. COVERS OF D-TYPE ARTIN GROUPS. 1. Introduction

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1 . COVERS OF D-TYPE ARTIN GROUPS MEIRAV AMRAM 1, ROBERT SHWARTZ 1 AND MINA TEICHER Abstract. We stdy certain qotients of Generalized Artin grops which hae natral map onto D-type Artin grops. In particlar the Generalized Artin grop A(T ) defined by a signed graph T. Then we find a certain qotient G(T ) according to the graph T, which hae a natral map onto A(D n ) too. We proe that G(T ) is isomorphic to a semidirect prodct of a grop K, with the Artin grop A(D n ), where K depends only on the nmber of cycles and on the nmber of edges of the graph T. 1. Introdction Coxeter and Artin grop are sed in many domains in mathematics, sch as dealing with reflections, symmetries, Lie Algebras classifications of finite simple grops, comptations in algebraic geometry and in many other aspects. The strctres of Coxeter and Artin grops are ery interesting since these grops are defined in a ery easy way, in terms of generators and relations. These grops can be described easily by diagrams which are called Dynkin diagrams, and the grops has interesting properties in terms of grop theory, like the cancelation property (see [4], [7]). The goal of this paper is to find a strctre for certain qotients of Artin grops which has natral map onto the finite type simply laced Artin grop A(D n ), sch that throgh that strctre the word problem is solble in that certain qotient. The terminate goal of the series of papers which we describe, is to find general strctres for certain qotients of Coxeter and Artin grops, in a way that it will be easy to sole the word problem throgh natral maps onto Coxeter or Artin grops. 1 Partially spported by the Emmy Noether Research Institte for Mathematics (center of the Minera Fondation of Germany), the Excellency Center Grop Theoretic Methods in the Stdy of Algebraic Varieties of the Israel Science Fondation, and EAGER (EU network, HPRN-CT ) Date: April 15, Key words and phrases. Artin Grops, Signed Graphs. MSC Classification: 20E22, 20E34, 20F05, 20F36, 20F55. 1

2 2 AMRAM, SHWARTZ, TEICHER The motiation of this paper comes from algebraic geometry. We take a projectie srface X, with a generic map of degree n to CP 2, and S is its branch cre in CP 2. There is a natral map from the fndamental grop π 1 (CP 2 S) to Artin s braid grop Br n. The kernel of π 1 (CP 2 S) Br n is a qotient of a map which was described in [3] in details. There is generalization of some important properties of Artin s Braid grops to D-type, and other finite type Artin grops in [1]. Hence it is interesting to look on the strctre of kernels of maps onto finite type Artin grops too as generalization of the kernel π 1 (CP 2 S) Br n. This paper is the forth in a series of papers which stdy certain qotients of Coxeter or Artin grops, which are defined by a graph T. In this section we shall describe the main reslts of those papers and the configration of the crrent paper. The first paper in the series, Coxeter coers of the symmetric grop [8], describes the strctre of a certain qotient of a Coxeter grop C(T ), which is defined by a graph T and has a natral map onto the symmetric grop S n. The main theorem of [8] is that there exists a certain qotient C Y (T ) of C(T ). This C Y (T ) is isomorphic to A t,n S n, where A t,n is a well defined grop whose only inariants are t (the nmber of cycles of T ) and n (the nmber of ertices of T ). Since the word problem is solable in A t,n, it is also solable in C Y (T ). The second paper Coxeter coers of the classical Coxeter grops [2] generalizes the reslts of [8]. This paper deals with a wider class of Coxeter grops that we can map onto B n or D n (the classical Coxeter grops). In [2], the graph T is generalized to a signed graph in which eery edge is labeled either by 1 or by, and which may inclde loops. Similar signed graphs were introdced in [5], Signed graphs, root lattices and Coxeter grops. The main theorem of [2] is as follows: there is a ceratin qotient C Y (T ) of C(T ), which is isomorphic to A t,n D n or A t,n B n, depending wether T contains loops, or not. The third paper [3], Artin coers of the braid grop, generalizes Coxeter coers to Artin coers. The graph T defines an Artin grop A(T ) (which means that the generators are not necessarily inoltions). The main theorem of [3] is that there exists a qotient G(T ) of A(T ) that is isomorphic to K t,n Br n, where t is the nmber of cycles in T and n is the nmber of ertices of T. Since the word problem is solable in K t,n,, it is solable in G(T ) as well.

