The BNS-Invariant for Artin Groups of Circuit Rank 2

J. Group Theory 21 (2018), 189–228 DOI 10.1515/jgth-2017-0029 © de Gruyter 2018

The BNS-invariant for Artin groups of circuit rank 2

Kisnney Almeida Communicated by Evgenii I. Khukhro

Abstract. The BNS-invariant or †1-invariant is the first of a series of geometric invariants of finitely generated groups defined in the eighties that are deeply related to finiteness properties of their subgroups, although they are very hard to compute. Meier, Meinert and VanWyk have obtained a partial description of †1 of Artin groups, but the complete description of the general case is still an open problem. Let the circuit rank of an Artin group be the free rank of the fundamental group of its underlying graph. Meier, in a previous work, obtained a complete description for Artin groups of circuit rank 0, i.e., whose underlying graphs are trees. In a previous work we have proved, in joint work with Kochloukova, the same description to be true for Artin groups of circuit rank 1. In this paper we prove the description to be true for every Artin group of circuit rank 2.

1 Introduction

The first †-invariant was introduced for the class of metabelian groups in [9], where it was used to classify all finitely presented metabelian groups. Later on it was defined for general finitely generated groups [7], in the form of the invariant now known as BNS-invariant or †1-invariant, which may be used to determine if a subgroup of G which contains G0 is finitely generated. Then a whole series of †-invariants †m.G; Z/ (resp. †m.G/), of homological (resp. homotopical) nature were defined in [8,15]. These higher invariants reveal which subgroups of G above the commutator have homological type FPm (resp. homotopical type Fm). It is worth mentioning that in general it is quite difficult to calculate these †-invariants, although some advances have been made. For example, for metabelian groups they are known in all dimensions only for the restricted class of metabelian groups of finite Prüfer rank [14]. Another relevant example is the case of the Thompson group F , for which the †-invariants in all dimensions were calculated in [6]. The †-invariants have also been applied in other contexts, as in [10]. The Artin group G with underlying finite simplicial graph G , with edges labeled by integer numbers greater than one, is given by a finite presentation, with generators corresponding to the vertices V.G / of G and relations given by

Œu; vn Œv; un for each edge u; v of G with label n; D ¹ º 190 K. Almeida where

Œa; bn abab for all a; b V.G /: WD „ ƒ‚ … 2 n factors

If G is connected and 1.G / is free of rank n, we say G is an Artin group of circuit rank n. The †-invariants have been studied for some classes of Artin groups, see for example [1–3, 11–13]. In [12] all homological †-invariants were calculated for the class of right-angled Artin groups. The proof in [12] generalizes the proof of the fact that some subgroups of right-angled Artin groups are of type FPm but are not finitely presented, see [5]. The BNS-invariant is known for Artin groups of circuit rank 0 [11]. In other joint works we have obtained †1 for Artin groups of circuit rank 1 [2], for some other subclasses of higher circuit ranks [1, 3] and for all Artin groups of finite type [4]. In this paper we obtain a complete description of †1 for Artin groups of circuit rank 2. Before we go into details, we note that all these results are particular true cases for Conjecture 2.2, which is a general conjecture on †1 of Artin groups. We have also obtained recently, in joint work with F. Lima, that the class of Artin groups that satisfy Conjecture 2.2 is closed for finite direct products [4]. Let G be a graph. A subgraph H of G is called dominant if, for each vertex v V.G / V.H/, there is an edge e E.G / such that .e/ v and .e/ V.H/. 2 n 2 D 2 Let be a character of an Artin group G with underlying graph G . An edge e of G is called dead if e has an even label greater than 2 and ..e// ..e//. D Let G be an Artin group with underlying graph G and let G R be a non- W ! zero real character of G. Define L L . / as the full subgraph of G gen- F D F erated by the vertices v V.G / such as .v/ 0. Define the living subgraph 2 ¤ L L. / L as the subgraph obtained from L after removing the dead D F F edges. In [11], †1.G/ was calculated when G is a tree and in [13] some sufficient and necessary conditions were obtained for an element Œ of the unit sphere S.G/ to be in †1.G/. In [2] we have shown, in a joint work with D. H. Kochloukova, the sufficient condition of [13] to be also necessary for Artin groups of circuit rank 1 and some other subclasses. In this paper we use a similar strategy to show the same result for Artin groups of circuit rank 2. Thus our main result is the following:

Theorem 1.1. Let G be an Artin group of circuit rank 2, with underlying graph G . Then

†1.G/ Œ S.G/ L. / is a connected dominant subgraph of G : D ¹ 2 j º The BNS-invariant for Artin groups of circuit rank 2 191

Our approach is to reduce the problem to some special cases that are discussed in Section 3. The proofs for the special cases are technical and reduce to show- ing that a particular normal subgroup N of G with G=N Z cannot be ﬁnitely ' generated. Though many of the calculations in these special cases are similar, the details in each case are slightly different. In Section 3.1 we outline the strategy of the proofs and explain the common notations and in Section 3.2 we describe the detailed calculations for each case. Finally, in the last section we show how the special cases imply our main result.

2 Preliminary results

2.1 †1-invariant of Artin groups The following result gives a partial description of the one-dimensional †-invariant of Artin groups

Theorem 2.1 ([13]). Let G be an Artin group with underlying graph G and let be a non-zero real character of G. (i) If L. / is a connected dominant subgraph of G , then Œ †1.G/. 2 (ii) If Œ †1.G/, then L . / is a connected dominant subgraph of G . 2 F We also believe the converse of the ﬁrst item to be true, which leads to the conjecture below.

Conjecture 2.2. Let G be an Artin group with underlying graph G . Then †1.G/ Œ S.G/ L. / is a connected dominant subgraph of G : D ¹ 2 j º We have been trying to prove the conjecture above to be true for as many particular cases as possible. By using Theorem 2.1, we can reduce Conjecture 2.2 to a simpler version.

Corollary 2.3 ([2, Corollary 2.9]). Let G be an Artin group and let G be its underlying graph. Assume that, for every discrete character of G, we have L . /is connected, L. / is disconnected Œ †1.G/c: F H) 2 Then †1.G/ Œ S.G/ L. / is a connected dominant subgraph of G : D ¹ 2 j º It is known that Conjecture 2.2 is true for Artin groups with a tree as underlying graph.

Theorem 2.4 ([11]). Let G be an Artin group with underlying tree G . Then †1.G/ Œ S.G/ L. / is a connected dominant subgraph of G : D ¹ 2 j º 192 K. Almeida

In previous joint work with Kochloukova we have proved the conjecture to be true for Artin groups of circuit rank 1.

Theorem 2.5 ([2]). Let G be an Artin group of circuit rank 1 with underlying graph G . Then †1.G/ Œ S.G/ L. / is a connected dominant subgraph of G : D ¹ 2 j º It is easy to see that the †1-invariant of Artin groups is symmetric.

Lemma 2.6 ([2, Lemma 2.10]). If G is an Artin group, then †1.G/ †1.G/. D We can use that to obtain an easy criterion.

Corollary 2.7 ([2, Corollary 2.11]). Let G be an Artin group and a discrete character of G. Then Œ †1.G/ if and only if ker is ﬁnitely generated. 2 It is also a well-established fact that the BNS-invariant behaves well on quo- tients.

Proposition 2.8. Let G G be a group epimorphism and a non-trivial W N N character of G. Then Œ †1.G/c implies Œ †1.G/c: N N 2 N N ı 2 We will also use the following lemma several times:

Lemma 2.9. Let G R be a non-trivial group homomorphism. Furthermore, W ! let N ker , and suppose that N N is a normal subgroup of G such that WD Ä G Z . Then N N . Besides, G N Ì . N ' D h i D ' h i Proof. Since is non-trivial, we have G G=N .G/ ' N ' N=N is inﬁnite. Then G Z implies N 0, hence N N . Since Im Z N ' N D D ' ' is free and N ker , we obtain G N Ì . h i D D h i 2.2 Reduction results To reduce the underlying graphs we need to use essentially two strategies. The ﬁrst one is “cutting”, based on Bass–Serre theory. Let H be a subgraph of G and H its corresponding Artin group. Then there is a natural homomorphism i H G such that the image of an arbitrary generator W ! of H corresponding to a vertex u V.H/ V.G / is the generator of G corre- 2 sponding to the vertex u. We recall that a subgraph H of G is full if whenever two vertices of H are linked by an edge e E.G /, then e E.H/. 2 2 The BNS-invariant for Artin groups of circuit rank 2 193

Lemma 2.10 ([17]). Let G be an Artin group with underlying graph G . Let H be a full subgraph of G and H its corresponding Artin group. Then the natural homomorphism i H G is a monomorphism. W ! By using the monomorphism above we can establish a decomposition of some Artin groups as amalgamated free products of Artin groups, and, by using a Bass– Serre theoretical result from [16], we establish the following.

Theorem 2.11 ([2, Theorem 2.13]). Let G be an Artin group with underlying graph G . Let GA, GB , GC be subgraphs of G such that GC GA GB , G GA GB D \ D [ and GC is a full subgraph of G . Let GA, GB , GC be the Artin groups with underlying graphs GA, GB , GC , respectively. Let G R be a character of G and let W ! A; B ; C be the restrictions of to GA;GB ;GC , respectively. Assume that C 1 1 is non-trivial, Œ C † .GC / and Œ † .G/. Then 2 2 © 1 1 G GA GB ; Œ A † .GA/ and Œ B † .GB /: ' 2 2 GC The second strategy is “shrinking”, based on an application of Proposition 2.8.

Lemma 2.12 ([2, Lemma 2.14]). Let G be an Artin group with underlying graph G , whose set of vertices is V.G / ui 1 i n. If there is an edge connecting ui D ¹ º Ä Ä and uj , with i < j , we denote the label of this edge as ˛ij . Let be a non-zero character of G. Let G be an Artin group with underlying graph G , obtained from N G after a ﬁnite number of steps of the following types: (i) Delete all the vertices v such that .v/ 0, as well as all the edges to which they belong. D (ii) Add an edge (with arbitrary label 2) connecting two vertices of G that are not connected by an edge.

