On the Bieri-Neumann-Strebel-Renz Σ 1 -invariant of even Artin groups
aa r X i v : . [ m a t h . G R ] S e p ON THE BIERI-NEUMANN-STREBEL-RENZ Σ -INVARIANT OFEVEN ARTIN GROUPS DESSISLAVA H. KOCHLOUKOVA
Abstract.
We calculate the Bieri-Neumann-Strebel-Renz invariant Σ ( G ) foreven Artin groups G with underlying graph Γ such that if there is a closedreduced path in Γ with all labels bigger than 2 then the length of such path isalways odd. We show that Σ ( G ) c is a rationally defined spherical polyhedron. Introduction
By definition the Σ-invariants Σ m ( G ) and Σ m ( G, Z ) are subsets of the charactersphere S ( G ), where S ( G ) is the set of equivalence classes of Hom ( G, R ) \{ } , wheretwo real characters are equivalent if one is obtained from the other by multiplicationwith a positive real number. S ( G ) can be identified with the unit sphere S n − in R n , where n is the torsion-free rank of G/G ′ . Whenever the invariants are definedwe have the inclusions S ( G ) = Σ ( G ) ⊇ Σ ( G ) ⊇ . . . ⊇ Σ m ( G ) ⊇ Σ m +1 ( G ) ⊇ . . . and S ( G ) = Σ ( G, Z ) ⊇ Σ ( G, Z ) ⊇ . . . ⊇ Σ m ( G, Z ) ⊇ Σ m +1 ( G, Z ) ⊇ . . . , where Σ m ( G ) is defined only for groups G of homotopical type F m and Σ m ( G, Z ) isdefined for any finitely generated group G but if G is not of homological type F P m then Σ m ( G, Z ) = ∅ . The homotopical finiteness property F m was first defined byWall in [30] and its homological version F P m by Bieri in [5]. By definition a group G is of type F P m if the trivial Z G -module Z has a projective resolution with finitelygenerated modules up to dimension m . The homotopical version of the property F P m is called F m and by definition a group G is of homotopical type F m if thereis a K ( G, m -skeleton. Alternatively for m ≥ F m if it is F P m and finitely presented. In [4] Bestvina and Brady constructedthe first examples of groups that are F P but are not F (i.e. finitely presented).Furthermore these examples include groups that are of homological type F P ∞ , i.e.are F P m for every m , but are not finitely presented.The first Σ-invariant was defined by Bieri and Strebel in [10] for the class offinitely generated metabelian groups and later the definiton was extended to homo-logical and homotopical versions by Bieri, Neumann, Strebel [8], Bieri and Renz [9],Renz [28]. In the case of a 3-manifold group the Σ-invariant defined in [8] is related Mathematics Subject Classification.
Key words and phrases.
Artin groups, Σ-invariants.The author was partially supported by CNPq grant 301779/2017-1 and by FAPESP grant2018/23690-6. to the Thurston semi-norm from [29]. Bieri and Renz showed in [9] that the homo-logical Σ-invariants control the homological finiteness properties of the subgroupsof G that contain the commutator. The same holds in homotopical setting.The Σ-invariants have been calculated for some classes of groups and sometimesin low dimensions only : Thompson group F [6], generalised Thompson groups[18], [32], braided Thompson groups [33], Houghton groups [32], some free-by-cyclicgroups [16],[17], fundamental groups of compact K¨ahler manifolds [15], metabeliangroups of finite Pr¨ufer rank [25], some Artin groups including right angled Artingroups [2], [3], [22], [23], [24], limit groups [19], pure symmetric automorphismgroups of finitely generated free groups [27], [34], some wreath products [26], someresidually free groups [20], the Lodha-Moore groups [21]. In this paper we add tothe list above a subclass of the class of even Artin groups.Let Γ be a finite simplicial graph with edges labeled by integers greater than one.The Artin group G Γ associated to the graph Γ has generators the vertices V (Γ) ofΓ and relations given by uvu · · · | {z } n factors = vuv · · · | {z } n factors