3 COVERS OF D-TYPE ARTIN GROUPS 3 Or paper contains a combined generalization of [2] and [3] to Artin grops that hae natral maps onto the simply laced finite type Artin grop A(D n ). According to [6], A(D n ) F n Br n and A(B n ) F n Br n. In Coxeter grops, it is known that D n B n, bt A(D n ) is not embeddable into A(B n ). Therefore it is impossible to combine Artin coers A(B n ) and A(D n ) as done in [2]. Hence, in this paper, we deal only with coers of A(D n ). This paper is in the spirit of the paper Artin coers of the braid grops ([3]) and we assme the definitions, theorems and reslts gien there, althogh this paper is a generalization of [3] and deals with a wider class of grops, e.g., grops whose Dynkin diagrams contain sbgraphs of the form (e.g., A(D n )). The paper is diided as follows. In Section 2, we define the grop that we obtain from the signed graph T. Then we define A Y (T ) as a qotient of A(T ), where the configration of the sbgraphs from which arise the relations of A Y (T ) are the same as in [3]. We recall the basic properties of A Y (T ). In Section 3 we consider a graph T that contains only one cycle and only one anti-cycle. Then, following [3], we define G(T ) as a qotient of A Y (T ). Using the properties of A Y (T ), we proe that G(T ) K A(D n ), where the strctre of K is ery similar to the strctre defined in [3][Section 6]. In Section 4 we proe that eery signed graph T is eqialent to a signed graph T (m) of the form Then we proe that m 2 _ σ 1 σ σ Figre 1 σ n G(T ) K m,n A(D n ), where n is the nmber of ertices of T and m 1 is the nmber of cycles and anti-cycles in T (T mst inclde at least one anti-cycle). Or reslt is a generalization of the main theorem in [3].

4 4 AMRAM, SHWARTZ, TEICHER Althogh the strctre of G(T ) is ery similar to the strctre of G(T ) in [3], sing signed graphs and mapping onto A(D n ) instead of mapping onto Br n allows s to deal with a mch wider class of simply laced Artin grops than those in [3]. 2. The qotient A Y (T ) Definition 1. We call a weighted graph T a signed graph if eery edge of T contained in a cycle is signed by 1 or by. For example, Figre 2 Here the edges that are not contained in any cycle are not signed, and we may assme that the signs of all sch edges are 1. We denote by s(e) the sign of the edge e. Let A(T ) be the generalized Artin grop that corresponds to the graph T (i.e., A(T ) is generated by the edges of T ). In [3], each edge in a graph is considered as a positie signed edge. In this paper, there is a generalization to signed graphs, where each edge can be signed by or -. Therefore, the relations in A(T ) in this paper are: 1 2 < 1, 2 >= 1 if 1 and 2 meet in a ertex 1 2 [ 1, 2 ] = 1 if 1 and 2 are disjoint 2 2 There is no relation between 1 and 2 if 1 and connect the same two ertices and s( 1 ) = s( 2 )

5 4 COVERS OF D-TYPE ARTIN GROUPS 5 In the case of a cycle with odd nmber of negatie signs, we hae an additional relation: [ 1... n2 n n2... 1, n ] = 1. We call sch a cycle an anti-cycle, (see [2]). 2 Note that an anti-cycle of length two has the form, where 1 and 2 are 1 two edges that connect the same two ertices bt are signed differently. Hence, the indced relation is [ 1, 2 ] = 1. Remark 2. The graph grop A(D n ). _ represents the finite type Artin Definition 3. Let T be a planar graph, A Y (T ) is the qotient of A(T ) by the following relations (similar to the relations in [3] with an additional case): (1) [w w, ] = 1 if,, w as in w (2) < w w, >= 1 if,, w as in w w (3) [w w, x] = 1 if,, w, x as in x w x w x w w x (4) < w w, x >= 1 if,, w, x as in Now we define irtal edges. w x w x

6 6 AMRAM, SHWARTZ, TEICHER Definition 4. Let x and y be paths in a signed graph T, sch that x and y intersect in no more than one point. Then we define a path x y, called a irtal edge, as: (1) x y = y if x and y do not intersect; (2) if x and y intersect in one ertex. x. y x Figre 3 y The sign of x y is 1 in the case when x and y hae the same sign (both 1 or both ). The sign of x y is if x and y are signed differently (one of them 1 and the other ). We note that the definition of x y is similar to Definition 3.5 in [3], bt we hae also introdced signs for irtal edges. Definition 5. Let T be a planar graph. We define ˆT as a graph with the same ertices as those of T. The edges of ˆT are either actal or irtal, and for eery ordered pair of edges x, y T, we hae the irtal edge x y in ˆT with the corresponding sign. (See [3], Definition 3.7). Theorem 6. Let T be a graph. There is a well defined map A( ˆT ) A Y (T ), that maps each actal edge x ˆT to x T and each irtal edge x y to x yx. Proof. The proof is the same as the proof of Theorem 3.8 of [3]. Remark 7. If T is a simple graph, or if any two edges that connect the same two ertices are signed differently, then A(T ) A Y (T ) Br n A(S n ) if there are no anti-cycles (i.e. T is necessarily a simple graph), see [3][Remark 3.3]. And A(T ) A Y (T ) A(D n ) if there exists at least one anti-cycle in T.