(iii) Change the label ˛ij of an edge with vertices ui and uj of G to ˇij > 1 such that ˇij ˛ij unless .ui / .uj / and ˇij is odd. j ¤ (iv) Identify two vertices ui ; uj such that .ui / .uj /. The resulting graph may not be simplicial, so we do the followingD process: If any two edges now have the same endpoints, with labels m; n, then identify those edges and replace their labels with gcd.m; n/. Now identify the endpoints of any edge that is labeled by 1. Repeat this process while there are still two edges with the same endpoints, as many times as necessary. After that delete all edges that are loops. Then there is a natural epimorphism G G and induces a real character W N N of G (i.e. ). In particular, Œ †1.G/c implies Œ †1.G/c. N DN ı N 2 N 2 194 K. Almeida

3 Reduced cases

3.1 The strategy of the proof of the reduced cases In this section we study some special cases of the Main Theorem. We start with some definition and notations that will appear in the results of this section. The proofs of the theorems in this section are similar but the technical details some- times are different, so we collect here the common definitions and properties. If there is any unclear notation, the reader is advised to consult Section 3.2, since some of the notation used here will be properly established in the course of the proofs. In all cases we suppose that Œ †1.G/. In each case that is equivalent to 2 N Ker. / to be finitely generated. So we suppose N to be finitely generated WD and find a contradiction. First of all, we establish the initial presentation for G directly from the definition. For the relations labeled by 2ki , ki > 1, i 1; 2; 3, we always assume D that ki is prime, for technical reasons (we can do that by Lemma 2.12). Then we find a presentation for N by adapting the presentation of G. That will lead us to an infinite presentation of N , for which we choose a specific quotient N (which is of N course finitely generated). Directly from the presentation of N we find a infinite presentation for N with generators xi ; yi ; zi , i Z. N 2 Then we consider the groups

k1 K1 xi 0 i 1 xi 1; 0 i 1 WD h¹ º Ä Ä j D Ä Ä i © k1 © xi xi 1 Zk1 ; D 0 i 1h j D i ' 0 i 1 Ä Ä Ä Ä k3 K3 zi 0 i Ã 1 zi 1; 0 i 1 WD h¹ º Ä Ä j D Ä Ä i © k3 © zi zi 1 Zk3 ; D 0 i Ã 1h j D i ' 0 i Ã 1 Ä Ä Ä Ä k2 K yi i Z yi 1; i Z WD h¹ º 2 j D 2 i © k2 © yi yi 1 Zk2 ; D i Zh j D i ' i Z 2 2 and M K1 K;D Z Z and A K D; WD WD k1 k3 WD where and are different numbers in each case, depending on the initial presentation. We also consider x and z as being the elements of D corresponding to the canonical generators of Zk1 and Zk3 , respectively. Note that by the Normal Form The BNS-invariant for Artin groups of circuit rank 2 195

Theorem for free products © KA yd A: (3.1) D h i i Ä i Z; d D 2 2 Now we define as the canonical homomorphism M M N; W k1 ki M ' N xi ; zi i Z h¹ º 2 i where xi i Z; zi i Z M extend xi 0 i 1; zi 0 i Ã 1, respectively. The ¹ º 2 ¹ º 2 ¹ º Ä Ä ¹ º Ä Ä isomorphism above is justified by the presentation of N , slightly different for N each case. We define M A as an epimorphism such that K IdK and projects W j D Ki onto Z for i 1; 3. That means induces an epimorphism N A and ki D N W N N a commutative diagram Â M / / A

ı N / / A, N Â N N where A ı A A k k A W .xi / 1 ; .zi / 3 i Z DW N h¹ º 2 i is the canonical projection and Ker.ı/ KA. Note that KA has finite index in A. Â Then B ı.KA/ WD has finite index in A, and since N is finitely generated, we deduce that B is finitely N N generated. Finally, we consider the abelianization Bab of B. As B is finitely generated, so is the Z-module Bab. Since A=B A=KA D, we can view Bab as a right N ' ' ZŒD-module where the action is induced by conjugation. By (3.1), Bab as a right

Zk2 ŒD-module is generated by the images of

ei ıyi ; i Z: WD 2 ab L Thus B is a quotient of the free Zk2 ŒD-module i Z ei Zk2 ŒD with basis ab 2 ei i Z and using this description we prove B cannot be finitely generated, ¹ º 2 a contradiction. Then N cannot be finitely generated, which means Œ †1.G/c. N 2 That finishes the proof. 196 K. Almeida

3.2 Proof of reduced cases

Theorem 3.1. Let G G.k1; k2; k3; l/ be the Artin group with underlying graph D 2 u1 u4 2k1 2k2 2k3

u2 u3 u5, 2l 1 2 C where k1; k2; k3; l are positive integers, with ki > 1 for all i. Furthermore, let Hom.G; R/ with .u1/ .u2/ .u3/ 1 and .u4/ .u5/ Q. 2 D D D D 2 Then Œ †1.G/c. 2 Proof. By Lemma 2.12 we can assume that ki is prime, for i 1; 2; 3. Define D N Ker. / and note that is discrete. By Corollary 2.7, Œ †1.G/ if and WD 2 only if N is finitely generated. Assume that N is finitely generated. By definition, G has a presentation

k1 k1 k2 k2 G u1; u2; u3; u4; u5 .u1u2/ .u2u1/ ; .u1u3/ .u3u1/ ; D h j D D l l u1u4 u4u1; .u2u3/ u2 .u3u2/ u3; D D k3 k3 u3u5 u5u3; .u4u5/ .u5u4/ : D D i p Let .u4/ , with gcd.p; q/ 1. Deﬁne D q D p q x0 u1u2; y0 u1u3; z0 u4u5; w u u ; WD WD WD WD 1 4 so 1 1 1 u2 u x0; u3 u y0; u5 u z0: D 1 D 1 D 4 As gcd.p; q/ 1, there are ˛; ˇ Z such that ˛p ˇq 1. Set uˇ u˛ G. D 2 C D D 1 4 2 Then q ˛ p ˇ ˇ p u1 w ; u4 w ; hence u5 w z0: (3.2) D D D G By construction G= x0; y0; z0; w is generated by the class of . By Lemma 2.9, hG i N x0; y0; z0; w and G N Ì , hence D h i ' h i i i i N xi ; yi ; zi ; w i Z with xi x0 ; yi y0 ; zi z0 ; (3.3) D h¹ º 2 i WD WD WD since w commutes with . Now we need to translate the relations of G to relations on the generators of N shown in (3.3). Firstly, it is easy to see that the relation u1u4 u4u1 is equivalent D to the combination of the following relations:

p ˇ ˛ q w w ; u4 w ; u1 w ; D D D The BNS-invariant for Artin groups of circuit rank 2 197 which also implies (3.2). By using these equations, we can translate the other relations:

k1 k1 k1 ˛ k1 ˛ .u1u2/ .u2u1/ xi w xi qw ; i Z; (3.4) D ” D C 2 k2 k2 k2 ˛ k2 ˛ .u1u3/ .u3u1/ yi w yi qw ; i Z; (3.5) D ” D C 2 l l ˛ ˛ ˛ .u2u3/ u2 .u3u2/ xi w yi qw xi 2q w xi 2lq D ” C C C ˛ ˛ yi w xi qw yi 2q D C C ˛ w yi 2lq; i Z; (3.6) C 2 q (by moving the factors to the right by conjugation, clearing common factors i q at the right and left, and conjugating both sides by ), ˛ ˇ ˇ ˛ u3u5 u5u3 w yi qw zi p q w zi pw yi p q; i Z; (3.7) D ” C C C D C C C 2 q p (by moving the and factors to the right, clearing common factors at the right and conjugating both sides by i )

k3 k3 k3 ˇ k3 ˇ .u4u5/ .u5u4/ zi w zi pw ; i Z: (3.8) D ” D C 2 So we have the following presentation:

N xi ; yi ; zi ; w i Z (3.4), (3.5), (3.6), (3.7), (3.8) : (3.9) D h¹ º 2 j i By hypothesis, N is ﬁnitely generated, then so is N N : N WD k1 k2 k3 N xi ; yi ; zi ; w i Z h¹ º 2 i Analyzing the relations of (3.9), we obtain the following presentation:

k1 k2 k3 N xi i Z xi yi zi 1; N D h¹ º 2 j ¹ D D D xi yi qxi 2q xi 2lq yi xi qyi 2q yi 2lq; C C C D C C C yi qzi p q zi pyi p q i Z : (3.10) C C C D C C C º 2 i Let 2lq and q. Consider the groups K1, K3, K, M , D and A deﬁned D D in Section 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i The isomorphism above is justiﬁed by the following: Conjugating by suitable powers of , we obtain the two last relations of (3.10) to be equivalent to, respectively, 1 1 1 1 xi yi qxi 2qyi 3q xi 2lqyi 2lqxi 2lq qyi 2lq 2q yi ; D C C (3.11) 1 zi yi pzi qyi ; D 198 K. Almeida which are also equivalent to 1 1 1 1 xi yi xi qyi 2q yi 2lqxi 2lqyi 2lq qxi 2lq 2q yi q; D C C C C C C C (3.12) 1 zi yi p qzi qyi q: D C C C So we can use (3.11) and (3.12) to see the elements zi and xi , for i Z, as 2 elements of M , constructing the isomorphism. We write by abuse of notation x , y and z for the images of x , y and z in N . i i i i i i N Now we need a technical lemma.