for every edge of Γ with vertices u, v and label n ≥ . A subgraph Γ of Γ is dominant if, for each v ∈ V (Γ) \ V (Γ ), there is an edge e ∈ E (Γ) that links v with a vertex from V (Γ ).Let χ : G Γ → R be a non-zero character of an Artin group G Γ . An edge e ∈ E (Γ)is called dead if χ ( σ ( e )) = − χ ( τ ( e )) and e has an even label greater than 2. Let L F = L F ( χ ) be the complete subgraph of Γ generated by the vertices v ∈ V (Γ) suchthat χ ( v ) = 0. Removing from L F all dead edges we obtain a new graph denotedby L = L ( χ ) ⊂ L F and called the living subgraph . Inspired by the Meier resultsin [22] Almeida and Kochloukova suggested in [2] the following conjecture. The Σ -Conjecture for Artin groups Let G = G Γ be an Artin group. Then Σ ( G ) = { [ χ ] ∈ S ( G ) | L ( χ ) is a connected dominant subgraph of Γ } . The Σ -Conjecture for Artin groups is known to hold when Γ is a tree [22], π (Γ) ≃ Z [2], π (Γ) is a free group of rank 2 [1], for Artin groups of finite type (i.e.the associated Coheter group is finite) [3]. We concentrate now on a special typeof Artin groups : the even Artin groups i.e. all the labels of the graph Γ are evennumbers. Though there is an extensive literature on Artin groups only recently evenArtin groups were considered with more details. In [14] Blasco-Garcia, Martinez-Perez and Paris showed that the even Artin groups of FC type are poly-free andresidually finite. The Σ-invariants of even Artin groups of FC type are studied in[13]. Our main result describes Σ ( G ) for specific even Artin groups G that are notrequired to be of FC type. Theorem A
Let G = G Γ be an even Artin group such that if there is a closedreduced path in Γ with all labels bigger than 2 then the length of such path is alwaysodd. Then Σ ( G ) = { [ χ ] ∈ S ( G ) | L ( χ ) is a connected dominant subgraph of Γ } . Our approach to prove Theorem A is homological in the sense that we obtaininformation about G ′ /G ′′ by calculating it as a certain homology group. In the N THE Σ -INVARIANT OF EVEN ARTIN GROUPS 3 calculation part we use Fox derivatives in order to construct the beginning of a freeresolution of the trivial Z G -module Z .In [7] using valuation theory Bieri and Groves showed that for a finitely generatedmetabelian group G the complement Σ ( G ) c = S ( G ) \ Σ ( G ) is a rationally definedspherical polyhedron i.e. a finite union of finite intersections of closed rationallydefined semi-spheres, where rationality means that each semisphere is defined by avector in R n with rational coordinates. In [17] Kielak considered a non-normalisedversion of Σ ( G ), where Σ ( G ) is a subset of Hom ( G, R ) \{ } and he showed that forone of the classes of groups : descending HNN extensions of finitely generated freegroups; fundamental groups of compact connected oriented 3-manifolds; agrarianPoincar duality groups of dimension 3 and type F; agrarian groups of deficiency oneΣ ( G ) is determined by an integral polytope [17]. Using the normalised definitionof Σ in many cases Σ ( G ) c is a rationally defined spherical polyhedron but this isnot true in general. In [8] there is an example of a group of PL automorphisms ofan interval where Σ ( G ) c contains a non-discrete isolated point, hence it is not arationally defined spherical polyhedron.For a subgroup H of G we set S ( G, H ) := { [ χ ] ∈ S ( G ) | χ ( H ) = 0 } . Note that S ( G, H ) is a rationally defined spherical polyhedron.
Proposition B
Let G = G Γ be an Artin group for which the Σ -Conjecture forArtin groups holds. Then there exist subgroups H . . . , H s of G such that Σ ( G ) c = ∪ ≤ i ≤ s S ( G, H i ) , in particular Σ ( G ) c is a rationally defined spherical polyhedron. As a corollary of the proof of Theorem A we obtain the following result.
