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Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Examples of Groups: Coxeter Groups

Mike Davis

OSU

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

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

1 Geometric reflection groups Some history Properties

2 Abstract reflection groups Coxeter systems The cell complex Σ Variation for Artin groups

3 Euler characteristics

4 Exercises

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Some history Properties

Dihedral groups A dihedral gp is any gp which is generated by 2 involutions, call them s, t . It is determined up to isomorphism by the order m of st (m is an integer ≥ 2 or the symbol∞). Let Dm denote the dihedral gp corresponding to m.

For m 6=∞, Dm can be represented as the subgp of O(2) which is generated by reflections across lines L, L′, making an angle of π/m.

r L

r L´

π/m

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Some history Properties

- 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-sphere was a spherical triangle with angles πp ,

π q ,

π r , with p, q, r

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

1 q +

1 r > 1.

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

- Later work by Riemann and Schwarz showed there were discrete gps of isometries of E2 or H2 generated by reflections across the edges of triangles with angles integral submultiples of π. Poincaré and Klein: similarly for polygons in H2.

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Some history Properties

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

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

- Around 1890, Killing and E. Cartan classified complex semisimple Lie algebras in terms of their root systems. In 1925, Weyl showed the symmetry gp of such a root system was a finite reflection gp.

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

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Some history Properties

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. For Hn there are other possibilities, eg, a right-angled pentagon in H2 or a right-angled dodecahedron in H3.

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Some history Properties

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

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

is a presentation for W .

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Some history Properties

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. Label the edge by m(s, t) if this is > 3 or =∞ and leave it unlabeled if it is = 3. (W ,S) is irreducible if Γ is connected. (The components of Γ give the irreducible factors of W .) The next slide shows Coxeter’s classification of irreducible spherical and cocompact Euclidean reflection gps.

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Some history Properties

Spherical Diagrams Euclidean Diagrams

I (p) p2

H 53

H 5 4

F 4 4

ωA1 ∼

4 4B 2 ∼

6G 2

∼

4F 4

∼

E 6

∼

E 7

∼

E 8

∼

E 6

E 7

E 8

An

B 4n

Dn

nA ∼

4B n ∼

4 4Cn ∼

Dn ∼

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Coxeter systems The cell complex Σ Variation for Artin groups

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

Two answers Let Cay(W ,S) be the Cayley graph (ie, its vertex set is W and {w , v} spans an edge iff v = ws for some s ∈ S). First answer: for each s ∈ S, the fixed set of s separates Cay(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!

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Coxeter systems The cell complex Σ Variation for Artin groups

If either answer holds, (W ,S) is a Coxeter system and W a Coxeter 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 of spherical subsets of S. (N.B. ∅ ∈ S.)

Definition The 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.)

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Coxeter systems The cell complex Σ Variation for Artin groups

Example In previous lecture, given simplicial cx L, we defined a right-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.)

Question Can every Coxeter system be geometrized?

Answer Yes. In fact, there are two different answers.

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Coxeter systems The cell complex Σ Variation for Artin groups

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

Moreover, WC (= ⋃

w∈W wC) is a convex cone and if I denotes the interior of the cone, then I = U(W ,Cf ), where Cf is the complement of the nonspherical 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).

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

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Coxeter systems The cell complex Σ Variation for Artin groups

One consequence W is virtually torsion-free. (This is true for any finitely generated linear gp.)

Disadvantage

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

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

Mike Davis Examples of Groups: Coxeter Groups

Geometric reflection groups Abstract reflection groups

Euler characteristics Exercises

Coxeter systems The cell complex Σ Variation for Artin groups

There are two constructions of Σ.

Geometric realization of a poset

Given a poset P, let Flag(P) denote the abstract simplicial complex with vertex set P and with simplices all finite, totally ordered subsets of P. The geometric realiza