K Lemma 3.2. For all j Z, there are unique xj K1, zj K3, vj K 1 and K 2 Q 2 Q 2 2 wj K 3 such that xj xj vj and zj zj wj . Note that xj xj , vj 1 if 2 DQ DQ Q D D 0 j < 2lq, and zj zj , wj 1 if 0 j < q. Besides, Ä Q D D Ä 1 1 1 xi xi 2qxi 4q xi 2lqxi 2lq qxi 2lq 3q xi q; (3.13) Q DQ Q Q Q C Q C Q 1 1 1 xi xi qxi 3q xi 2lq qxi 2lqxi 2lq 2q xi 2q (3.14) Q DQ C Q C Q C Q C Q C Q C and zi zi q (3.15) Q DQ for all i Z. Finally, 2 1 K1 vj yj yk 0 k

Let M A be the epimorphism given as follows: K IdK and projects W j D K1 and K3 onto D Z Z such that D k1 k3 .xj / x Z D for 0 j < 2lq q; D 2 k1 Ä 2 .xj / x Z D for 2lq q j < 2lq D 2 k1 Ä and .z0/ z Z D. Let D 2 k3 A A : k k A WD .zi / 3 ; .xi / 1 i Z h¹ º 2 i Then induces an epimorphism N A and a commutative diagram, as noted N W N N in Section 3.1. We also deﬁne A, ı and B as in the same subsection and we recall The BNS-invariant for Artin groups of circuit rank 2 199 that © KA yd A: (3.18) D h i i Ä i Z; d D 2 2 Consider the abelianization Bab of B. Since B is ﬁnitely generated, so is the Z-module Bab. As A=B A=KA D, we can view Bab as a right ZŒD-module N ' ' where the action is induced by conjugation. Note that

k 1 k 2 k3 k3 z 3 z 3 z .zi / ..zi wi / / .wi / .wi / .wi / .wi /: D Q D Now we have two cases to consider. If k1 2, then, by (3.13) and (3.14), ¤ ab .xi / is always a generator of Z , since k1 is prime. So in B , Q k1 k 1 k 2 k1 k1 x 1 x 1 x .xi / .xi vi / .vi / .vi / .vi / .vi /; D Q D by permutating the factors if necessary. Let

L j Z j is not congruent to any x q 1; q; : : : ; 2 mod 2lq 1 : D ¹ 2 j 2 ¹ º C º

If k1 2, then, by (3.13) and (3.14), .xi / is always a generator of Zk1 except D Q ab when i L; in this case .xi / 1. Then in B , … Q D ´ 2 2 .vi / ; if i L; .xi / x … D ..vi / /.vi /; otherwise:

ab By (3.18), B as a right Zk2 ŒD-module is generated by the images of

ei ıyi ; i Z: WD 2 ab L It follows that B is a quotient of the free Zk2 ŒD-module i Z ei Zk2 ŒD with ab 2 basis ei i Z. By the deﬁnition of B there is an isomorphism of Zk2 ŒD-modules ¹ º 2 L ab i Z ei .Zk2 ŒD/ B 2 ; ' J I I C C 0 where ei Z ŒD Z ŒD, J is the Z ŒD-submodule generated by k2 ' k2 k2 k3 1 wi .z z 1/ for all i; O C C C

I is the Zk2 ŒD-submodule generated by

Suppose first that k1 2. Consider the ring ¤ Z Œx; z R k2 ; WD .1 x xk1 1; 1 z zk3 1/ C C C C C C which is also a Z -vector space of dimension .k1 1/.k3 1/ 1. Then we k2 have I 0 0. So D ab M B ei R WD i Z 2 is a quotient of Bab (as a Z-module), hence finitely generated (as a Z-module), a contradiction with R 0. ¤ ab Suppose now that k1 2. Consider B as a Z -vector space. Consider the D k2 Zk2 -vector space Bab V : WD vi i L h¹ O º 2 i ab As I 0 vi i L , it follows that V is a quotient of B (as Zk2 -vector space), h¹ O º 2 i hence finite dimensional. Consider the ring

Zk2 Œx; z R Zk2 : D .1 x; 1 z zk3 1/ ' C C C C Note that L i Z ei R V 2 ; ' vi i L h¹ O º 2 i where ei R R. By (3.16) and (3.17), ' M V ei Z : ' k2 i Z L 2 n As Z L is inﬁnite and R 0, it follows that V has inﬁnite dimension, a contra- n ¤ diction.

Theorem 3.3. Let G G.k1; k2; k3/ be the Artin group with underlying graph D 2 2 u1 u2 u3

2k1 2k2 2k3

u6 u5 u4, 2 2 where k1, k2 and k3 are positive integers, with ki > 1 for all i. Furthermore, let Hom.G; R/ with .G/ Z, .u1/ .u6/ p, .u2/ .u5/ q 2 D D D D1 c D and .u3/ .u4/ t such that gcd.p; q; t/ 1. Then Œ † .G/ . D D D 2 The BNS-invariant for Artin groups of circuit rank 2 201

Proof. By Lemma 2.12 we can assume that ki is prime, for i 1; 2; 3. Define D N Ker. / and note that is discrete. By Corollary 2.7, Œ †1.G/ if and WD 2 only if N is finitely generated. Assume that N is finitely generated. By definition, G has a presentation

k1 k1 k2 k2 G u1; u2; u3; u4; u5; u6 .u1u6/ .u6u1/ ; .u2u5/ .u5u2/ ; D h j D D k3 k3 .u3u4/ .u4u1/ ; .u1u2/ .u2u1/; D D u2u3 u3u2; u4u5 u5u4; u5u6 u6u5 : D D D i Deﬁne x0 u1u6; y0 u2u5; z0 u3u4 WD WD WD and p q r q r p u u a; u u b; u u c; 2 1 DW 1 3 DW 1 3 DW so 1 1 1 u6 u x0; u5 u y0; u4 u z0: D 1 D 2 D 3 Since gcd.p; q; t/ 1, there are ˛; ˇ; Z such that ˛p ˇq r 1. Deﬁne ˇ Á D 2 C C D u˛u u G. Note that . / 1, which means . / generates .G/. Also D 1 2 3 2 D p Á ˇ ˛ Á q ˛ ˇ r u1 c a ; u2 a b ; u3 c b ; D D D (3.19) r ˇ ˛ q Á ˛ ˇ Á p u4 b c z0; u5 b a y0; u6 a c x0; D D D G hence G= x0; y0; z0; a; b; c is generated by the class of . By Lemma 2.9, we h i G have N x0; y0; z0; a; b; c and G N Ì , hence D h i ' h i i i i N xi ; yi ; zi ; a; b; c i Z with xi x0 ; yi y0 ; zi z0 ; (3.20) D h¹ º 2 i WD WD WD since a; b; c commute with . Now we need to translate the relations of G to relations on the generators of N shown in (3.20). Firstly, it is easy to see that the relations ui uj uj ui , D i; j 1; 2; 3, are equivalent to the combination of the following relations: D a a ; b b ; c c ; D D D p ˇ Á q ˛ Á r ˛ ˇ u1a c ; a b u2; c b u3; D D D which also implies (3.19). By using these equations, we can translate the other relations: ˇ ˛ Á ˛ u4u5 u5u4 b c zi r b a yi r q D ” C C C Á ˛ ˇ ˛ b a yi qb c zi q r ; i Z; (3.21) D C C C 2 Á ˛ ˇ Á u5u6 u6u5 b a yi qa c xi p q D ” C C C ˇ Á Á ˛ a c xi pb a yi p q; i Z; (3.22) D C C C 2 202 K. Almeida and

k1 k1 k1 ˇ Á Á ˇ k1 .u1u6/ .u6u1/ xi a c xi pc a ; i Z; (3.23) D ” D C 2 k2 k2 k2 Á ˛ ˛ Ák2 .u2u5/ .u5u2/ yi b a yi qa b ; i Z; (3.24) D ” D C 2 k3 k3 k3 ˇ ˛ ˛ ˇ k3 .u3u4/ .u4u3/ zi b c zi r c b ; i Z: (3.25) D ” D C 2 So we have the following presentation:

N xi ; yi ; zi ; a; b; c i Z (3.21), (3.22), (3.23), (3.24), (3.25) : (3.26) D h¹ º 2 j i By hypothesis, N is ﬁnitely generated, then so is N N : N WD k1 k2 k3 N xi ; yi ; zi ; a; b; c i Z h¹ º 2 i Analyzing the relations of (3.26), we obtain the following presentation:

N xi ; yi ; zi i Z zi r yi r q yi qzi q r ; N D h¹ º 2 j ¹ C C C D C C C yi qxi p q xi pyi p q; C C C D C C C k1 k2 k3 xi yi zi 1 i Z : (3.27) D D D º 2 i We will show that the above presentation contradicts N being finitely generated. N Let q. Consider the groups K1, K3, K, M , D and A defined in Sec- D D tion 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i The isomorphism above is justified by the following: By conjugating by suitable powers of , the first and second relations of (3.27) are equivalent to, respectively, 1 1 zi yi r zi qyi and xi yi pxi qyi ; (3.28) D D which is also equivalent to 1 1 zi yi q r zi qyi q and xi yi q pxi qyi q: (3.29) D C C C D C C C So we can use (3.28) and (3.29) to see the elements zi and xi , for i Z, as 2 elements of M , constructing the isomorphism. We write by abuse of notation x , y and z for the images of x , y and z in N . i i i i i i N By using (3.28) and (3.29) and some easy induction arguments, we can see that, for K K each j Z, there are unique xj K1, vj K 1 , zj K3 and wj K 3 such 2 Q 2 2 Q 2 2 that xj xj vj and zj zj wj ; DQ DQ such that xi xi and zi zi for each i 0; 1; : : : ; q 1. Through (3.28) and Q D Q D D (3.29) we can also conclude xi xi q and zi zi q for all i Z. Q DQ C Q DQ C 2 The BNS-invariant for Artin groups of circuit rank 2 203

Let M A be the epimorphism such that K IdK , .xi / x and W j D D .zi / z for all i 0; 1 : : : ; q 1. Then .xi / x and .zi / z for all i Z. D D Q D Q D 2 So k 1 k 2 k1 k1 x 1 x 1 x .xi / .xi vi / .vi / .vi / .vi / .vi / (3.30) D Q D and k 1 k 2 k3 k3 z 3 z 3 z .zi / ..zi wi / / .wi / .wi / .wi / .wi / (3.31) D Q D Let A A : k k A WD .zi / 3 ; .xi / 1 i Z h¹ º 2 i Then induces an epimorphism N A and a commutative diagram, as noted N W N N in Section 3.1. We also deﬁne A, ı and B as in the same subsection and we recall that © KA yd A: (3.32) D h i i Ä i Z; d D 2 2 Consider the abelianization Bab of B. Since B is ﬁnitely generated, so is the Z-module Bab. As A=B A=KA D, we can view Bab as a right ZŒD-module N ' ' ab where the action is induced by conjugation. By (3.32), B as a right Zk2 ŒD-module is generated by the images of

ei ıyi ; i Z: WD 2 ab L It follows that B is a quotient of the free Zk2 ŒD-module i Z ei Zk2 ŒD with 2 basis ei i Z. By (3.30) and (3.31) we have an isomorphism of Zk2 ŒD-modules ¹ º 2 L ab i Z ei Zk2 ŒD B 2 ; ' I where ei Z ŒD Z ŒD and I is the Z ŒD-submodule generated by k2 ' k2 k2 k1 1 k3 1 vi .1 x x /; wi .1 z z / i Z: ¹ C C C C C C º 2 Consider the ring Z Œx; z R k2 ; WD .1 x xk1 1; 1 z zk3 1/ C C C C C C which is also a Z -vector space of dimension .k1 1/.k3 1/ 1. Then the k2 free R-module with basis ei i Z ¹ º 2 ab M B ei R WD i Z 2 is a quotient of Bab (as Z-module), hence ﬁnitely generated (as Z-module), a contradiction with R 0. ¤ 204 K. Almeida