Corollary C
Let G = G Γ be an even Artin group such that if there is a closedreduced path in Γ with all labels bigger than 2 then the length of such path is alwaysodd. Then for [ χ ] ∈ S ( G ) we have [ χ ] ∈ Σ ( G ) if and only if [ χ ] ∈ Σ ( G/G ′′ ) ,where χ is the real character of G/G ′′ induced by χ . In Section 7 we explain why the approach we use in the proof of Theorem Adoes not work for more general Artin groups. The point is that for more generalArtin groups (including some even Artin groups) Corollary C does not necessaryhold. We provided concrete examples (one is an even Artin group and the otheris not) where the expected behaviour of Σ ( G ), as suggested by the Σ -Conjecturefor Artin groups, differs from the structure of Σ ( G/G ′′ ).2. Preliminary results
On the Σ -invariants. Let G be a finitely generated group. By definition S ( G ) is the set of equivalence classes [ χ ] of non-zero real characters χ : G → R with respect to the equivalence relation ∼ , given by: χ ∼ χ if and only if thereis a positive real number r such that χ = rχ . Thus [ χ ] = R > χ . For a fixedcharacter χ : G → R consider the monoid G χ = { g ∈ G | χ ( g ) ≥ } . DESSISLAVA H. KOCHLOUKOVA
Bieri and Renz defined in [9] the homological Σ-invariants asΣ m ( G, A ) = { [ χ ] ∈ S ( G ) | A has type F P m as Z [ G χ ]-module } , where A is a finitely generated Z G -module. In this paper all considered modulesare left ones.There is a homotopical version of Σ m ( G, Z ) where Z is the trivial Z G -module,denoted by Σ m ( G ). By definition [ χ ] ∈ Σ ( G ) if and only if for a Cayley graph Γ of G , that corresponds to a finite generating set of G , the subgraph Γ χ spanned by allvertices in G χ is connected. In some papers the invariants are defined by left actionsand in others by right actions, thus they might differ by sign but if all actions comefrom the same side the known definitions of Σ-invariants agree. By [28] for anygroup G we have Σ ( G ) = Σ ( G, Z ) and for m ≥
2, Σ m ( G ) = Σ m ( G, Z ) ∩ Σ ( G ).Furthermore by [24] the inclusion Σ m ( G ) ⊆ Σ m ( G, Z ) could be strict. Theorem 2.1. [9]
Let G be a group of type F P m and H be a subgroup of G containing the commutator. Then H is of type F P m if and only if S ( G, H ) = { [ χ ] ∈ S ( G ) | χ ( H ) = 0 } ⊆ Σ m ( G, Z ) . Theorem 2.2. [9]
Let G be a finitely generated group and A a finitely generated Z G -module. Then Σ m ( G, A ) is open in S ( G ) . If G is of type F m then Σ m ( G ) isopen in S ( G ) . For a subset T ⊆ S ( G ) we write T c for the complement S ( G ) \ T of T in S ( G ).The following result is a well-known fact that follows from the definition of Σ . Lemma 2.3.
Let π : G ։ H be a group epimorphism and µ be a non-trivialcharacter of H . Then [ µ ] ∈ Σ ( H ) c implies [ µ ◦ π ] ∈ Σ ( G ) c . -invariants of Artin groups. Consider the living subgraph L = L ( χ ) thatwas defined in the introduction. Theorem 2.4. [23]
Let G = G Γ be an Artin group and let χ be a non-zero realcharacter of G . (1) If L ( χ ) is a connected dominant subgraph of Γ then [ χ ] ∈ Σ ( G ) . (2) If [ χ ] ∈ Σ ( G ) then L F ( χ ) is a connected dominant subgraph of Γ . Using Theorem 2.4 the Σ -Conjecture for Artin groups was reduced in [2] to thecase of discrete characters. Lemma 2.5. [2]
Let G = G Γ be an Artin group. Assume that for every discretecharacter χ of G such that L F ( χ ) is connected and L ( χ ) is disconnected then [ χ ] ∈ Σ ( G ) c . Then Σ ( G ) = { [ χ ] ∈ S ( G ) | L ( χ ) is a connected dominant subgraph of Γ } . We state some easy facts that were observed in [2].
