Examples of Groups: Coxeter Groups

Transcription

1 Examples of Groups: Coxeter Groups OSU May 31, mdavis/

2 1 Geometric reflection groups Some history Properties 2 Coxeter systems The cell complex Σ Variation for Artin groups 3 4

3 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 D m denote the dihedral gp corresponding to m. For m, D m can be represented as the subgp of O(2) which is generated by reflections across lines L, L, making an angle of π/m. π/m r L r L

4 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 π p, π q, π r, with p, q, r integers 2. - Since the sum of the angles is > π, we have 1 p + 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 E 2 or H 2 generated by reflections across the edges of triangles with angles integral submultiples of π. Poincaré and Klein: similarly for polygons in H 2.

5 Some history Properties In 2 nd half of the 19 th century work began on finite reflection gps on S n, n > 2, generalizing Möbius results for n = 2. It developed along two lines. - Around 1850, Schläfli classified regular polytopes in R n+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 S n 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 S n or E n.

6 Some history Properties Let K be a fundamental polytope for a geometric reflection gp. For S n, K is a simplex. For E n, K is a product of simplices. For H n there are other possibilities, eg, a right-angled pentagon in H 2 or a right-angled dodecahedron in H 3.

7 Some history Properties Conversely, given a convex polytope K in S n, E n or H n 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), and for s 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.

8 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) 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.

9 Some history Properties Spherical Diagrams Euclidean Diagrams A n A n B n 4 B n 4 D n C n 4 4 I 2(p) p D n H3 H A 1 B 2 G 2 ω F 4 4 F 4 4 E 6 E 6 E 7 E 7 E 8 E 8

10 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!

11 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 W T := T (called a special subgp). T is spherical if W T 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.)

12 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 (W L, S) (meaning that s t, m(s, t) is 2 or ). L is the nerve of (W L, S) iff L is a flag cx. (Pf: W T 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.

13 Coxeter systems The cell complex Σ Variation for Artin groups First answer: Tits showed a linear action of W on R S generated by (not necessarily orthogonal) reflections across the faces of the standard simplicial cone C R S giving a faithful linear representation W GL(R S ). Moreover, WC (= w W wc) is a convex cone and if I denotes the interior of the cone, then I = U(W, C f ), where C f is the complement of the nonspherical faces of C and U(W, ) is defined as before, ie, U(W, C f ) = (W C f )/, where (w, x) (w, x ) x = x and w 1 w W S(x). Advantages I is contractible (since it is convex) and W acts properly (ie, with finite stabilizers) on it.

14 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 C f 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 = P L, the complex from the previous lecture.

15 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 realization of Flag(P) is denoted P. First construction of Σ Recall S is the poset of spherical subsets of S. The fundamental chamber K is defined by K := S. (K is the cone on the barycentric subdivision of L.) Mirror structure: K s := S {s}. Σ := U(W, K ).

16 Coxeter systems The cell complex Σ Variation for Artin groups Second construction Let W S denote the disjoint union of all spherical cosets (partially ordered by inclusion: W S := T S W /W T and Σ := W S. Coxeter polytopes Suppose W T is finite reflection gp on R T. Choose a point x in the interior of fundamental simplicial cone and let P T be convex hull of W T x. P T is determined up to isometry once we specify the distance of x from each bounding hyperplane.

17 Coxeter systems The cell complex Σ Variation for Artin groups s st t s 1 s t sts= tst t s t ts There is a cell structure on Σ with {cells} = W S. This follows from fact that poset of cells in P T is = W T S T. The cells of Σ are defined as follows: the geometric realization of subposet of cosets ww T is = barycentric subdiv of P T.

18 Coxeter systems The cell complex Σ Variation for Artin groups 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 P L discussed in the last lecture. - Moussong: the induced piecewise Euclidean metric on Σ is CAT(0).

19 Coxeter systems The cell complex Σ Variation for Artin groups 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 W T ). - 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 reflection gp on X n = E n or H n, then K can be identified with the fundamental polytope, Σ with X n and the cell structure is dual to the tessellation of Σ by translates of K.