Theorem 3.4. Let G G.k1; k2; k3/ be the Artin group with underlying graph D 2k2 u1 u2

2 2 u5 2 2k1

u4 u3, 2k3 where k1; k2; k3 are integers, with ki > 1 for all i. Besides, let Hom.G; R/ 2 with .G/ Z, .u1/ .u2/ q, .u4/ .u5/ .u3/ p such that D D D D D D gcd.p; q/ 1. Then Œ †1.G/c. D 2 Proof. By Lemma 2.12 we can assume that ki is prime, for i 1; 2; 3. Define D N Ker. / and note that is discrete. By Corollary 2.7, Œ †1.G/ if and WD 2 only if N is finitely generated. Assume that N is finitely generated. By definition, G has a presentation

k2 k2 G u1; u2; u3; u4; u5 .u1u2/ .u2u1/ ; .u1u5/ .u5u1/; D h j D D 2k3 2k3 .u2u3/ .u3u2/; .u3u4/ .u4u3/ ; D D 2k1 k1 .u3u5/ .u5u3/ : D i Deﬁne p q x0 u3u5; y0 u1u2; z0 u3u4; w u u : WD WD WD WD 2 3 As gcd.p; q/ 1, there are ˛; ˇ Z such that ˛p ˇq 1. Set u˛uˇ G. D 2 C D D 2 3 2 Note that . / 1, which means . / generates .G/. Also D q ˛ ˛ p ˇ p u1 y0 w ; u2 w ; u3 w ; D D D (3.33) p ˇ p ˇ u4 w z0; u5 w y0; D D G hence G= x0; y0; w is generated by the class of . By Lemma 2.9 we have h iG N x0; y0; z0; w and G N Ì , hence D h i ' h i i i i N xi ; yi ; zi ; w i Z with xi x0 ; yi y0 ; zi z0 ; (3.34) D h¹ º 2 i WD WD WD since w commutes with . Now we need to translate the relations of G to relations on the generators of N shown in (3.34). Firstly, it is easy to see that the relation u2u3 u3u2 is equiva- D lent to the combination of the following relations: p ˛ p ˇ w w ; w u2; w u3; D D D The BNS-invariant for Artin groups of circuit rank 2 205 which also implies (3.33). By using these equations, we can translate the other relations: ˇ ˛ ˇ ˛ u1u4 u4u1 yi w zi p q w zi pyi pw ; i Z; (3.35) D ” C C D C C 2 ˇ ˛ ˇ ˛ u1u5 u5u1 yi w xi p q w xi pyi pw ; i Z; (3.36) D ” C C D C C 2 and

k2 k2 k2 ˛ ˛k2 .u1u2/ .u2u1/ yi w yi qw ; i Z; (3.37) D ” D 2 k3 k3 k3 ˇ ˇ k3 .u3u4/ .u4u3/ zi w zi pw ; i Z; (3.38) D ” D C 2 k1 k1 k1 ˇ ˇ k1 .u3u5/ .u5u3/ xi w xi pw ; i Z: (3.39) D ” D C 2 So we have the following presentation:

N xi ; yi ; zi ; w i Z (3.35), (3.36), (3.37), (3.38), (3.39) : (3.40) D h¹ º 2 j i By hypothesis, N is ﬁnitely generated, then so is N N : N WD k1 k2 k3 N xi ; yi ; zi ; w i Z h¹ º 2 i Analyzing the relations of (3.40), we obtain the following presentation:

N xi ; yi ; zi i Z yi zi p q zi pyi p; N D h¹ º 2 j ¹ C C D C C yi xi p q xi pyi p; C C D C C k1 k2 k3 xi yi zi 1 i Z : (3.41) D D D º 2 i We will show that the above presentation contradicts N being finitely generated. N Let q. Consider the groups K1, K3, K, M , D and A defined in Sec- D D tion 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i The isomorphism above is justified by the following: By conjugating by suitable powers of , the first and second relations of (3.41) are equivalent to, respectively, 1 1 zi yi p qzi qyi q and xi yi p qxi qyi q; (3.42) D D which is also equivalent to 1 1 zi yi pzi qyi and xi yi pxi qyi : (3.43) D C D C So we can use (3.42) and (3.43) to see the elements zi and xi , for i Z, as 2 elements of M , constructing the isomorphism. 206 K. Almeida

We write by abuse of notation x , y and z for the images of x , y and z in N . i i i i i i N By using (3.42) and (3.43) and some easy induction arguments, we can see that, for K K each j Z, there are unique xj K1, vj K 1 , zj K3 and wj K 3 such 2 Q 2 2 Q 2 2 that xj xj vj and zj zj wj DQ DQ such that xi xi and zi zi for each i 0; 1; : : : ; q 1. Through (3.42) and Q D Q D D (3.43) we can also conclude xi xi q and zi zi q for all i Z. We proceed Q DQ C Q DQ C 2 exactly like the end of the proof of Theorem 3.3 to ﬁnd a contradiction.

Theorem 3.5. Let G G.k1; k2; k3; l/ be the Artin group with underlying graph D

u2 2k1 2l 1 C 2k2 u1 u3

2 2k3 u4, where k1; k2; k3; l are positive integers, with ki > 1 for all i. Let Hom.G; R/ 1 2 c with .u1/ .u2/ .u3/ .u4/ 1. Then Œ † .G/ . D D D D 2 Proof. By Lemma 2.12 we can assume that ki is prime, for i 1; 2; 3. Define D N Ker. / and note that is discrete. By Corollary 2.7, Œ †1.G/ if and WD 2 only if N is finitely generated. Assume that N is finitely generated. By definition, G has a presentation

k1 k1 k2 k2 G u1; u2; u3; u4 .u1u2/ .u2u1/ ; .u1u3/ .u3u1/ ; D h j D D k3 k3 l l .u1u4/ .u4u1/ ; .u2u3/ u2 .u3u2/ u3; D D u3u4 u4u3 : D i Deﬁne x0 u1u2; y0 u1u3; z0 u1u4; WD WD WD so 1 1 1 u2 u x0; u3 u y0; u4 u z0: D 1 D 1 D 1 G By construction G= x0; y0; z0 is generated by the class of u1. By Lemma 2.9, h G i we have N x0; y0; z0 and G N Ì u1 , hence D h i ' h i i i i u1 u1 u1 N xi ; yi ; zi i Z with xi x0 ; yi y0 ; zi z0 : (3.44) D h¹ º 2 i WD WD WD The BNS-invariant for Artin groups of circuit rank 2 207

Now we need to translate the relations of G to relations on the generators of N shown in (3.44):

k1 k1 k1 k1 .u1u2/ .u2u1/ x x ; i Z; (3.45) D ” 0 D i 2 k2 k2 k2 k2 .u1u3/ .u3u1/ y y ; i Z; (3.46) D ” 0 D i 2 k3 k3 k3 k3 .u1u4/ .u4u1/ z z ; i Z; (3.47) D ” 0 D i 2 and l l .u2u3/ u2 .u3u2/ u3 xi yi 1xi 2yi 3 yi 2l 1xi 2l D ” C C C C C yi xi 1yi 2xi 3 D C C C xi 2l 1yi 2l ; i Z; (3.48) C C 2 1 (by moving the u1 factors to the right by conjugation, clearing common factors i 1 at the right, and conjugating both sides by u1 ),

u3u4 u4u3 yi zi 1 zi yi 1; i Z; (3.49) D ” C D C 2 1 (by moving the u1 factors to the right, clearing common factors at the right and i 1 conjugating both sides by u1 ). So we have the following presentation:

N xi ; yi ; zi i Z (3.45), (3.46), (3.47), (3.48), (3.49); i Z : D h¹ º 2 j 2 i By hypothesis, N is ﬁnitely generated, then so is N N ; N WD k1 k2 k3 N xi ; yi ; zi i Z h¹ º 2 i for which we obtain the following presentation:

k1 k2 k3 N xi ; yi ; zi i Z xi yi zi 1; (3.48), (3.49); i Z : (3.50) N D h¹ º 2 j D D D 2 i Let 2l and 1. Consider the groups K1, K3, K, M , D and A deﬁned in D D Section 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i The isomorphism above is justiﬁed by the following: Conjugating by suitable powers of u1, we obtain the two last relations of (3.50) to be equivalent to, respectively, 1 1 1 1 xi yi 1xi 2yi 3 xi 2l yi 2l xi 2l 1yi 2l 2 yi ; D C C (3.51) 1 zi yi 1zi 1yi ; D which are also equivalent to 1 1 1 1 xi yi xi 1yi 2 yi 2l xi 2l yi 2l 1xi 2l 2 yi 1; D C C C C C C C (3.52) 1 zi yi zi 1yi 1: D C C 208 K. Almeida

So we can use (3.51) and (3.52) to see the elements zi and xi , for i Z, as 2 elements of M , constructing the isomorphism. We write by abuse of notation x , y and z for the images of x , y and z in N . i i i i i i N Now we need a technical lemma.

K Lemma 3.6. For all j Z, there are unique xj K1, zj K3, vj K 1 and K 2 Q 2 Q 2 2 wj K 3 such that xj xj vj and zj zj wj . Note that z0 z0, w0 1 and, 2 DQ DQ Q D D if 0 j 2l 1, we have xj xj , so vj 1. Besides, Ä Ä Q D D 1 1 1 xi xi 2xi 4 xi 2l xi 2l 1xi 2l 3 xi 1; (3.53) Q DQ Q Q Q C Q C Q 1 1 xi xi 1xi 3 xi 2l 1xi 2l xi 2l 2 xi 2 (3.54) Q DQ C Q C Q C Q C Q C Q C and zi z0 (3.55) Q DQ for all i Z. Finally, 2 1 K1 vj yj yk 0 k

Let M A be the epimorphism given by K IdK and projects K1 W j D and K3 onto D Z Z such that D k1 k3

.x0/ .x1/ .x2l 2/ x Zk1 D; D D D D 2 2 .x2l 1/ x Zk1 D and .z0/ z Zk3 D. Then induces an epi- D 2 D 2 morphism N A and a commutative diagram, as noted in Section 3.1. We N W N N also deﬁne A, ı and B as in the same subsection and we recall that © KA yd A: (3.58) D h i i Ä i Z; d D 2 2 Consider the abelianization Bab of B. Since B is ﬁnitely generated, so is the Z-module Bab. As A=B A=KA D, we can view Bab as a right ZŒD-module N ' ' where the action is induced by conjugation. Note that

k 1 k 2 k3 k3 z 3 z 3 z .zi / ..zi wi / / .wi / .wi / .wi / .wi /: D Q D The BNS-invariant for Artin groups of circuit rank 2 209

Now we have two cases to consider. If k1 2, then, by (3.53) and (3.54), ¤ ab .xi / is always a generator of Z , since k1 is prime. So in B , Q k1 k 1 k 2 k1 k1 x 1 x 1 x .xi / .xi vi / .vi / .vi / .vi / .vi /; D Q D by permutating the factors if necessary.