7 COVERS OF D-TYPE ARTIN GROUPS 7 3. Graphs with a single cycle Let T be a planar graph and let x and y be two edges in T that hae a common ertex q. p x x q Figre 4 Let x be a path connecting p and r; then (as in Section 8 of [3]) we can define a graph T where we replace x by x and the sign of x is the prodct of the signs of x and y, i.e., if x and y hae the same sign then the sign of x is 1, and if x and y hae the different signs then the sign of x is. The map x x yx defines an isomorphism between A Y (T ) and A Y (T ), as described in Section 8 of [3]. Now we classify the graphs T that inclde a single cycle connected by a path to an anticycle or graphs that inclde only two anti-cycles connected by a path. Theorem 8. Let T be an anti-cycle of length n. Then T is eqialent to a graph T that contains an anti-cycle of length two connected by a path. y Figre 5 r Proof. Let T be an anti-cycle with edges: σ 1, σ 2,... σ n sch that n i=1 s(σ i) =. Let σ be σ 2 σ 3... σ nσ n σ n... σ 2 which we get from the trianglation property. Then s(σ) = n i=2 s(σ i) and since n i=1 s(σ i) =, s(σ) = s(σ 1 ). Hence we get T which incldes the anti-cycle of length two σ 1 and σ, where they are connected by a path. Since the only edges of T that are inoled in a cycle or in an anti-cycle are σ 1 and σ, we can omit the signs from σ 2,..., σ n. Corollary 9. All the anti-cycles of length n are eqialent, and A Y (T ) A(D n ).

8 8 AMRAM, SHWARTZ, TEICHER Proof. By Theorem 8, eery anti-cycle of length n is eqialent to an anti-cycle of length two connected to a path. All the anti-cycles are eqialent to the same graph T. Since A Y (T ) A(D n ) for eery eqialent graph T, A Y (T ) A Y (T ) A(D n ). Theorem 10. Let T be a graph consisting of two anti-cycles connected to a path. Then T is eqialent to T, where T is a cycle with an additional negatiely signed edge connected to two adjacent ertices of the cycle. Figre 6 σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 σ 7 σ 1 Proof. By Theorem 8, T is eqialent to T, where T consists of two anti-cycles of length two connected by a path. Let σ 1, σ and σ n, σ n σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 σ 7 σ 1 σ 7 be the edges of the two anti-cycles in T and σ 2,..., σ n be the edges of the path connecting them. By the trianglation property, there is a graph T that is eqialent to T, sch that σ n is replaced by = σ σ 2... σnσ n σ n... σ 2 σ. Theorem 11. Let T be a graph consisting of an anti-cycle C connected to a cycle by a path. Then T is eqialent to T, where T is a cycle with an additional negatiely signed edge, which is connected to two adjacent ertices of the cycle. Figre 7 σ 1 σ 2 σ3 σ 4 σ 5 σ 6 σ 7 σ 1 Proof. By the same proof as in Lemma 9.3 in [3], we hae Figre 8. Figre 8

9 COVERS OF D-TYPE ARTIN GROUPS 9 Figre 9 Then sing Theorem 10, the anti-cycle C is eqialent to an anti-cycle of length two connected by a path. Then, combining this with the cycle, we get T as in Figre 9. Corollary 12. Eery graph T with a path connecting either a cycle with an anti-cycle or two anti-cycles is eqialent to T of a form:. Proposition 13. The length of the cycle in T is one less than the nmber of edges in T. Proof. T contains a cycle and an additional edge signed by. Hence, the length of the cycle in T is one less than the length of the nmber of edges in T. Since we get T from T by trianglation, and trianglation preseres the nmber of the edges, the proposition follows. Now we define the grop G(T ) for a graph T, where T consists of either two anti-cycles connected by a path or an anti-cycle connected to a cycle by a path. By Theorems 10 and 11, T is eqialent to T, where T is a cycle and an additional edge signed by that is connected to two adjacent ertices in the cycle in T, as in Figre 10. Hence, A Y (T ) A Y (T ). α σ 1 σ 2 σ 3 σ σ 4 5 σ n 1 σ n n Figre 10 The edges of T are labelled by σ i, 1 i n 1, σ and, the generators of A(T ). We denote by α the element α = L(σ 1,..., σ n ) = σ 1 σ 2... σ n2σ n σ n2... σ 2 σ 1.