20 Coxeter systems The cell complex Σ Variation for Artin groups 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.

21 Coxeter systems The cell complex Σ Variation for Artin groups If W is finite, then a K (π, 1) for the corresponding Artin gp can be constructed as a quotient space of the Coxeter polytope P for W. One identifies any two faces of P of the same type. Specifically, if F and F are faces of type T corresponding to the cosets 1W T and ww T, then identify F with F via translation by u, where u is the shortest element in the coset ww T. Result is a CW complex with one cell of dim = Card(T ) for each T S. Similarly, for an arbitrary (W, S), there is a CW cx X with one cell of dim Card(T ) for each T S. One glues together copies of X T, T S, where X T is the K (π, 1) for the Artin gp for W T. Conjecturally, X is a K (π, 1) for the big Artin gp. When W is right-angled, the X T are tori and this is the previously constructed K (π, 1).

22 The Euler characteristic of a group - Suppose G is a gp and Γ G a torsion-free subgp of finite index 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) = ( 1) dim σ G σ orbits of cells σ

23 Applying this to action of W on Σ (cellulated by Coxeter cells): χ(w ) = ( 1) T W T. T S When W is right-angled, χ(w ) = T S ( 1 2) T. Example If W is the right-angled pentagon gp, then χ(w ) = 1 # edges 2 + # vertices 4 = = 1 4.

24 The Euler Characteristic Conjecture (Hopf-Chern-Thurston) Suppose M 2n is a closed aspherical mfld. Then ( 1) n χ(m 2n ) 0. Suppose L = L(W, S) is a triangulation of S 2n 1, then Σ is contractible 2n-mfld and for any torsion-free, finite index subgp Γ W, χ(σ/γ) = [W : Γ]χ(W ) and χ(w ) = 1 + σ L ( 1) dim σ+1. W σ So, if κ(l) is the RHS, the conjecture is ( 1) n κ(l) 0.

25 For any simplicial cx L, put κ(l) := T S(L) ( 1 2) T. The Charney-Davis Conjecture Conjecture If L is a triangulation of S 2n 1 as a flag complex, then ( 1) n κ(l) 0. True for Flag triangulations of S 1 and S 3. The barycentric subdiv of (convex polytope).

26 1 Suppose n S n R n+1 is a spherical simplex with its {codim 1 faces} indexed by S and all dihedral angles of form π/m(s, t). Let {u s } s S be the set of outward-pointing unit normals. Show that the angle beween u s and u t is π π/m(s, t) and hence, u s u t = cos π/m(s, t). Show that such a n exists iff its (S S) cosine matrix, ( cos π/m(s, t)) is positive definite. 2 Suppose n E n has dihedral angles of form π/m(s, t). Show n exists iff its cosine matrix is positive semidefinite with a 1-dimensional null space and each principal n n submatrix is posiive definite. 3 Formulate the corresponding condition and do the same problem for hyperbolic simplices n H n.

27 More exercises 4 Using the Coxeter diagrams of spherical and Euclidean Coxeter groups, classify the possible diagrams for hyperbolic simplices. (The answer is provided on the next page.) 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 simplicial Coxeter groups of rank 5 (i.e., those listed under n = 4 on the next page).

28 Hyperbolic Simplicial Diagrams n = 2 p r q with (p -1 + q -1 + r -1 ) < 1 n = n =

29 References Geometric reflection groups N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springer, New York and Berlin, R. Charney and M.W. Davis, The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold, Pac. J. Math. 171 (1995), , Finite K (π, 1) s for Artin groups, Annals of Math Studies, vol 138, pp , Princeton Univ Press, M.W. Davis, Exotic aspherical manifolds, School on High-Dimensional Topology, ICTP, Trieste, 2002., The Geometry and Topology of Coxeter Groups, London Math. Soc. Monograph Series, vol. 32, Princeton Univ. Press, 2007.