If k1 2, then, by (3.53) and (3.54), .xi / is always a generator of Zk1 except D Q ab when i 2 mod 2l 1, in this case .xi / 1. Then in B , Á C Q D ´ 2 2 .vi / if i 2 mod 2l 1; .xi / x Á C D ..vi / / .vi / otherwise:

ab By (3.58), B as a right Zk2 ŒD-module is generated by the images of

ei ıyi ; i Z: WD 2 ab L Thus B is a quotient of the free Zk2 ŒD-module i Z ei Zk2 ŒD with basis ab 2 ei i Z. By the deﬁnition of B there is an isomorphism of Zk2 ŒD-modules ¹ º 2 L ab i Z ei Zk2 ŒD B 2 ; ' J I I C C 0 where ei Z ŒD Z ŒD, J is the Z ŒD-submodule generated by k2 ' k2 k2 k3 1 wi .z z 1/ for all i; O C C C

I is the Zk2 ŒD-submodule generated by

k1 1 vi .x x 1/ for all i such that .xi / 1; O C C C Q ¤ L where vi ; wi are respectively the preimages of bi ; ci in i Z ei Zk2 ŒD, with O abO ab 2 bi ; ci B being the images of ıvi ; ıwi B in B , and I is the Z ŒD-sub- 2 2 0 k2 module generated by k1vi for all i such that .xi / 1. O Q D Suppose ﬁrst that k1 2. Consider the ring ¤ Z Œx; z R k2 ; WD .1 x xk1 1; 1 z zk3 1/ C C C C C C which is also a Z -vector space of dimension .k1 1/.k3 1/ 1. Then we k2 have I 0 0. So D ab M B ei R WD i Z 2 is a quotient of Bab (as a Z-module), hence ﬁnitely generated (as a Z-module), a contradiction with R 0. ¤ 210 K. Almeida

ab Suppose now that k1 2. Consider B as a Z -vector space. Let D k2 L i Z i 2 mod 2l 1 WD ¹ 2 j Á C º and consider the Zk2 -vector space Bab V : WD vi i L h¹ O º 2 i ab As I 0 vi i L , it follows that V is a quotient of B (as Zk2 -vector space), h¹ O º 2 i hence ﬁnite dimensional. Consider the ring

Zk2 Œx; z R Zk2 : D .1 x; 1 z zk3 1/ ' C C C C Note that L i Z ei R V 2 ; ' vi i L h¹ O º 2 i where ei R R. By (3.56) and (3.57), ' M V ei Z : ' k2 i Z L 2 n As Z L is inﬁnite and R 0, it follows that V has inﬁnite dimension, a contra- n ¤ diction.

Theorem 3.7. Let G G.k1; k2; k3; l/ be the Artin group with underlying graph D

2k1 u1 u2 2k2 2 2l 1 C u4 u3, 2k3 where k1; k2; k3; l are positive integers, with ki > 1 for all i. Let Hom.G; R/ 1 2c with .u1/ .u2/ .u3/ .u4/ 1. Then Œ † .G/ . D D D D 2 Proof. By Lemma 2.12 we can assume that ki is prime, for i 1; 2; 3. Define D N Ker. / and note that is discrete. By Corollary 2.7, Œ †1.G/ if and WD 2 only if N is finitely generated. Assume that N is finitely generated. By definition, G has a presentation

k1 k1 k2 k2 G u1; u2; u3; u4 .u1u2/ .u2u1/ ; .u1u3/ .u3u1/ ; D h j D D k3 k3 l l .u3u4/ .u4u3/ ; .u2u3/ u2 .u3u2/ u3; D D u1u4 u4u1 : D i The BNS-invariant for Artin groups of circuit rank 2 211

Deﬁne x0 u1u2; y0 u1u3; z0 u3u4; WD WD WD so 1 1 1 u2 u x0; u3 u y0; u4 y u1z0: D 1 D 1 D 0 G By construction, G= x0; y0; z0 is generated by the class of u1. By Lemma 2.9, G h i N x0; y0; z0 and G N Ì u1 , hence D h i ' h i i i i u1 u1 u1 N xi ; yi ; zi i Z with xi x0 ; yi y0 ; zi z0 : (3.59) D h¹ º 2 i WD WD WD Now we need to translate the relations of G to relations on the generators of N shown in (3.59):

k1 k1 k1 k1 .u1u2/ .u2u1/ x x ; i Z; (3.60) D ” 0 D i 2 k2 k2 k2 k2 .u1u3/ .u3u1/ y y ; i Z; (3.61) D ” 0 D i 2 k3 k3 k3 1 k3 .u3u4/ .u4u3/ zi 1 yi 1zi yi 1; i Z; (3.62) D ” C D C C 2 and l l .u2u3/ u2 .u3u2/ u3 xi yi 1xi 2yi 3 yi 2l 1xi 2l D ” C C C C C yi xi 1yi 2xi 3 D C C C xi 2l 1yi 2l ; i Z; (3.63) C C 2 1 (by moving the u1 factors to the right by conjugation, clearing common factors i 1 at the right, and conjugating both sides by u1 ), 1 1 u1u4 u4u1 yi 1zi yi 2zi 1; i Z; (3.64) D ” C D C C 2 (by moving the u1 factors to the left, clearing common factors at the left and i conjugating both sides by u1). So we have the following presentation:

N xi ; yi ; zi i Z (3.60), (3.61), (3.62), (3.63), (3.64); i Z : D h¹ º 2 j 2 i By hypothesis, N is ﬁnitely generated, then so is N N ; N WD k1 k2 k3 N xi ; yi ; zi i Z h¹ º 2 i for which we obtain the following presentation:

k1 k2 k3 N xi ; yi ; zi i Z xi yi zi 1; (3.63), (3.64); i Z : (3.65) N D h¹ º 2 j D D D 2 i Let 2l and 1. Consider the groups K1, K3, K, M , D and A deﬁned in D D Section 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i 212 K. Almeida

The isomorphism above is justiﬁed by the following: Conjugating by suitable powers of u1, we obtain the two last relations of (3.65) to be equivalent to, respectively, 1 1 1 1 xi yi 1xi 2yi 3 xi 2l yi 2l xi 2l 1yi 2l 2 yi ; D C C (3.66) 1 zi yi 1yi zi 1; D C which are also equivalent to 1 1 1 1 xi yi xi 1yi 2 yi 2l xi 2l yi 2l 1xi 2l 2 yi 1; D C C C C C C C (3.67) 1 zi yi 1yi 2zi 1: D C C C So we can use (3.66) and (3.67) to see the elements zi and xi , for i Z, as 2 elements of M , constructing the isomorphism. From now on, the proof continues just like the one of Theorem 3.5.

Theorem 3.8. Let G G.k1; k2; k3; l/ be the Artin group with underlying graph D u2 2k1 2 2k2 u1 u3

k1 k1 k2 k2 G u1; u2; u3; u4 .u1u2/ .u2u1/ ; .u1u3/ .u3u1/ ; D h j D D k3 k3 .u1u4/ .u4u1/ ; u2u3 u3u2; u3u4 u4u3 : D D D i Deﬁne x0 u1u2; y0 u1u3; z0 u1u4; WD WD WD so 1 1 1 u2 u x0; u3 u y0; u4 u z0: D 1 D 1 D 1 G By construction G= x0; y0; z0 is generated by the class of u1. By Lemma 2.9, h G i we have N x0; y0; z0 and G N Ì u1 , hence D h i ' h i i i i u1 u1 u1 N xi ; yi ; zi i Z with xi x0 ; yi y0 ; zi z0 : (3.68) D h¹ º 2 i WD WD WD The BNS-invariant for Artin groups of circuit rank 2 213

Now we need to translate the relations of G to relations on the generators of N shown in (3.68):

k1 k1 k1 k1 .u1u2/ .u2u1/ x x ; i Z; (3.69) D ” 0 D i 2 k2 k2 k2 k2 .u1u3/ .u3u1/ y y ; i Z; (3.70) D ” 0 D i 2 k3 k3 k3 k3 .u1u4/ .u4u1/ z z ; i Z; (3.71) D ” 0 D i 2 u2u3 u3u2 xi yi 1 yi xi 1; i Z (3.72) D ” C D C 2 1 (by moving the u1 factors to the right by conjugation, clearing common factors i 1 at the right, and conjugating both sides by u1 ),

u3u4 u4u3 yi zi 1 zi yi 1 for i Z; (3.73) D ” C D C 2 1 (by moving the u1 factors to the right, clearing common factors at the right and i 1 conjugating both sides by u1 ). So we have the following presentation:

N xi ; yi ; zi i Z (3.69), (3.70), (3.71), (3.72), (3.73); i Z : D h¹ º 2 j 2 i By hypothesis, N is ﬁnitely generated, then so is N N ; N WD k1 k2 k3 N xi ; yi ; zi i Z h¹ º 2 i for which we obtain the following presentation:

k1 k2 k3 N xi ; yi ; zi i Z xi yi zi 1; (3.72), (3.73); i Z : (3.74) N D h¹ º 2 j D D D 2 i Let 1. Consider the groups K1, K3, K, M , D and A deﬁned in Sec- D D tion 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i The isomorphism above is justiﬁed by the following: Conjugating by suitable powers of u1, we obtain the two last relations of (3.74) to be equivalent to, respectively,

1 1 xi yi 1xi 1yi ; zi yi 1zi 1yi (3.75) D D which are also equivalent to

1 1 xi yi xi 1yi 1; zi yi zi 1yi 1: (3.76) D C C D C C So we can use (3.75) and (3.76) to see the elements xi and zi , for i Z, as 2 elements of M , constructing the isomorphism. 214 K. Almeida

