K , which is compact (so the action is cocompact)- If W is finite, then Σ is a Coxeter polytope

If W is a geometric reflectiongp on Xn = En or Hn, then Kcan be identified with thefundamental polytope, Σ withXn and the cell structure isdual to the tessellation of Σby translates of K .





Groups, Graphs and Trees

An Introduction to the Geometry of Infinite Groups

John Meier

Jon McCammond

Σ is the cocompact core of the Tits cone I.





If W is finite, then a K (π,1) for the corresponding Artin gpcan be constructed as a quotient space of the Coxeterpolytope P for W . One identifies any two faces of P of thesame type. Specifically, if F and F ′ are faces of type Tcorresponding to the cosets 1WT and wWT , then identify Fwith F ′ via translation by u, where u is the shortestelement in the coset wWT . Result is a CW complex withone cell of dim = Card(T ) for each T ⊂ S.Similarly, for an arbitrary (W ,S), there is a CW cx X withone cell of dim Card(T ) for each T ∈ S. One gluestogether copies of XT , T ∈ S, where XT is the K (π,1) forthe Artin gp for WT . Conjecturally, X is a K (π,1) for thebig Artin gp.When W is right-angled, the XT are tori and this is thepreviously constructed K (π,1).




The Euler characteristic of a group

- Suppose G is a gp and Γ ⊂ G a torsion-free subgp of finiteindex which has a finite K (Γ,1) complex.

-χ(Γ) := χ(K (Γ,1)) and χ(G) :=

χ(Γ)

[G : Γ]∈ Q.

- If G acts on a contractible cell cx with finite cell stabilizers,then

χ(G) =∑

orbits ofcells σ

(−1)dimσ

|Gσ|




Applying this to action of W on Σ (cellulated by Coxetercells):

χ(W ) =∑T∈S

(−1)T

|WT |.

When W is right-angled,

χ(W ) =∑T∈S

(−12)|T |

.

ExampleIf W is the right-angled pentagon gp, then

χ(W ) = 1− # edges2

+# vertices

4= 1− 5

2+

54

= −14.




The Euler Characteristic Conjecture (Hopf-Chern-Thurston)

Suppose M2n is a closed aspherical mfld. Then

(−1)nχ(M2n) ≥ 0.

Suppose L = L(W ,S) is a triangulation of S2n−1, then Σ iscontractible 2n-mfld and for any torsion-free, finite index subgpΓ ⊂W , χ(Σ/Γ) = [W : Γ]χ(W ) and

χ(W ) = 1 +∑σ⊂L

(−1)dimσ+1

|Wσ|.

So, if κ(L) is the RHS, the conjecture is (−1)nκ(L) ≥ 0.




For any simplicial cx L, put

κ(L) :=∑

T∈S(L)

(−12)|T |

.

The Charney-Davis Conjecture Conjecture

If L is a triangulation of S2n−1 as a flag complex, then

(−1)nκ(L) ≥ 0.

True for

Flag triangulations of S1 and S3.The barycentric subdiv of ∂(convex polytope).




Exercises1 Suppose ∆n ⊂ Sn ⊂ Rn+1 is a spherical simplex with its{codim 1 faces} indexed by S and all dihedral angles ofform π/m(s, t). Let {us}s∈S be the set of outward-pointingunit normals. Show that the angle beween us and ut isπ − π/m(s, t) and hence, us · ut = − cosπ/m(s, t). Showthat such a ∆n exists iff its (S × S) cosine matrix,(− cosπ/m(s, t)) is positive definite.

2 Suppose ∆n ⊂ En has dihedral angles of form π/m(s, t).Show ∆n exists iff its cosine matrix is positive semidefinitewith a 1-dimensional null space and each principal n × nsubmatrix is posiive definite.

3 Formulate the corresponding condition and do the sameproblem for hyperbolic simplices ∆n ⊂ Hn.




More exercises4 Using the Coxeter diagrams of spherical and Euclidean

Coxeter groups, classify the possible diagrams forhyperbolic simplices. (The answer is provided on the nextpage.)

5 What is the Euler characteristic of a (p,q, r) triangle group(i.e., the angles are π/p, π/q and π/r )?

6 Answer the same question for the hyperbolic simplicialCoxeter groups of rank 5 (i.e., those listed under n = 4 onthe next page).




Hyperbolic Simplicial Diagrams

n = 2

with (p-1 + q-1 + r-1) < 1p q

r

n = 35

5 4

5 5

5

4

4 5 4

5 4

5 5

n = 4

5

5

5

5

4

5 4




References

N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6,Springer, New York and Berlin, 2002.

R. Charney and M.W. Davis, The Euler characteristic of anonpositively curved, piecewise Euclidean manifold, Pac. J.Math. 171 (1995), 117-137.

, Finite K (π,1) s for Artin groups, Annals of MathStudies, vol 138, pp. 110–124, Princeton Univ Press, 1995.

M.W. Davis, Exotic aspherical manifolds, School onHigh-Dimensional Topology, ICTP, Trieste, 2002.

, The Geometry and Topology of Coxeter Groups,London Math. Soc. Monograph Series, vol. 32, PrincetonUniv. Press, 2007.


Geometric reflection groupsSome historyProperties

Abstract reflection groupsCoxeter systemsThe cell complex Variation for Artin groups


Examples of Groups: Coxeter Groups · Coxeter systems The cell complex Variation for Artin groups...

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