10 10 AMRAM, SHWARTZ, TEICHER Similarly to [3][Section 6], we define the path α as a directed path with starting point at the ertex 1 and ending point at the ertex n. We notice that the path σ i is a path with starting point at i and ending point at i1, where σ i is the path with opposite direction to σ i, which means that starting point of σ i is i1, and the ending point of σ i is i. Now we define signed paths for the action of A(D n ) in T. σ is the negatiely signed path with starting point 1 and ending point 2, and similarly, σ is the negatiely signed path with starting point 2 and ending point 1. By Definition 3, A Y (T ) is the grop generated by σ 1,... σ n, σ and with the relations [σ i, σ j ] = 1 for i j > 1 [, σ i ] = 1 for 2 i n 2 [σ, σ j ] = 1 for j 2 < σ i, σ i1 >= 1 for 1 i n 2 < σ, σ 2 >= 1 < σ 1, >= 1 < σ, >= 1 < σ n, >= 1 [σ σ, σ1 σ 2 σ 1 ] = 1 [σ 1 σ 1, σ σ 2σ] = 1. Note that A Y (T ) = A(D n ),, where A(D n ) is the parabolic sbgrop of A Y (T ) generated by σ i, 1 i n 1 and σ. As in [3][Section 6], we define x α to be x α = α, and x α = α x α α = α. The stabilizer of α in the action of A(D n ) is generated by σ 2,..., σ n2, σ n σ 1 ασ 1 σ n, α 2 as in [3][Remark 6.2], and moreoer, σ n σ elements that inole σ). ασσ n, σσ 1 ασ 1 σ and σ 1 σ ασσ In the spirit of [3][Section 6], we define G(T ) = A(D n ), x α with the relations < σ 1, x α α >= 1, < σ, x α α >= 1, < σ n, x α α >= 1, [x α, σ i ] = 1, for 2 i n 2 which hold also in A(T ), and fie additional relations concerning the stabilizer of α : [x α, α 2 ] = 1, 1 (three

11 [x α, σ n σ 1 ασ 1 σ n] = 1, [x α, σσ 1 ασ 1 σ ] = 1, [x α, σ 1 σ [x α, σ n σ ασσ 1 ] = 1, ασσ n] = 1. COVERS OF D-TYPE ARTIN GROUPS 11 Since σ i, 1 i n 1 and σ are conjgates to α, we can easily define x σi and x σ1 as they are defined in [3][Section 8]. Proposition 14. [, σ n σ 1 ασ 1 σ n] = 1, [, σσ 1 ασ 1 σ ] = 1, [, σ 1 σ [, σ n σ ασσ 1 ] = 1, ασσ n] = 1. Proof. Since x α = α, the relation [x α, σ n σ 1 ασ 1 σ n] = 1 implies [α, σ n σ 1 ασ 1 σ n] = 1. Now, [ασ n α, ασ 1 α ] = 1, which is eqialent to [σ nασ n, σ 1 ασ 1 ] = 1 which implies [α, σ n σ 1 ασ 1 σ n], Then we get [, σ n σ 1 ασ 1 σ n] = 1. The proof for the other three commtation relations is ery similar. Proposition 15. x σi = (σ i) σ i where (σ i) = L(σ i1,..., σ n,, σ 1,..., σ i ) and (σ) = L(σ 2,..., σ n, σ σ 1σ1 σ) Proof. Since α = L(σ 1,..., σ n ) = σ n... σ 2 σ 1 σ 2... σ n, σ 1 = σ 2... σ nασ n... σ 2. It ths follows that σ 1 is the conjgate of α by the element σ n... σ 2. Hence (σ 1) = σ 2... σ nσ n... σ 2, which is by definition L(σ 2,..., σ n, ). Now assme, by indction on i, that (σ i) = L(σ i1,..., σ n,, σ 1,..., σ i ). Since σ i1 = σ i σ i1 σ i σ i1 σ i, we get that (σ i1) = σ i σ i1 L(σ i1,..., σ n,, σ 1,..., σ i )σ = σ i σ i1 σ i1 σ i2... σ n σ 1... σ i2 σ iσ i2... σ 1 σ n... σ i2 σ i1 σ i1 σ i = = σ i2... σ n σ 1... σ i2 σ iσ i σ i σ i2... σ 1 σ n... σ i2 = = σ i2... σ n σ 1... σ i2 σ i σ iσ i σ i2... σ 1 σ n... σ i2 = = L(σ i2,..., σ n,, σ 1,..., σ i ). Now we proe the expression for (σ). σ = σ 2 σ σ 2σσ 2, hence (σ) = σ = σ 2 σ σ 2 σ (σ 2) σσ 2 = σ 2 σ 3... σn σ 1 σ n... σ 3 σσ 2 = σ L(σ 3,..., σ n,, σ 1 )σσ 2 = 2 σ 3... σ nσ i1 σ i = σ 1 σσ n... σ 3 σ 2 =

