Simplicial Complexes. Definition. an Abstract Simplicial Complex K

Home , Simplicial complex

5. SOME CLASSESOF GROUPS Simplicial complexes. Deﬁnition. An abstract simplicial complex K consists of a set V and a set S of ﬁnite non-empty subsets of V, satisfying (1) if v ∈ V then {v} ∈ S; (2) if σ ∈ S and τ ⊆ σ, then τ ∈ S. (V =set of vertices, S =set of simplices.) If σ ∈ S has n+1 elts, n is the dimension of σ and σ is called an n-simplex.

Let W = ∏v∈V R, a real vector space. Identify v ∈V with the elt of W having 1 in the v-coordinate and 0 in all other coords. Give each finite dimensional linear subspace of W the Euclidean topology. Give W the weak topology: a subset is closed iff its intersection with every finite dimensional subspace is closed. For a simplex σ = {v0,v1,...,vn}, let |σ| be the convex hull of {v0,v1,...vn} in W. Let |K|0 = V, and [ |K|n = |K|n−1 ∪ {|σ| | σ is an n-simplex of K}, (n ≥ 1) S n and let |K| = n≥0 |K| , so |K| ⊆ W. Give |K| the relative topology. Then |K|0 ⊆ |K|1 ⊆ ... and this gives |K| the structure of a CW-complex. ⊆ is a partial order on S, satisfying: (i) any two elts σ, τ have a glb in S (σ ∩ τ); (ii) For any σ ∈ S, the set {τ ∈ S | τ ⊆ σ} is isomorphic to the poset of subsets of {1,2,...,r}, for some r ≥ 0. Conversely, given a poset (P,≤) satisfying (i) and (ii), get a simplicial complex by defining V to be the set of rank 1 elts (rank being the integer r in (ii)). For A ∈ P, define A0 = {v ∈ V | v ≤ A}, and let S = {A0 | A ∈ P}. This gives a simplicial complex, isomorphic as a poset to P, via A 7→ A0. Hyperbolic groups. Let (X,d) be a metric space. Definition. A geodesic in (X,d) is the image of an isometry α : [0,a] → X, where a ≥ 0. It joins α(0) to α(a); (X,d) is geodesic if ∃ a geodesic joining any two pts. A geodesic joining x, y is denoted [x,y], even though it may not be unique. A geodesic triangle ∆ = ∆(x,y,z) is the union of three geodesics [x,y] ∪ [y,z] ∪ [z,x]. A comparison triangle for ∆ in R2 is a triangle ∆ in R2 with verticesx ¯,y ¯, z¯ such that kx¯ − y¯k = d(x,y), etc. There are isometries [x¯,y¯] → [x,y] etc, which give a map p∆ : ∆ → ∆. 1 2

Deﬁnition. If u ∈ ∆, any pointu ¯ such that p∆(u¯) = u is called a comparison point for u.

Let cx, cy, cz be the interior pts of ∆, i.e. pts of contact of the sides with the incircle. x¯

cy cz

z¯ c x y¯

Let T∆ be the “tripod” obtained by isometrically identifying [x¯,cy] and [x¯,cz], etc. x ¯

z¯ c

y¯

Let q : ∆ → T∆ be the quotient map. There is an induced map f∆ : ∆ → T∆, with f∆ ◦ p∆ = q.

Definition . Let δ ≥ 0; ∆ is δ-thin if, for all u, v ∈ ∆, f∆(u) = f∆(v) ⇒ d(u,v) ≤ δ. Suppose (X,d) is a metric space. Choose a point z ∈ X and, for x, y ∈ X, define 1 (x · y)z = 2 (d(x,z) + d(y,z) − d(x,y) . (In the comparison triangle above, (x · y)z = kz¯− cxk = kz¯− cyk.) Definition (Gromov). Let (X,d) be a metric space, let v ∈ X, and let δ ∈ R with δ ≥ 0. Then (X,d) is called δ-hyperbolic with respect to v if ∀x,y,z ∈ X, x · y ≥ min{x · z,y · z} − δ, where x · y means (x · y)v, etc. Lemma 5.1. If (X,d) is δ-hyperbolic with respect to v, and t is any other point of X, then (X,d) is 2δ-hyperbolic with respect to t. 3

Definition. (X,d) is δ-hyperbolic if it is δ-hyperbolic with respect to all points of X, and is hyperbolic if it is δ-hyperbolic for some δ ≥ 0. Proposition 5.2. Let (X,d) be a geodesic metric space. (1) If (X,d) is δ-hyperbolic, then all geodesic triangles are 4δ-thin. (2) If all geodesic triangles are δ-thin, then (X,d) is δ-hyperbolic. √ 3+ 5 Examples. (1) In hyperbolic space, triangles are δ-thin for δ = loge 2 . (2) Complete simply connected Riemannian manifolds with sectional cur- vature κ satisfying κ ≤ c < 0 for some c are hyperbolic. (3) A geodesic 0-hyperbolic metric space is an R-tree. (4) A bounded metric space (X,d) is hyperbolic (δ = diam(X)). If Γ is a connected graph, the path metric d on V(Γ) is defined by: d(x,y) =length of a shortest path joining x and y. Eg Cayley graph: path metric on V(Γ(G,X)) = G is given by d(g,h) = L(g−1h), where L(g) =length of a shortest word in X±1 representing g. Definition. If X is a finite set of gens for a gp G, G is word hyperbolic if (Γ(G,X),d) is hyperbolic, where d =path metric on G. Can show this is independent of choice of generating set X. Examples. (1) Finite gps. (2) Finitely generated free gps (Cayley graph is a tree, and trees are 0- hyperbolic). (3) Surface gps with negative Euler characteristic. (4) Any gp ctg a free abelian subgp of rank 2 is NOT word hyperbolic.

Definition . Let (X,d) be a metric space; the Rips complex Pk(X) is the simplicial complex with vertex set X, where a finite subset is a simplex ⇔ its diameter ≤ k. Theorem 5.3. If (X,d) is a geodesic δ-hyperbolic metric space and k > 0, then |Pk(X)| is contractible for k ≥ 4δ. CAT(0) spaces. Definition. Let ∆ be a geodesic triangle in (X,d); ∆ is CAT(0) if, for all u, v ∈ ∆, and comparison ptsu ¯,v ¯ for u, v resp., d(u,v) ≤ ku¯ − v¯k. The space (X,d) is CAT(0) if it is geodesic and all triangles are CAT(0). Fact: CAT(0) spaces are contractible. A group is CAT(0) if it has a proper, cocompact action on a CAT(0) space. A gp action on metric space (X,d) is proper if ∀x ∈ X, ∃r > 0 such that {g ∈ G | gB(x,r) ∩ B(x,r) 6= /0} is finite. (B(x,r) = {y ∈ X | d(x,y) < r}.) Action is cocompact if ∃ a compact K ⊆ X such that X = GK. 4

Coxeter groups. Let I be a set. For i, j ∈ I let mi j ∈ {1,2,3,...} ∪ {∞}. Suppose mi j = m ji ≥ 2, for all i 6= j, and mii = 1 for all i. Deﬁnition. The corresponding Coxeter group is the gp W with presn

mi j hsi (i ∈ I) | (sis j) = 1 (i, j ∈ I)i.

(If mi j = ∞, the relation is omitted.)

2 m ji Note that for i = j, relation is si = 1. Also, (s jsi) = 1 is a consequence m of (sis j) i j = 1. Put S = {si | i ∈ I}; (W,S) is called a Coxeter system. 2 2 n Examples. (1) The dihedral gps Dn. hy,u | y = 1,u = 1,(yu) = 1i (2) The infinite dihedral gp C2 ∗C2 (3) The symmetric gps Sn (see exercises). Let V be an R-vector space with basis {ei | i ∈ I}. Define a symmetric bilin- ear form (−,−) on V by (ei,e j) = −cos(π/mi j). Let σi be the “reflection” 2 σi(v) = v − 2(v,ei)ei. Then σi = 1, and σi is clearly linear, so σi ∈ GL(V). Also, σi(ei) = −ei, so σi has order 2. Can show: σiσ j has order mi j. Fol- lows that the map si 7→ σi induces a homom W → GL(V). Hence si has order 2 in W and sis j has order mi j. Theorem 5.4. This homom is injective.

If T ⊆ S, let WT = hTi.

Theorem 5.5. WT is itself a Coxeter gp., and (WT ,T) is a Coxeter system with the mi j the same as in (W,S). Let (W,S) be a Coxeter system, with I finite. A special coset is a coset of W of the form wWT , where w ∈ W and T ⊆ S. Order the set of special cosets by: A ≤ B iff A ⊇ B. This poset satisfies the conditions (i) and (ii) above, so defines a simplicial cx Σ(W,S), called the Coxeter complex of (W,S). There is a way of associating a simplicial complex to any poset (P,≤); the flag complex of (P,≤), has vertex set P and n-simplices all chains (totally ordered subsets) with n + 1 elts. Let P = {wWT | w ∈ W, T ⊆ S, WT finite}, ordered by ⊆. Corr. simplicial cx is called the Davis complex of (W,S), denoted DW . W acts on DW , hence on |DW |, by left mult.