We write by abuse of notation x , y and z for the images of x , y and z in N . i i i i i i N By the Normal Form Theorem for free products, we have that for all j Z there K K 2 are unique xj K1, zj K3, vj K 1 and wj K 3 such that xj xj vj and Q 2 Q 2 2 2 DQ zj zj wj . Note that x0 x0, v0 1, z0 z0 and w0 1. Besides, by con- DQ Q D D Q D D jugating, applying an induction argument and the uniqueness above on (3.75), we obtain xi x0 x0; zi z0 z0 for all i Z: Q DQ D Q DQ D 2 Let M A be the epimorphism given by K IdK and projects K1 W j D and K3 onto D Z Z such that D k1 k3 .x0/ x Z D and .z0/ z Z D: D 2 k1 D 2 k3 Let A A : k k A WD .zi / 3 ; .xi / 1 i Z h¹ º 2 i Then induces an epimorphism N A and a commutative diagram, as noted N W N N in Section 3.1. We also define A, ı and B as in the same subsection and we recall that © KA yd A: (3.77) D h i i Ä i Z; d D 2 2 As KA has finite index in A, it follows that B ı.KA/ WD has finite index in A, and since N is finitely generated we deduce that B is finitely N N generated. Consider the abelianization Bab of B. Since B is finitely generated, so is the Z-module Bab. As A=B A=KA D, we can view Bab as a right N ' ' ZŒD-module where the action is induced by conjugation. Note that

k 1 k 2 k1 k1 x 1 x 1 x .xi / .xi vi / .vi / .vi / .vi / .vi /; D Q D and

k 1 k 2 k3 k3 z 3 z 3 z .zi / ..zi wi / / .wi / .wi / .wi / .wi / D Q D for all i Z. By (3.77), Bab as a right Z ŒD-module is generated by the images 2 k2 of ei ıyi ; i Z: WD 2 ab L It follows that B is a quotient of the free Zk2 ŒD-module i Z ei Zk2 ŒD with ab 2 basis ei i Z. By the deﬁnition of B there is an isomorphism of Zk2 ŒD-modules ¹ º 2 L ab i Z ei .Zk2 ŒD/ B 2 ; ' J I C The BNS-invariant for Artin groups of circuit rank 2 215

where ei Z ŒD Z ŒD, J is the Z ŒD-submodule generated by k2 ' k2 k2 k3 1 wi .z z 1/ for all i O C C C and I is the Zk2 ŒD-submodule generated by

k1 1 vi .x x 1/ for all i; O C C C L where vi ; wi are respectively the preimages of bi ; ci in i Z ei Zk2 ŒD, with O abO ab 2 bi ; ci B being the images of ıvi ; ıwi B in B . Consider the ring 2 2 Z Œx; z R k2 ; WD .1 x xk1 1; 1 z zk3 1/ C C C C C C which is also a Z -vector space of dimension .k1 1/.k3 1/ 1. So k2 ab M B ei R WD i Z 2 is a quotient of Bab (as a Z-module), hence ﬁnitely generated (as a Z-module), a contradiction with R 0. ¤

Theorem 3.9. Let G G.k1; k2; k3; l/ be the Artin group with underlying graph D

2k1 u1 u2 2k2 2 2

u4 u3, 2k3 where k1; k2; k3; l are positive integers, with ki > 1 for all i. Let Hom.G; R/ 1 2c with .u1/ .u2/ .u3/ .u4/ 1. Then Œ † .G/ . D D D D 2 Proof. By Lemma 2.12 we can assume that ki is prime, for i 1; 2; 3. Define D N Ker. / and note that is discrete. By Corollary 2.7, Œ †1.G/ if and WD 2 only if N is finitely generated. Assume that N is finitely generated. By definition, G has a presentation

k1 k1 k2 k2 G u1; u2; u3; u4 .u1u2/ .u2u1/ ; .u1u3/ .u3u1/ ; D h j D D k3 k3 .u3u4/ .u4u3/ ; u2u3 u3u2; u1u4 u4u1 : D D D i Deﬁne x0 u1u2; y0 u1u3; z0 u3u4; WD WD WD so 1 1 1 u2 u x0; u3 u y0; u4 y u1z0: D 1 D 1 D 0 216 K. Almeida

G By construction G= x0; y0; z0 is generated by the class of u1. By Lemma 2.9, h G i we have N x0; y0; z0 and G N Ì u1 , hence D h i ' h i i i i u1 u1 u1 N xi ; yi ; zi i Z with xi x0 ; yi y0 ; zi z0 : (3.78) D h¹ º 2 i WD WD WD Now we need to translate the relations of G to relations on the generators of N shown in (3.78):

k1 k1 k1 k1 .u1u2/ .u2u1/ x x ; i Z; (3.79) D ” 0 D i 2 k2 k2 k2 k2 .u1u3/ .u3u1/ y y ; i Z; (3.80) D ” 0 D i 2 k3 k3 k3 1 k3 .u3u4/ .u4u3/ zi yi 1zi 1yi 1; i Z; (3.81) D ” D C C C 2 u2u3 u3u2 xi yi 1 yi xi 1; i Z; (3.82) D ” C D C 2 1 (by moving the u1 factors to the right by conjugation, clearing common factors i 1 at the right, and conjugating both sides by u1 ), 1 1 u1u4 u4u1 yi 1zi yi 2zi 1; i Z; (3.83) D ” C D C C 2 (by moving the u1 factors to the left, clearing common factors at the left and i conjugating both sides by u1). So we have the following presentation:

N xi ; yi ; zi i Z (3.79), (3.80), (3.81), (3.82), (3.83); i Z : D h¹ º 2 j 2 i By hypothesis, N is ﬁnitely generated, then so is N N ; N WD k1 k2 k3 N xi ; yi ; zi i Z h¹ º 2 i for which we obtain the following presentation:

k1 k2 k3 N xi ; yi ; zi i Z xi yi zi 1; (3.82), (3.83); i Z : (3.84) N D h¹ º 2 j D D D 2 i Let 1. Consider the groups K1, K3, K, M , D and A deﬁned in Sec- D D tion 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i The isomorphism above is justiﬁed by the following: Conjugating by suitable powers of u1, we obtain the two last relations of (3.84) to be equivalent to, respectively, 1 1 xi yi 1xi 1yi ; zi yi 1yi zi 1 (3.85) D D C which are also equivalent to 1 xi yi xi 1yi 1; zi yi 1yi 2zi 1: (3.86) D C C D C C C The BNS-invariant for Artin groups of circuit rank 2 217

So we can use (3.85) and (3.86) to see the elements xi and zi , for i Z, as 2 elements of M , constructing the isomorphism. From now on, the proof continues just like the one of Theorem 3.8.

Theorem 3.10. Let G G.k1; k2; k3; l/ be the Artin group with underlying graph D 2 u1 u4 2k1 2k2 2k3

u2 u3 u5, 2 2 where k1; k2; k3; l are positive integers, with ki > 1 for all i. Let Hom.G; R/ 2 1 c with .u1/ .u2/ .u3/ 1, .u4/ .u5/ Q. Then Œ † .G/ . D D D D 2 2 Proof. By Lemma 2.12 we can assume that ki is prime, for i 1; 2; 3. Define D N Ker. / and note that is discrete. By Corollary 2.7, Œ †1.G/ if and WD 2 only if N is finitely generated. Assume that N is finitely generated. By definition, G has a presentation

k1 k1 k2 k2 G u1; u2; u3; u4; u5 .u1u2/ .u2u1/ ; .u1u3/ .u3u1/ ; D h j D D u1u4 u4u1; u2u3 u3u2; D D k3 k3 u3u5 u5u3; .u4u5/ .u5u4/ : D D i Deﬁne p q x0 u1u2; y0 u1u3; z0 u4u5; w u u ; WD WD WD DW 1 4 so 1 1 1 u2 u x0; u3 u y0; u5 u z0: D 1 D 1 D 4 p Let .u4/ q , with gcd.p; q/ 1. Then there are two integers ˛ and ˇ such D Dˇ that ˛p ˇq 1. Deﬁne u u˛ G. Then C D WD 1 4 2 q ˛ p ˇ ˇ p u1 w ; u4 w ; hence u5 w z0: (3.87) D D D G By construction, G= x0; y0; z0; w is generated by the class of . Lemma 2.9 h i G then implies that N x0; y0; z0; w and G N Ì , hence D h i ' h i i i i N xi ; yi ; zi ; w i Z with xi x0 ; yi y0 ; zi z0 ; (3.88) D h¹ º 2 i WD WD WD since w commutes with . Now we need to translate the relations of G to relations on the generators of N shown in (3.88). Firstly, it is easy to see that the relation u1u4 u4u1 is equiva- D 218 K. Almeida lent to the combination of the following relations:

p ˇ ˛ q w w ; u4 w ; u1 w ; D D D which also implies (3.87). By using these equations, we can translate the other relations:

k1 k1 k1 ˛ k1 ˛ .u1u2/ .u2u1/ xi w xi qw ; i Z; (3.89) D ” D C 2 k2 k2 k2 ˛ k2 ˛ .u1u3/ .u3u1/ yi w yi qw ; i Z; (3.90) D ” D C 2 and

˛ ˛ ˛ ˛ u2u3 u3u2 w xi w yi q w yi w xi q; i Z; (3.91) D ” D 2 (by moving the q factors to the right by conjugation, clearing common factors at i q the right, and conjugating both sides by C ), ˛ ˇ u3u5 u5u3 w yi qw zi p q D ” C C C ˇ ˛ w zi pw yi p q; i Z; (3.92) D C C C 2 q p (by moving the and factors to the right, clearing common factors at the right and conjugating both sides by i ), and

k3 k3 k3 ˇ k3 ˇ .u4u5/ .u5u4/ zi w zi pw ; i Z: (3.93) D ” D C 2 So we have the following presentation:

N xi ; yi ; zi ; w i Z (3.89), (3.90), (3.91), (3.92), (3.93) : (3.94) D h¹ º 2 j i By hypothesis, N is ﬁnitely generated, then so is N N : N WD k1 k2 k3 N xi ; yi ; zi ; w i Z h¹ º 2 i Analyzing the relations of (3.94), we obtain the following presentation:

k1 k2 k3 N xi i Z xi yi zi 1; xi yi q yi xi q; N D h¹ º 2 j ¹ D D D D yi qzi p q zi pyi p q i Z : (3.95) C C C D C C C º 2 i Let q. Consider the groups K1, K3, K, M , D and A deﬁned in Sec- D D tion 3.1. Let be the canonical homomorphism M M N: W k1 k3 M ' N xi ; zi i Z h¹ º 2 i The BNS-invariant for Artin groups of circuit rank 2 219