Lemma 2.6. [2]
Let G Γ be an Artin group.a) Σ ( G Γ ) = − Σ ( G Γ ) ;b) if χ a discrete character of G then [ χ ] ∈ Σ ( G Γ ) if and only if Ker( χ ) isfinitely generated. N THE Σ -INVARIANT OF EVEN ARTIN GROUPS 5 Fox derivatives
Let G be a group and M be a left Z G -module. A function d : G → M is called a derivation if d ( gh ) = d ( g ) + gd ( h ) for every g, h ∈ G. Let F = F ( X ) be a free group with a basis X , x ∈ X and consider the derivation ∂∂x : F → Z F given by ∂∂x ( y ) = δ x,y the Kroniker symbol for every y ∈ X .Suppose now that G has a presentation h X | r , . . . , r s i . Then by [12, Prop. 5.4, Chapter II + Exer. 3,p. 45] there is an exact complex ofleft Z G -modules(3.1) ⊕ ≤ i ≤ s Z Ge r i ∂ −→ ⊕ x ∈ X Z Ge x ∂ −→ Z G ∂ −→ Z −→ , where ∂ = ǫ is the augmentation map, ∂ ( e x ) = x − Z F in Z G and for r ∈ { r , . . . , r s } ∂ ( e r ) = X x ∈ X ∂∂x ( r ) e x . We fix the commutator notation [ a, b ] = a − b − ab . Lemma 3.1.
Let s ∈ X and a, b ∈ F ( X ) . Then (3.2) ∂∂s ([ a, b ]) = a − ( b − − ∂∂s ( a ) + a − b − ( a − ∂∂s ( b ) . In particular for x, y ∈ X with x = y and r = [ x, y ] ∈ { r , . . . , r s } we have ∂ ( e r ) = x − y − (( x − e y − ( y − e x ) . Proof. ∂∂s ([ a, b ]) = ∂∂s ( a − b − ab ) = ∂∂s ( a − )+ a − ∂∂s ( b − )+ a − b − ∂∂s ( a )+ a − b − a ∂∂s ( b ) = − a − ∂∂s ( a ) − a − b − ∂∂s ( b ) + a − b − ∂∂s ( a ) + a − b − a ∂∂s ( b ) = a − ( b − − ∂∂s ( a ) + a − b − ( a − ∂∂s ( b ) . (cid:3) Lemma 3.2.
Suppose x, y ∈ X with x = y and m ≥ an integer. Then for r = [( xy ) m , x ] we have ∂∂y ( r ) = ( xy ) − m ( x − − xy + . . . + ( xy ) m − ) x and ∂∂x ( r ) = ( xy ) − m ( y − xy + . . . + ( xy ) m − ) . DESSISLAVA H. KOCHLOUKOVA
In particular if r ∈ { r , . . . , r s } we have that the image λ of ∂ ( e r ) in Z Qe x ⊕ Z Qe y is λ = ( x y ) − m (1 + x y + . . . + ( x y ) m − )(( y − e x − ( x − e y ) , where Q = G/G ′ , x is the image of x in Q and y is the image of y in Q .Proof. By (3.2) ∂∂s ( r ) = ∂∂s ([( xy ) m , x ]) =( xy ) − m ( x − − ∂∂s (( xy ) m ) + ( xy ) − m x − (( xy ) m − ∂∂s ( x ) =( xy ) − m ( x − − xy + . . . + ( xy ) m − ) ∂∂s ( xy ) + ( xy ) − m x − (( xy ) m − ∂∂s ( x ) . Then ∂∂x ( r ) = ( xy ) − m ( x − − xy + . . . + ( xy ) m − ) ∂∂x ( xy )+( xy ) − m x − (( xy ) m − ∂∂x ( x ) =( xy ) − m ( x − − xy + . . . + ( xy ) m − ) + ( xy ) − m x − (( xy ) m −
1) =( xy ) − m ( x − − x − ( xy − xy + . . . + ( xy ) m − ) =( xy ) − m ( y − xy + . . . + ( xy ) m − )and ∂∂y ( r ) = ( xy ) − m ( x − − xy + . . . + ( xy ) m − ) ∂∂y ( xy )+( xy ) − m x − (( xy ) m − ∂∂y ( x ) = ( xy ) − m ( x − − xy + . . . + ( xy ) m − ) x. Then for the image λ of ∂ ( e r ) in Z Qe x ⊕ Z Qe y we have λ = ( x y ) − m ( y − x y + . . . + ( x y ) m − ) e x +( x y ) − m ( x − − x y + . . . + ( x y ) m − ) x e y =( x y ) − m (1 + x y + . . . + ( x y ) m − ) γ, where γ = ( y − e x + ( x − − x e y = ( y − e x − ( x − e y . (cid:3) One algebraic lemma
Lemma 4.1.