Theorem 5.6 (Davis). |DW | is contractible.

Theorem 5.7 (Moussong). In fact, |DW | has a natural CAT(0) metric. m Artin groups. Given Coxeter system (W,S), rewrite (sis j) i j = 1 (i 6= j) as

−1 −1 −1 sis jsi ... = s j si s j ... | {z } | {z } mi j mi j 5

2 and since si = 1, as sis jsi ... = s jsis j ... | {z } | {z } mi j mi j 2 Call this relation ri j. Now omit the relations si = 1 to obtain hS | ri j (i, j ∈ I,i 6= j)i.

Deﬁnition . The group A with this presentation is called the Artin group associated to (W,S).

There is a homom. A → W induced by si 7→ si. There are two analogues of the Davis complex for Artin gps, the Deligne complex and the Salvetti complex; A acts freely on the Salvetti cx. A well-studied special case: A is right-angled if all mi j = 2, i 6= j. Relations then say that certain pairs of generators commute. In particular, ﬁnitely gend free gps and free abelian gps are Artin gps. Braid groups. Let M be a connected manifold of dim. ≥ 2. Deﬁne Fn(M) = (x1,...,xn) ∈ M × ... × M | xi 6= x j for i 6= j .

Then Fn(M) is path connected. Symmetric gp Sn acts on Fn(M): σ(x1,...,xn) = (xσ(1),...,xσ(n)) (σ ∈ Sn). Let Cn(M) = Fn(M)/Sn (set of orbits, with quotient topology), and let p : Fn(M) → Cn(M) be the quotient map. Choose c0 ∈ Cn(M).

Deﬁnition (Fox). Bn(M) = π1(Cn(M),c0).

(Indep. of c0 as Cn(M) is path connected.) Let β ∈ π1(Cn(M),c0) be represented by f : [0,1] → Cn(M), so f (0) = f (1) = c0. Choosec ˜0 ∈ Fn(M) such that p(c˜0) = c0; p is a covering map, so ∃! f˜ : [0,1] → Fn(M) s.t. p ◦ f˜ = f and f˜(0) = c˜0.

Then f˜(t) = ( f˜1(t),..., f˜n(t)), where f˜i : [0,1] → M.

Lemma 5.8. (i) f˜i(t) 6= f˜j(t) for i 6= j and t ∈ [0,1]; ˜ ˜ (ii) fi(1) = fτ(i)(0) for 1 ≤ i ≤ n, where τ ∈ Sn.

Deﬁne αi : [0,1] → M × [0,1] by αi(t) = ( f˜i(t),t).

Deﬁnition. A := (α1,...,αn) is an n-string braid in M × [0,1].

Denote the permutation τ by τA. 2 2 Specialise to M = R ; Bn(R ) is denoted Bn, the braid group on n strings. Takec ˜0 = (x1,...,xn), where xi = (i,0). An n-string braid looks like: 6 (1,0,0)(2,0,0)(... n,0,0) r r rr r [Here, τA(1) = 3, τA(2) = 1, τA(3) = 4 etc.] ... (1,0,1)(r 2,0,r1)(rr rn,0,1) (The strings run between planes z = 0 and z = 1, where z-axis points down.) 2 Corresponding to homotopy of paths in Cn(R ) there is a notion of homotopy of braids. 2 Multiplication in π1(Cn(R ),c0) (c0 = p(c˜0)) corresponds to foll. operation. If A1, A2 are n-string braids, move A2 under A1 so the top plane of A2 is the bottom plane of A1 and the endpts match up. Then squeeze this system so it lies between z = 0 and z = 1. Deﬁne A1A2 to be the resulting braid. (Artin’s original defn of Bn.) e.g.

r r r r r r r r r r r r times equals

r r r r r r r r r r r r

Note: τA1A2 = τA1 τA2 . Inverse of a braid is its mirror image wrt a horizontal plane between z = 0 and z = 1. Let σi be the n-string braid: ... (i,0,0)(i + 1,0,0)... r r r r r r

...... r r r r rr Then homotopy classes of σ1,...,σn−1 generate Bn; to see this, make sure every crossing in a braid occurs at a different level.

Theorem 5.9 (Artin). Bn has presn

hσ1,...,σn−1 | σiσ j = σ jσi (|i − j| ≥ 2),σiσi+1σi = σi+1σiσi+1 (1 ≤ i ≤ n − 2)i.

Thus Bn is an Artin group. Corresponding Coxeter gp is Sn and the homo- morphism Bn → Sn sends σi to transposition (i,i + 1) (see exercises). This

is the permutation τσi , hence the homom. sends any elt. σ of Bn to τσ .