The isomorphism above is justiﬁed by the following: Conjugating by suitable powers of , we obtain the two last relations of (3.95) to be equivalent to, respectively,

1 1 xi yi xi qyi q; zi yi pzi qyi for all i Z; (3.96) D D 2 which are also equivalent to

1 1 xi yi qxi qyi ; zi yi p qzi qyi q for all i Z: (3.97) D C C D C C C 2 So we can use (3.96) and (3.97) to see the elements zi and xi , for i Z, as 2 elements of M , constructing the isomorphism. From now on, the proof continues just like the one of Theorem 3.3

Theorem 3.11. Let G G.k1; k2; k3; l1; l2/ be the Artin group with underlying graph D

2k1 u1 u2 2k2 2l2 1 2l1 1 C C u4 u3. 2k3 where k1; k2; k3; l1; l2 are positive integers, ki > 1 for all i. Let Hom.G; R/ 1 2c with .u1/ .u2/ .u3/ .u4/ 1. Then Œ † .G/ . D D D D 2 Proof. This is essentially a corollary of the main result (Proposition A) of [3]. In fact, suppose that k1; k2; k3 are not all powers of 2. Then by Lemma 2.12 we can add an edge labeled by 2k4 lcm.2k1; 2k3/ and apply [3, Proposition A]. WD So we can assume k1; k2; k3 to be powers of 2. By Lemma 2.12 we can assume k1 k2 k3 2. Define N Ker. / and note that is discrete. By Corol- D D D WD lary 2.7, Œ †1.G/ if and only if N is finitely generated. Assume that N is 2 finitely generated. By definition, G has a presentation

2 2 2 2 G u1; u2; u3; u4 .u1u2/ .u2u1/ ; .u1u3/ .u3u1/ ; D h j D D 2 2 l1 l1 .u3u4/ .u4u3/ ; .u2u3/ u2 .u3u2/ u3; D D l2 l2 .u1u4/ u1 .u4u1/ u4 : D i Deﬁne x0 u1u2; y0 u1u3; z0 u3u4; WD WD WD so 1 1 1 u2 u x0; u3 u y0; u4 y u1z0: D 1 D 1 D 0 220 K. Almeida

G By construction, G= x0; y0; z0 is generated by the class of u1. Lemma 2.9 h i G then implies that N x0; y0; z0 and G N Ì u1 , hence D h i ' h i i i i u1 u1 u1 N xi ; yi ; zi i Z with xi x0 ; yi y0 ; zi z0 : (3.98) D h¹ º 2 i WD WD WD Now we need to translate the relations of G to relations on the generators of N shown in (3.98): 2 2 2 2 .u1u2/ .u2u1/ x x ; i Z; (3.99) D ” 0 D i 2 2 2 2 2 .u1u3/ .u3u1/ y y ; i Z; (3.100) D ” 0 D i 2 2 2 2 1 2 .u3u4/ .u4u3/ zi 1 yi 1zi yi 1; i Z; (3.101) D ” C D C C 2 and

l1 l1 .u2u3/ u2 .u3u2/ u3 xi yi 1xi 2yi 3 yi 2l1 1xi 2l1 D ” C C C C C yi xi 1yi 2xi 3 D C C C

xi 2l1 1yi 2l1 ; i Z; (3.102) C C 2 1 (by moving the u1 factors to the right by conjugation, clearing common factors i 1 at the right, and conjugating both sides by u1 ),

l2 l2 1 1 1 .u1u4/ u1 .u4u1/ u4 yi zi 1yi 2zi 3 yi 2l 2zi 2l 1 D ” C C 1 1 yi 1zi yi 1zi 2 D C 1 yi 2l 1zi 2l ; i Z; (3.103) C 2 (by moving the u1 factors to the right, clearing common factors at the right and i 1 conjugating both sides by u1C ). So we have the following presentation:

N xi ; yi ; zi i Z (3.99), (3.100), (3.101), (3.102), (3.103); i Z : D h¹ º 2 j 2 i By hypothesis, N is ﬁnitely generated, then so is N N ; N 2 2 2 N WD xi ; yi ; zi i Z h¹ º 2 i for which we obtain the following presentation: 2 2 2 N xi ; yi ; zi i Z xi yi zi 1; (3.102), (3.103); i Z : (3.104) N D h¹ º 2 j D D D 2 i Let 2l1 and 2l2. Consider the groups K1, K3, K, M , D and A deﬁned D D in Section 3.1. Let be the canonical homomorphism M M N: 2 2 M N W xi ; zi i Z ' h¹ º 2 i The BNS-invariant for Artin groups of circuit rank 2 221

The isomorphism above is justiﬁed by the following: Conjugating by suitable powers of u1, we obtain the two last relations of (3.104) to be equivalent to, respectively,

x y 1 x 1 y 1 x 1 y i i 1 i 2 i 3 i 2l1 i 2l1 D

xi 2l1 1yi 2l1 2 yi for all i Z (3.105) C C 2 and

z y y 1z y 1 z y 1 z z 1 y i i 1 i i 1 i 2 i 3 i 2l2 2 i 2l2 1 i 2l2 i 2l2 1 D C C C C z 1 y z 1 y for all i Z; (3.106) i 2l2 2 i 2l2 3 i 2 i 1 C C 2 which are also respectively equivalent to

x y x y y x 1 i i i 1 i 2 i 2l1 i 2l1 D C C C C y 1 x 1 y 1 for all i Z (3.107) i 2l1 1 i 2l1 2 i 1 C C C 2 and

z y z 1 y z 1 y y 1 z 1 i i 1 i 2 i 3 i 4 i 2l2 1 i 2l2 i 2l2 1 D C C C C C C C C y 1 z z for all i Z: (3.108) i 2l2 2 i 2l2 3 i 1 C C C 2 So for each i Z we can use (3.105), (3.107) and (3.106), (3.108) to see the 2 elements xi and zi , respectively, as elements of M , constructing the isomorphism. From now on, the proof continues just like the one of Theorem 3.5.

Theorem 3.12. Let G G.k1; k2; k3; l1; l2/ be the Artin group with underlying graph D u2 2k1 2l1 1 C 2k2 u1 u3

2l2 1 2k3 C u4, where k1; k2; k3; l1; l2 are positive integers, ki > 1 for all i. Let Hom.G; R/ 1 2 c with .u1/ .u2/ .u3/ .u4/ 1. Then Œ † .G/ . D D D D 2 Proof. This is a particular case of [1, Proposition A]. 222 K. Almeida

4 The Main Theorem follows from the reduced cases

In this section, we prove the Main Theorem by using the reduced cases proven in the last section.

Theorem 4.1. Let G be an Artin group of circuit rank 2, with underlying graph G . Then

†1.G/ Œ S.G/ L. / is a connected dominant subgraph of G : D ¹ 2 j º

Proof. Let be a discrete character of G such that LF . / is connected and L. / is disconnected. By Corollary 2.3, it is enough to prove that Œ †1.G/c. Sup- 2 pose that Œ †1.G/. 2 First, we claim that we can assume .v/ 0 for all v V.G /, hence we have ¤ 2 G L . /. Indeed, by using Step 1 from Lemma 2.12 we can substitute G with D F a new graph G by deleting all the vertices v V.G / such that .v/ 0, along with N 2 D every edge that contains them and substitute the group G with the Artin group G N of the new graph and with the character of G induced by . Our hypotheses N N are still true: G L . / is connected, L. / is disconnected and Œ †1.G/. N D F N N N 2 N Furthermore, the graph GN either is a tree or it has fundamental group Z or it has fundamental group Z Z. For the ﬁrst two cases, Theorem 2.4 and Theorem 2.5 give us a contradiction. Then G has fundamental group Z Z and we can assume N that G G , so G G and . We are going to use Lemma 2.12 in this fashion N D D N N D several times. From now on, we are also going to use the cutting technique of Theorem 2.11 several times, as well as its notation. As 1.G / Z Z, we have that G contains two reduced loops I e1 : : : en ' D (where ei are edges) and II. There are also (possibly trivial) disjoint trees Ti , i 1; : : : ; n, “attached” to I , such that Ti I ei , with ei denoting the D \ D ¹ º beginning of ei . A similar phenomenon may occur on II. Let GC Z be the ' Artin group with underlying graph formed by the single vertex v1 e1. Let GA D be the Artin group with underlying graph GA G .T1 v1 /, and let GB be WD n n ¹ º the Artin group with underlying tree T1 and so GC v1 . Let A; B ; C be D ¹ º the restrictions of to GA;GB ;GC , respectively. Note that C is non-trivial and 1 1 GC Z, hence C † .GC / S.GC /. So by Theorem 2.11, Œ A † .GA/ ' 1 2 D 2 and Œ B † .GB /. 2 As L. / is disconnected, either L. A/ is disconnected or L. B / is discon- 1 nected. By Theorem 2.4, we have that L. B / is connected, since Œ B † .GB / 2 and T1 is a tree. So L. A/ is disconnected. Therefore we may “cut” the tree T1 out of the graph without loss of generality. By using the same argument several times, we can consider that all the trees attached to I and II are trivial. The BNS-invariant for Artin groups of circuit rank 2 223

Suppose that I and II have no intersection. That will lead to a contradiction we explain below. In that case, there is a single path (possibly with trees attached) connecting the two reduced loops. We may assume that the trees are trivial, by cutting them off if necessary, just like we did to the trees attached to I and II. If has a dead edge, we use Theorem 2.11 with GA I , GB II and 1 D [ D GC II to conclude A † .GA/. Since GA is an Artin group of circuit D \ 2 rank 1 and L. A/ GA is disconnected, we have a contradiction by Theorem 2.5. If has no dead edges, then – as L. / is disconnected – (at least) one of I , II has two dead edges. Then, by using GA I , GB II and GC , in D [ 1 D [ 1 D the notations of Theorem 2.11, we have Œ A † .GA/ and Œ B † .GB /. That 2 2 contradicts Theorem 2.5, since at least one of GA; GB has two dead edges. Then we assume that I and II have an intersection, a (possibly trivial, for now) path that we will call 0. From now on, we will see this graph as a union of three paths: 0, 1 and 2, where 1 is I minus the interior points of 0 and 2 is II minus the interior points of 0. Note that the three paths have the same vertices as extremities.