Let T be a forest, where the set of vertices is a disjoint union V ∪ W and each edge links a vertex of V with a vertex of W that is labelled by an evenpositive integer m v,w > . Let Q be the free abelian group with free basis V ∪ W and consider the Z Q -module K T = ( ⊕ v ∈ V,w ∈ W Z Qf v,w ) /J, where Z Qf v,w is a free Z Q -module and J is the Z Q -submodule generated by { (1 + vw + . . . + ( vw ) m v,w − ) f v,w | v ∈ V, w ∈ W, v, w are vertices of an edge in T } , { ( t − f v,w − ( v − f t,w | v, t ∈ V, w ∈ W } and { ( s − f v,w − ( w − f v,s | v ∈ V, w, s ∈ W } . Let χ : Q → Z N THE Σ -INVARIANT OF EVEN ARTIN GROUPS 7 be a character (i.e. homomorphism of groups) such that for every vertex u ∈ V (Γ) we have χ ( u ) = 0 and for every edge in T with vertices v and w we have χ ( v ) = − χ ( w ) . Then K T is not finitely generated as Z Ker( χ ) -module.Proof. Suppose that there is an edge in T with vertices v ∈ V, w ∈ W . We define λ v,w ∈ C as a primitive m v,w -th root of 1.We decompose T as a disjoint union ∪ i T i where each T i is a tree. Let v i be afixed base point of T i . Consider the Laurent polynomial ring C [ x ± ] and the ringhomomorphism µ : C Q → C [ x ± ]that sends1) vw to λ v,w for all v ∈ V, w ∈ W vertices of an edge of T ,2) v i to x χ ( v i ) ,3) is the identity on C .Note that µ is well-defined because T is a forest and that µ (Ker( χ )) ⊆ C . Furthermore for q ∈ Q we have that µ ( q ) ∈ C x χ ( q ) , hence µ ( Q ) ⊆ ( ∪ j ∈ Z \{ } C x j ) . Then µ induces an epimorphism of C Q -modules b µ : C ⊗ Z K T → ( ⊕ v ∈ V,w ∈ W C [ x ± ] f v,w ) /R =: b K T that is the identity on f v,w , where R is the C [ x ± ]-submodule generated by { ( µ ( t ) − f v,w − ( µ ( v ) − f t,w | v, t ∈ V, w ∈ W } and { ( µ ( s ) − f v,w − ( µ ( w ) − f v,s | v ∈ V, w, s ∈ W } and we view b K T as C Q -module via µ .Let D be the localised ring C [ x ± , { cx j − | j ∈ Z \ { } , c ∈ C } ]. Since thecoefficients of C Q that appear in the generators of R are invertible in D , we havethat D ⊗ C [ v ± ] b K T is a cyclic D -module isomorphic to D . Hence for the annihi-lator of D ⊗ C [ x ± ] b K T in D we have ann D ( D ⊗ C [ x ± ] b K T ) = ann D ( D ) = 0, thus ann C [ x ± ] ( b K T ) = 0. This implies that b K T is infinite dimensional over C . Finally if K T is finitely generated as Z Ker( χ )-module then C ⊗ Z K T is finitelygenerated as C Ker( χ )-module. Then applying b µ we obtain that b K T is finitely gener-ated as µ ( C Ker( χ ))-module. But since µ (Ker( χ )) ⊆ C we deduce that µ ( C Ker( χ )) = C µ (Ker( χ )) = C . Thus b K T is finite dimensional over C , a contradiction. (cid:3) Proof of Theorem A