12 12 AMRAM, SHWARTZ, TEICHER = σ 2 σ 3... σ n(σ σ 1σ1 σ)σ n... σ 3 σ 2 = L(σ 2, σ 3,..., σ n, σ σ 1σ1 σ). Now we proe that x σ i and x σ satisfy similar relations as in [3][Section 6]. Proposition 16. (1) [x σ i, σ j ] = 1 for i j > 1 (2) [x σ i, x σ j ] = 1 for i j > 1 (3) [x, σ σ j] = 1 for j 2 (4) [x σ j, σ] = 1 for j 2 (5) [x, x σ σ j ] = 1 for j 2 Proof. The proof of (1) and (2) is in [3]. We now proe(3). First we proe [x σ, σ j] = 1 for j 4. Since x σ [σ, σ j] = 1, it is enogh to proe that [ (σ), σ j ] = 1. [ (σ), σ j ] = [L(σ 2, σ 3,..., σ n, σ σ 1σ1 σ), σ j ] = = [σ 2 σ 3... σ = [σ j σ j = [σ j... σ = [σ j... σ... σ = [σ j1... σ nσ nσ nσ nσ nσ since 3 j 1 n 2. σ 1σ1 σσ n... σ 3 σ 2, σ j ] = σ 1σ1 σσ n... σ j σ j, σ j ] = σ 1σ1 σσ n... σ j, σ j σ j σ j ] = σ 1σ1 σσ n... σ j, σ j σ j σ j ] = σ 1σ 1 σσ n... σ j1, σ j ] = 1, = ( σ ) σ, and Now we proe that [ (σ), σ 1 ] = 1. [ (σ), σ 1 ] = [σ 2 σ 3... σ nσ σ 1σ1 σσ n... σ 3 σ 2, σ 1 ] = = [σ σ 1σ1 σ, σ n... σ 3 σ 2 σ 1 σ2 σ3... σn] = = [σ σ 1σ1 σ, α] = [, σσ 1 ασ 1 σ ] = 1 by Proposition 14. Now we proe that [ (σ), σ 3 ] = 1. [ (σ), σ 3 ] = [σ 2 σ 3... σ = [σ 3... σ = [σ 3... σ = [σ 4... σ nσ nσ nσ nσ = [σ σ 1σ1 σ, σ 2 ] = [σ σ 1σ1 σσ n... σ 3 σ 2, σ 3 ] = σ 1σ1 σσ n... σ 3, σ 2 σ 3 σ2 ] = σ 1σ1 σσ n... σ 3, σ3 σ 2 σ 3 ] = σ 1σ1 σσ n... σ 4, σ 2 ] = σ, σ 1 σ 2 σ 1 ] = 1 according to Relation (3) in Definition 3 (see Figre 11). Hence, case (3) is proed.