1 0 2

z If one of these three paths, let us say 1, has two dead edges, then, in view of Lemma 2.12, we may add an edge e, labeled by 2, such that .e/ . i / and D .e/ . i / for i 0; 1; 2. Then we can use Theorem 2.11, with GA 1 e, D D 1 D [ GB 0 e 2 and GC e, to conclude that Œ A † .GA/, which is a con- D [ [ D 2 tradiction by Theorem 2.5. If one of these three paths, let us say 0, has no dead edge, then one of the others, let us say 1, has two dead edges (since L. / is disconnected), so we again ﬁnd a contradiction. So we conclude 0, 1 and 2 have exactly one dead edge each. Now we need to analyze several cases, and for that we will establish some notations: each path i , i 0; 1; 2, will be of one of four forms. If the path is D formed by just a single dead edge, we will call it direct. If the path has at least one odd edge and no even edges besides the dead one, we will call it straight. If it has one or more even edges but at only one side of the dead edge (before or after the dead edge), we will call it twisted. If it has one or more even edges in both sides of the dead edges (before and after the dead edge), we will call it double-twisted. From now on we will use this notation freely. If one of the paths i is direct, we will assume it is 0, reindexing if necessary. Note that it is not possible that two of them are direct, or the graph would not be simplicial. 224 K. Almeida

If 0 has any odd edges, we collapse them by using Step 4 of Lemma 2.12 as many times as necessary. We still have a graph G such that L. / is disconnected and all our hypothesis hold. So we may consider 0 as not being straight. We have some other cases to consider.

Case 1: 0 is direct. Then we have three subcases.

Case 1.A: 1 and 2 are straight. As 1 and 2 are straight, their only even edge is also dead. Then by using Step 4 of Lemma 2.12 as many times as necessary we may collapse all the odd edges of 1 and 2 but one of each and still obtain a graph G such that L. / is disconnected. This will give us the graph of Theorem 3.12 or the one of Theorem 3.11. They both lead to a contradiction.

Case 1.B: 1 is straight and 2 is not straight (or vice-versa). Again we have that the even edge of 1 is dead. By using Step 4 of Lemma 2.12 as many times as necessary we may collapse all the odd edges of 1 and 2 except for one of 1 and still obtain a graph G such that L. / is disconnected. Now we use Step 3 of Lemma 2.12 as many times as necessary to change all the labels of the even edges of 2 that are not dead to 2. Then we obtain a graph that contains two vertices connected by three paths: 1, formed by a single dead edge and a single odd edge; 0, formed by a single dead edge; and 2, formed by a dead edge and some edges labeled by 2. If there is more than one edge labeled by 2 in sequence, we need to reduce it to a single edge still labeled by 2. We do it by using a technique we have used in [2] and we describe below. For example, suppose that 2 e0e1e2 em, where e0 D is a dead edge and e1; e2 : : : ; em are edges labeled by 2. So G is as below.

1 ,/ 0 >

e0 em / / e1 e2

Let G1 be the graph obtained from G after adding an edge f , labeled by 2, such that .f / .e1/ and .f / .em/, as below. D D 1

0,/ 0 > f e0 em / / e1 e2 The BNS-invariant for Artin groups of circuit rank 2 225

Then G1, the Artin group with underlying graph G1, is obviously a quotient of G 1 and induces a character 1 G1 R. By Lemma 2.12, Œ 1 † .G1/. W ! 2 Now we use Theorem 2.11, with

GA 1 0 e0 f; GB e1 e2 em f; GC f ; D [ [ [ D [ [ [ [ D 1 to conclude that Œ A † .GA/ which reduces the 2-labeled edges e1; e2; : : : ; em 2 to a single 2-labeled edge f as we intended. We name the above technique as 2-cutting, for further reference. Back to the problem, by using 2-cutting one or two times, we get one of the following graphs: either one on the statement of Theorem 3.7, or the one on Theo- rem 3.5 (if 2 is twisted) or the one on Theorem 3.1 (if 2 is double-twisted). All these cases lead to contradictions.

Case 1.C: 1 and 2 are not straight. By using Step 4 of Lemma 2.12 as many times as necessary, we collapse all the odd edges of 1 and 2 and still obtain a graph G such that L. / is disconnected. Recall that exactly one edge of 1 and one edge of 2 are dead. We use Step 3 of Lemma 2.12 as many times as necessary to change all the labels of the even edges of 1 and 2 that are not dead to 2. Then we obtain a graph that contains two vertices connected by three paths: 0, formed by a single dead edge; and 1 and 2, both formed by a dead edge and some edges labeled by 2 (at least one, since the original graph is simplicial). By using 2-cutting technique one to four times (depending on the graph), we get a graph in the fashion of the statements of Theorem 3.8, Theorem 3.9 (if 1 and 2 are twisted), Theorem 3.10 (if 1 is twisted and 2 is double-twisted, or vice-versa) or Theorem 3.3 (if 1 and 2 are both double-twisted). All these cases lead to contradictions.

Case 2: 0 is twisted. In this case we change all the labels of the even edges of 0 that are not the dead one to 2 (by using Step 3 of Lemma 2.12) and reduce the 2-labeled edges in sequence to single 2-labeled edges by 2-cutting. Again we have some cases to consider:

Case 2.A: 1 (or 2, without loss of generality) is straight. Then we can collapse all the odd edges of 2 by using Step 4 of Lemma 2.12 and replace the indexes of 1 and 0, so that the new 0 is direct and we are back to case 1.

Case 2.B: 1 and 2 are twisted. We again reduce the paths by changing even labels to 2 and 2-cutting, so that we have two subcases: If the three twists are at 226 K. Almeida the same side, we can 2-cut them, as below (the unlabeled edges are dead edges), so that we get the hypothesis of Theorem 3.8, which leads to a contradiction. ( ( 2 2 2 2 2 2 2 2 2 2

If one of the twists is on a different side, we obtain the hypothesis of Theorem 3.4, which also leads to a contradiction.

Case 2.C: 1 is twisted and 2 is double-twisted (or vice-versa). We again reduce the paths by changing even labels to 2 and 2-cutting. Then one of the sides of the dead edges has three twists, so we can 2-cut them as below (the unlabeled edges are dead edges), obtaining the hypothesis of Theorem 3.10, which leads to a contradiction. ( ( 2 2 2 2 2 2 2 2 2 2

2 2 2

Case 2.D: 1 and 2 are double-twisted. We again reduce the paths by changing even labels to 2 and 2-cutting. Now we 2-cut the twists from the side of 0’s twist, as in the previous subcase, reaching the hypothesis of Theorem 3.3, which leads to a contradiction.

Case 3: If 0 is double-twisted If 1 (or 2, without loss of generality) is straight, then we collapse all its odd edges by using Step 4 of Lemma 2.12 and replace the indexes of 1 and 0, so that the new 0 is direct and we are back to Case 1. If 1 (or 2, without loss of generality) is twisted, we reduce the graph as before, by collapsing all the odd edges, changing all the not-dead even labels to 2 and The BNS-invariant for Artin groups of circuit rank 2 227

2-cutting the sequenced 2-labeled edges in the paths. Then we replace the indexes of 1 and 0, so that the new 0 is twisted and we are back to case 2. If 1 and 2 are double-twisted, we reduce the graph as before, by collapsing all the odd edges, changing all the not-dead even labels to 2 and 2-cutting the sequenced 2-labeled edges in the paths. Then we cut the twists from both sides, as we did in case 2, reaching the hypothesis of Theorem 3.3, which leads to a contradiction.

Acknowledgments. We thank Professor Dessislava H. Kochloukova for the valu- able advice and the anonymous referee for the great corrections and suggestions.

Bibliography

[1] K. Almeida, The BNS-invariant for some Artin groups of arbitrary circuit rank, J. Group Theory 20 (2017), no. 4, 793–806. [2] K. Almeida and D. Kochloukova, The †1-invariant for Artin groups of circuit rank 1, Forum Math. 5 (2015), 2901–2925. [3] K. Almeida and D. Kochloukova, The †1-invariant for some Artin groups of rank 3 presentation, Comm. Algebra 43 (2015), no. 2, 702–718. [4] K. Almeida and F. Lima, Finite direct products closeness for a conjecture on the BNS-invariant of Artin groups, in preparation. [5] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. [6] R. Bieri, R. Geoghegan and D. H. Kochloukova, The sigma invariants of Thompson’s group F , Groups Geom. Dyn. 4 (2010), no. 2, 263–273. [7] R. Bieri, W. D. Neumann and R. Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987), no. 3, 451–477. [8] R. Bieri and B. Renz, Valuations on free resolutions and higher geometric invariants of groups, Comment. Math. Helv. 63 (1988), no. 3, 464–497. [9] R. Bieri and R. Strebel, Valuations and finitely presented metabelian groups, Proc. Lond. Math. Soc. (3) 41 (1980), no. 3, 439–464. [10] D. Gonçalves and D. H. Kochloukova, Sigma theory and twisted conjugacy classes, Pacific J. Math. 247 (2010), no. 2, 335–352. [11] J. Meier, Geometric invariants for Artin groups, Proc. Lond. Math. Soc. (3) 74 (1997), no. 1, 151–173. [12] J. Meier, H. Meinert and L. VanWyk, Higher generation subgroup sets and the †-invariants of graph groups, Comment. Math. Helv. 73 (1998), 22–44. 228 K. Almeida

[13] J. Meier, H. Meinert and L. VanWyk, On the †-invariants of Artin groups, Topology Appl. 110 (2001), no. 1, 71–81. [14] H. Meinert, The homological invariants for metabelian groups of ﬁnite Prüfer rank: A proof of the †m-conjecture, Proc. Lond. Math. Soc. (3) 72 (1996), no. 2, 385–424. [15] B. Renz, Geometrische Invarianten und Endlichkeitseigenschaften von Gruppen, Dissertation, Johann Wolfgang Goethe-Universität Frankfurt am Main, 1988. [16] S. Schmitt, Über den Zusammenhang der geometrischen Invarianten von Grup- pen und Untergruppen mit Hilfe von variablen Modulokoefﬁzienten, Diplomarbeit, Johann Wolfgang Goethe-Universität Frankfurt am Main, 1991. [17] H. Van der Lek, The homotopy type of complex hyperplane complements, Ph.D. the- sis, University of Nijmegan, 1983.

Received July 11, 2017.

Author information Kisnney Almeida, Departamento de Ciências Exatas, Universidade Estadual de Feira de Santana (UEFS), 44036-900, Feira de Santana, BA, Brazil. E-mail: [email protected]