Let G = G Γ be an even Artin group such that if there is a closed reduced pathin Γ with all labels bigger than 2 then the length of such path is always odd. Let χ : G → Z be a non-zero character such that L F ( χ ) is connected and L ( χ ) isnot connected. By Lemma 2.5 to prove Theorem A is equivalent to show that[ χ ] / ∈ Σ ( G ). DESSISLAVA H. KOCHLOUKOVA
I. We suppose first that χ ( v ) = 0 for every v ∈ V (Γ). Since the graph L ( χ ) isnot connected V (Γ) = V (Γ ) ∪ V (Γ ) , where the union is disjoint, Γ and Γ arecomplete subgraphs of Γ ( i.e. if an edge e connects two vertices from Γ i in Γ thenthis edge belongs to E (Γ i )) and there is a disjoint union E (Γ) = E (Γ ) ∪ E (Γ ) ∪ E, where E is a set of dead edges (with respect to the character χ ) i.e. if u, v are thevertices of such an edge then χ ( u ) = − χ ( v ). Let γ be a closed reduced path withedges in E . The bipartite structure of the graph Γ implies that γ has even length,a contradiction with the assumption that a closed reduced path in Γ with all labelsbigger than 2 has always odd length. Hence no such path γ exists i.e. E is a forest.Let Γ be the graph with a set of vertices V (Γ ) = V (Γ), for i = 1 , V (Γ i ) are linked in Γ by an edge with label 2 and the edges in Γ thatlink some vertices of V (Γ ) with some vertices of V (Γ ) are precisely the edges from E with their original labels in Γ. Let G = G Γ be the Artin group with underlyinggraph Γ . Our aim is to calculate G ′ /G ′′ via the isomorphism G ′ /G ′′ ≃ H ( G ′ , Z ) . By (3.1) there is an exact complex P : ⊕ r Z G e r ∂ −→ ⊕ v Z G e v ∂ −→ Z G → Z → , where each relation r corresponds to one edge of Γ and v runs through the vertices V (Γ ). We have two type of relations : r = [ v , v ] where both v , v are verticesof V (Γ i ) for a fixed i ∈ { , } or r = [( vw ) m , v ], where v ∈ V (Γ ), w ∈ V (Γ ) andthere is an edge in E with label 2 m > v and w . By (3.1) ∂ ( e v ) = v − ∈ Z G and ∂ ( e r ) = X v ∂∂v ( r ) e v . Thus for Q = G /G ′ free abelian group with a free abelian basis X = V (Γ ) wehave the complex S = Z ⊗ Z G ′ P : ⊕ r Z Qe r e ∂ −→ ⊕ v Z Qe v e ∂ −→ Z Q −→ Z −→ H ( S , Z ) = Ker ( e ∂ ) /Im ( e ∂ ) . By abuse of notation we write v for the image of v ∈ V (Γ ) ⊂ G in Q = G /G ′ .Thus e ∂ ( e v ) = v − . By Lemma 3.1 for r = [ v , v ] for v , v ∈ V (Γ i ), v = v , i = 1 , e ∂ ( e r ) = v − v − (( v − e v − ( v − e v )and by Lemma 3.2 for r = [( v v ) m , v ] for some v , v ∈ V (Γ), where v and v are linked by an edge from E with label 2 m , we have e ∂ ( e r ) = ( v v ) − m (1 + v v + . . . + ( v v ) m − )(( v − e v − ( v − e v ) . On the other hand we have the exact Koszul complex R : . . . → ⊕ v 2) and (1+ vw + . . . +( vw ) m − ) e v ∧ e w if v ∈ V (Γ ) , w ∈ V (Γ ) and v, w are vertices of an edge from E with label2 m > π : ⊕ v 2) and is theidentity on e v ∧ e w if v ∈ V (Γ ) , w ∈ V (Γ ). Note that Ker( π ) ⊆ d − ( Im ( e ∂ )).Then π induces the isomorphism M ≃ ( ⊕ v ∈ V (Γ ) ,w ∈ V (Γ ) Z Qe v ∧ e w ) /N, where N = π ( d − ( Im ( e ∂ ))) is the Z Q -submodule generated by { (1 + vw + . . . + ( vw ) m − ) e v ∧ e w | v ∈ V (Γ ) , w ∈ V (Γ ) , v, w are vertices of e ∈ E with label 2 m > }∪ { π ( d ( e v ∧ e w ∧ e t )) | v ∈ V (Γ ) , w ∈ V (Γ ) , t ∈ V (Γ) = V (Γ ) ∪ V (Γ ) } . Thus if t ∈ V (Γ ) we have π ( d ( e v ∧ e w ∧ e t )) = π (( t − e v ∧ e w + ( v − e w ∧ e t + ( w − e t ∧ e v ) =( t − e v ∧ e w + ( v − e w ∧ e t and if t ∈ V (Γ ) we have π ( d ( e v ∧ e w ∧ e t )) = π (( t − e v ∧ e w + ( v − e w ∧ e t + ( w − e t ∧ e v ) =( t − e v ∧ e w + ( w − e t ∧ e v . Note that using the notation K T from Lemma 4.1 for T = E , V = V (Γ ) and W = V (Γ ) we have proved by now that(5.1) G ′ /G ′′ ≃ H ( G , Z ) ≃ ( ⊕ v ∈ V (Γ ) ,w ∈ V (Γ ) Z Qe v ∧ e w ) /N ≃ K E . Consider the epimorphism of groups θ : G = G Γ → G = G Γ which is the identity on V (Γ) and note that Ker ( θ ) ⊆ G ′ . Let χ : G → Z be thecharacter induced by χ , i.e. χ ◦ θ = χ , and ν : G /G ′ → Z and χ : G /G ′′ → Z be the characters induced by χ . By Lemma 4.1 K E is not finitely generated as Z Ker( ν )-module, hence by (5.1) G ′ /G ′′ is not finitely generated as Z Ker( ν ) − module.Furthermore since G has an automorphism ϕ that sends each vertex v ∈ V (Γ )to v − , we note that ϕ induces an automorphism of G /G ′′ . This implies thatΣ ( G /G ′′ ) = − Σ ( G /G ′′ ). Hence(5.2) [ χ ] / ∈ Σ ( G /G ′′ )otherwise [ χ ] ∈ Σ ( G /G ′′ ) ∩− Σ ( G /G ′′ ), Ker( χ ) is finitely generated and hence G ′ /G ′′ is finitely generated as Z Ker( ν )-module, a contradiction.Let χ : G/G ′′ → Z be the character induced by χ . Note that since G /G ′′ is aquotient of G/G ′′ , Lemma 2.3 and (5.2) imply that(5.3) [ χ ] / ∈ Σ ( G/G ′′ ) . By (5.3) and Lemma 2.3 applied for the canonical epimorphism G → G/G ′′ we getthat [ χ ] / ∈ Σ ( G ) as required.II. Now we consider the general case when χ might have 0 values on some verticesof Γ. In this case we consider b Γ = L F ( χ ) the graph obtained from Γ by deleting allvertices v with χ ( v ) = 0 and all edges that have a vertex v as above. Consider theepimorphism π : G = G Γ → b G = G b Γ that is identity on v ∈ V ( b Γ) and sends v ∈ V (Γ) \ V ( b Γ) to 1. Let b χ : b G → Z be thecharacter induced by χ and µ : b G/ b G ′′ → Z be the character induced by b χ . Thenby case I we have [ µ ] / ∈ Σ ( b G/ b G ′′ ) and [ b χ ] / ∈ Σ ( b G ) . Since b G/ b G ′′ is a quotient of G/G ′′ , for the character χ : G/G ′′ → Z induced by χ we have by Lemma 2.3 that(5.4) [ χ ] / ∈ Σ ( G/G ′′ ) . Applying again Lemma 2.3 for the canonical epimorphism G → G/G ′′ we obtainthat [ χ ] / ∈ Σ ( G ) . Proofs of Proposition B and Corollary