13 COVERS OF D-TYPE ARTIN GROUPS 13 σ 1 σ 1 σ 2 Figre 11 Now we proe case (4), [x σ i, σ] = 1 for i 2. Since x σ i = (σ i) σ i i 2, it is enogh to proe that [ (σ i), σ] = 1 for i 2., and [σ, σ] = 1 for For the case i = 1, [ (σ i), σ] = 1 holds by [3][Lemma 3.9], since (σ 1) = L(σ 2,..., σ n, ) and σ is disjoint from the irtal edge L(σ 2,..., σ n, ). If i 3, [ (σi), σ] = [σ i1... σ n σ1 σ2... σ i... σ 2 σ 1 σ n... σ i1, σ] = = [ σ 1 σ 2... σ i... σ 2 σ 1, σ] = [σ i... σ 2 σ 1 σ 1 σ 2... σ i, σ] = = [σ 2 σ 1 σ1 σ2, σ] = [σ 1 σ1, σ2 σσ 2 ] = [σ 1 σ1, σσ 2 σ ] = [σ A Y (T ). Hence, (4) holds. It remains to proe (5), [x σ, x σ i ] for i 2. Since x σ and [σ, σ i i σ, σ 1 σ 2 σ 1 ] = 1 in = (σ) σ and x σ i = (σ i) σ i, ] = 1 for i 2, and by (3) and (4), [ (σ), σ ] = 1, [ (σi), σ ] = 1, it is enogh to proe: [ (σ ), (σ i) ] = 1 for i 2. In the case i = 1, [ (σ), (σ 1) ] = [σ 2... σ nσ i σ 1σ1 σσ n... σ 2, σ2... σnσ n... σ 2 ] = = [σ σ 1σ1 σ, ] = [σ 1 σ1, σσ ] = [ σ 1, σ] = 1. In the case i 4, [ (σ), (σ i) ] = [σ 2... σ = [(σ 2... σ nσ 1 σ = [σ2 σ 1 σ σ 2 (σ1) σ In the case i = 3, nσ 1 σ σσ 1 σ n... σ 2, (σ i) ] = σ n... σ 2 ) (σ1) (σ2... σ 2 σ nσ 1 σσ n... σ 2 ), (σ i) ] = 1 σσ 2, (σ i) ] = [ (σ 1), (σ i) ]. Since i 4, [ (σ 1), (σ i) ] = 1 by [3]. [ (σ), (σ3) ] = [σ2 σ 1 σ σ 2 (σ1) σ2 σ1 σσ 2, (σ3) ] = = [σ 1 σ σ 2 (σ1) σ2 σσ 1, σ 2 (σ3) σ2 ] = [σ 1 σ σ 1 (σ2) σ 1 σσ 1, σ3 (σ2) σ 3 ] = = [σ (σ 2) σ, σ3 (σ2) σ 3 ] = [ (σ2) σ( (σ2) ), (σ2) σ 3 ( (σ2) ) ] = [σ, σ 3 ] = 1. Proposition 17. (1) σ i x σ i1 σ i (2) σ i x σ i1 σ i = σ i1 x σ i σ i1, 1 i n 2 (3) σ 2 x σ σ 2 = σ x σ 2 σ = σ i1 x σ i σ i1, 1 i n 2

14 14 AMRAM, SHWARTZ, TEICHER (4) σ 2 x σ σ 2 = σx σ 2 σ Proof. (1) and (2) hae been proed in [3][Section 6]; hence, we proe (3) and (4). Proof of (3). σ 2 x σ σ 2 = σ 2 (σ) σ σ = (σ 2 (σ2 σ 1 σ σ 2) (σ1) (σ = (σ 1 σ σ 2 (σ1) σ2 σσ = (σ (σ 2) σ)(σ σ Proof of (4). 2 = (σ 2 (σ) σ2 )(σ 2 σ σ 2 ) = 2 σ1 σσ 2 )σ2 )(σ σ 2 σ) = 1 )(σ σ 2 σ) = σ 2 σ) = (σ 1 σ (σ 2) σ σ 2 σ = σ 1 (σ 2) σ 1 σσ x σ 2 σ. 1 )(σ σ 2 σ) = We want to proe that σ2 x σ σ 2 = σx σ 2 σ. It is enogh to proe that σ 2 (σ) σ 2 = σ (σ2) σ, since σ 2 σ σ 2 = σσ 2 σ, x σ = (σ) σ and x σ 2 = (σ2) σ2. We know that (σ) = (σ2 σ 1 σ σ 2) (σ1) (σ, 2 σσ 1 σ 2 ). Ths, from σ 2 (σ 1) σ 2 = σ 1 (σ 2) σ 1 (see (1)), we conclde that (σ ) = σ 2 σ (σ 2) σσ 2. Hence, we need to proe σ 2 2 σ (σ 2) σσ 2 2 = σ (σ 2) σ. Since < σ, (σ2) >= 1, this is eqialent to σ 2 2 (σ 2) σ( (σ 2) ) σ 2 2 = ( (σ 2) ) σ (σ 2), which is eqialent to proing ( (σ 2) σ 2 2 (σ 2) )σ( (σ 2) ) σ 2 2( (σ 2) ) = σ. Since [σ 2 2, (σ 2) ] = 1 ([3][Section 6]), it is eqialent to proe that ( (σ 2) ) 2 σ 2 2 σσ 2 2( (σ 2) ) 2 = σ. Bt [x σ i, x σ i1 ] = z. [3][Section 6] implies that [x σ i, x σ i1 ] = ( (σ i) ) 2 σ 2 i = z for each 1 i n 1. Hence, ( (σ 2) ) 2 σ 2 2 = ( (σ 1) ) 2 σ 2 1. Hence the eqation we want to proe can be written as ( (σ 1) ) 2 σ 2 1 σσ 2 1( (σ 1) ) 2 = σ. By Proposition 16, [ (σ 1), σ] = 1, and [σ, σ 1 ] = 1, we get the reslt and this completes the proof of (4). Proposition 18. [x σ i, x σ i1 ] = [x σ, x σ 2 ] = z, where z 2 = 1 and z C(G(T )). Proof. By [3][Proposition 6.8], [x σ i, x σ i1 ] = [x σ j, x σ j1 ] for each 1 i n 2, 1 j n 2. Let G (T ) be the sbgrop of G(T ) generated by σ, σ 2,..., σ n and. By Propositions 16 and 17 : [σ, (σ i) ] = 1 for i 2 and σ (σ 2) σ = σ2 (σ ) σ 2, σ (σ 2) σ = σ 2 (σ ) σ Hence, G (T ) is isomorphic to the grop G that is defined in [3][Section 6] as a qotient of A(T (1) ), where T (1) is a single cycle. The map ϕ : G (T ) G is an isomorphism, where ϕ(σ) = σ 1, ϕ(σ i ) = σ i 2. for 2 i n 1 and ϕ(x σ ) = x σ 1. Since [x σ 1, x σ 2 ] = [x σ i, x σ i1 ] = z in G, by

15 COVERS OF D-TYPE ARTIN GROUPS 15 the isomorphism: [x, x σ σ 2 ] = [x σ i, x σ i1 ] in G (T ). Hence [x, x σ σ 2 ] = [x σ i, x σ i1 ] in G(T ). Since [x, x σ σ 2 ] is a central element in G (T ) and [x σ i, x σ i1 ] = [x, x σ σ 2 ], [[x σ i, x σ i1 ], σ] = 1, and from [3][Section 6], [[x σ i, x σ i1 ], σ i ] = 1 for 1 i n 1, we hae [x σ i, x σ i1 ] = [x σ 1, x σ 2 ] = z, where z C(G(T )) and z 2 = 1. Proposition 19. x σ i = zx σ i for 1 i n 1 and x σ = zx σ. Proof. x σ i = zx σ i is proed in [3][Proposition 6.8]. And x σ = zx σ can also be dedced from [3][Proposition 6.8] by considering G (T ), which was defined in the proof of Proposition 18. Theorem 20. G(T ) K A(D n ), where K is the grop generated by x σ i, 1 i n 1 and x σ with the relations [x σ i, x σ j ] = 1, i j > 1, [x σ, x σ j ] = 1, j 2, [x σ i, x σ j ] = z, i j = 1, [x σ, x σ 2 ] = z, z is a central element and z 2 = 1. Proof. The sbgrop generated by σ 1,..., σ n and σ is A(D n ). Then sing Propositions 16-19, we get the reslt from the same argment as in [3][Theorem 6.11]. 4. The general case Theorem 21. Eery graph T that incldes at least one anti-cycle is eqialent to a graph T (m), where T (m) consists of m cycles inclding the edges σ 1,..., σ n, i for 1 i m and a negatiely signed edge σ that connects the ertices 1 and 2 (see Figre 12). Remark: Here m 1 is the nmber of the cycles and anti-cycles in T, i.e., m 1 is the nmber of cycles in T, where T is the graph obtained from T by omitting the signs. Proof. The proof is by indction on m. In the case m = 0, T contains an anti-cycle. Hence the nmber of negatie signs in T is odd. By Theorem 9, T is eqialent by trianglation to

16 16 AMRAM, SHWARTZ, TEICHER m 2 _ σ 1 1 σ σ 1 2 Figre 12 σ n a cycle since T contains jst one cycle. Since trianglation can only conert an anti-cycle to another anti-cycle and not to a cycle, T is eqialent to an anti-cycle connected to a path, _ which is T (0). Assme by indction that for m k the theorem holds. Then T, with k 1 cycles and anti-cycles, is eqialent to T (m). Now assme m = k 1. If we consider only k cycles (i.e., the sbgraph T obtained by omitting one edge from one of the cycles or one of the anti-cycles of T ), T is eqialent to T (k) by the indction hypothesis. Since trianglation preseres the nmber of the edges of T, T contains one more edge e which does not appear in T (k). The edge e forms one more cycle or one more anti-cycle, since trianglation preseres the nmber of cycles. Hence, T is eqialent to the graph T (k),e (see Figre 13), where the edge e connects two ertices i and j. k 2 _ σ 1 σ σ Figre 13 e σ n Then we look at the sbgraph T (0),e of T (k),e that contains the edges σ i, 1 i n 1, σ and e (i.e. T (0),e we get from T (k),e by omitting i for 1 i k). By Theorems 10 and 11, T (0),e is eqialent to T (1). Hence, T (k),e is eqialent to T (k1) (adding the edges i to T (1) ). Proposition 22. In A Y (T (m) ), the following relation holds for 1 i < j m: (1) < σ 1 i σ 1, j >= 1

17 (2) < σ i σ, j >= 1 (3) < σ n i σ n, j >= 1 (4) [σ 1 i σ 1, σ n j σ n] = 1 (5) [σ i σ, σ n j σn] = 1 (6) [σ 1 i σ 1, σ j σ ] = 1 (7) [σ 1 j σ 1, σ i σ ] = 1 COVERS OF D-TYPE ARTIN GROUPS 17 Proof. The proof is deried directly from the definition of A Y (T (m) ), see Figre 13. In the spirit of [3], we define x (j) α = j α and x (j) α = α j for 1 j m. Definition 23. Let G(T (m) ) be a qotient of A Y (T (m) ) by the following relations: [x (j) α, σ n σ 1 ασ 1 σ n] = 1, [x (j) α, σσ 1 ασ 1 σ ] = 1, [x (j) α, σ 1 σ [x (j) α, σ n σ ασσ 1 ] = 1, ασσ n] = 1, [x (j) α, α 2 ] = 1, for 1 j m. Theorem 24. G(T (m) ) K (m,n) A(D n ), where K (m,n) is the grop generated by x (i) σ k and x (i) σ, 1 i m, 1 k n 1 with the following relations: [x (i) σ k,x (j) σ l ] = 1, k l > 1, [x (i) σ,x (j) σ l ] = 1, l 2, [x (i) σ k,x (i) σ l ] = z i, k l = 1, [x (i) σ,x (i) σ 2 ] = z i, [x (i) σ k,x (j) σ l ] = [x (i) σ p,x (j) σ q ] = [x (i) σ,x (j) σ 2 ], k l = 1, p q = 1, z 2 i = 1. Proof. The relations in K (m,n) that do not inole x σ were proed in [3][Section 10]. Now look at the sbgrop K (m,n) of K (m,n), where K (m,n) is generated by x σ and x σ l where 2 l n 1 (i.e., withot the generator x σ 1 ). Then K (m,n) is isomorphic to K (m,n), where K (m,n) is the grop K (m,n) from [3][Section 10], where ϕ(x (i) σ ) = x (i) σ 1, ϕ(x (i) σ 1 ) = x (i) σ 1. Hence, by [3][Section 10], [x (i) σ, x (j) σ p ] = 1 for p 3 and [x (i) σ, x (j) σ 2 ] = [x (i) σ p, x (j) σ q ] for p q = 1. It remains to proe the relation between x (i) σ and x (i) σ 1. By Proposition 22 ((5) and (6)), [σ 1 i σ1, σ jσ] = 1 implies that [σ 1 i α σ1, σ jα σ] = 1,

18 18 AMRAM, SHWARTZ, TEICHER since [σ 1 ασ1, σασ ] = 1. Hence [σ 1x (i) α σ1, σx (j) α σ ] = 1. Then, from Proposition 17, [α x (i) σ 1 α, α x (j) σ α] = 1 for each i and j. This completes the proof. Note that by similar argments as in [3], the word problem is solble in K (m,n) and solble in A(D n ) too, we get Corollary 25. The word problem is solble in G(T ). References [1] D. Allcock, Braid pictres for Artin grops, Trans. A.M.S , (2002). [2] M. Amram, R. Shwartz, M. Teicher, Coxeter coers of classical Coxeter grops, preprint. [3] M. Amram, R. Lawrence, U. Vishne, Artin coers of the braid grops, preprint. [4] A. Bjorner, F. Brenti, Combinatorics of Coxeter Grops, Gradate Texts in Math. 231, Springer, New York, (2005). [5] P. J. Cameron, J. J. Siedel, S. V. Tsarano, Signed Graphes, Root Lattices and Coxeter Grops, J. of Algebra 164, , (1994). [6] J. Crisp, L. Psris, Artin grops of type B and D, Ad. Geom, 5(4), , (2005). [7] J. E. Hmphreys, Reflection Grops and Coxeter Grops, Cambridge stdies in adanced mathematics 29, (1990). [8] L. Rowen, M. Teicher, U. Vishne, Coxeter coers of the symmetric grops, J. Grop Theory, 8, , (2005). Meira Amram, Robert Shwartz, Mina Teicher, Department of Mathematics, Bar-Ilan Uniersity, Ramat-Gan 52900, Israel address: meira,shwartr1,teicher@macs.bi.ac.il