C Proof of Proposition B. Let X be the set of vertices of the graph Γ. Sincethe Σ -Conjecture for Artin groups holds for G we have that [ χ ] / ∈ Σ ( G ) if andonly if L ( χ ) is either disconnected or not dominant in Γ. In the latter case it meansthat for X = { x ∈ X | χ ( x ) = 0 } and X = { x ∈ X | χ ( x ) = 0 } there is a vertex x ∈ X that is not connected by an edge with any vertex from X .Define K = { Y | Y is a subset of X such that there is an element of Y that is not linked by an edge with any element of X \ Y } . Define ∆( Y ) = { [ χ ] ∈ S ( G ) | χ ( Y ) = 0 } . N THE Σ -INVARIANT OF EVEN ARTIN GROUPS 11 Note that if [ χ ] ∈ ∆( Y ) for some Y ∈ K then Y ⊆ { v ∈ X | χ ( v ) = 0 } andthere is an element of Y that is not linked by an edge with any element of { v ∈ X | χ ( v ) = 0 } ⊆ X \ Y . Thus L ( χ ) is not dominant in Γ and hence ∪ Y ∈ K ∆( Y ) ⊆ Σ ( G ) c = S ( G ) \ Σ ( G ) . Now we analyse the condition that L ( χ ) is disconnected. We define N = { ( Y , e , . . . , e m ) | Y ⊂ X, e , . . . , e m are edges with even labels bigger than 2 in thecomplete subgraph L of Γ spanned by X \ Y and L \{ e , . . . , e m } is disconnected } . Define for ( Y , e , . . . , e m ) ∈ N ∆ ( Y , e , . . . , e m ) = { [ χ ] ∈ S ( G ) | χ ( Y ) = 0 , χ ( y ) = 0 for y ∈ X \ Y ,χ ( u i v i ) = 0 , where u i , v i are the vertices of e i , ≤ i ≤ m } and ∆( Y , e , . . . , e m ) = { [ χ ] ∈ S ( G ) | χ ( Y ) = 0 ,χ ( u i v i ) = 0 , where u i , v i are the vertice of e i , ≤ i ≤ m } . Since the Σ -Conjecture for Artin groups holds for G ∪ ( Y ,e ,...,e m ) ∈ N ∆ ( Y , e , . . . , e m ) ⊆ Σ ( G ) c . Since Σ ( G ) c is a closed subset of S ( G ) we deduce that ∪ ( Y ,e ,...,e m ) ∈ N ∆( Y , e , . . . , e m ) ⊆ Σ ( G ) c . Using again that the Σ -Conjecture for Artin groups holds for G we haveΣ ( G ) c = ( ∪ ( Y ,e ,...,e m ) ∈ N ∆ ( Y , e , . . . , e m )) ∪ ( ∪ Y ∈ K ∆( Y )) ⊆ ( ∪ ( Y ,e ,...,e m ) ∈ N ∆( Y , e , . . . , e m )) ∪ ( ∪ Y ∈ K ∆( Y )) ⊆ Σ ( G ) c . Thus Σ ( G ) c = ( ∪ ( Y ,e ,...,e m ) ∈ N ∆( Y , e , . . . , e m )) ∪ ( ∪ Y ∈ K ∆( Y ))and by construction each ∆( Y , e , . . . , e m ) and ∆( Y ) is of the type S ( G, H ) forappropriate subgroups H of G .6.2. Proof of Corollary C. I. Let [ χ ] ∈ S ( G ) and χ be the character of G/G ′′ induced by χ . By Lemma 2.3 if [ χ ] ∈ Σ ( G ) then [ χ ] ∈ Σ ( G/G ′′ ).II. Suppose now that [ χ ] / ∈ Σ ( G ). We aim to show that [ χ ] / ∈ Σ ( G/G ′′ ). ByTheorem A either L ( χ ) is disconnected or L ( χ ) is not dominant in Γ. We considerseveral cases.1) We suppose first that L ( χ ) is not dominant in Γ. Then there is a vertex t ofΓ such that χ ( t ) = 0 and t is not connected by an edge with any vertex of L ( χ )i.e. with any vertex v in Γ with χ ( v ) = 0. Let S be the graph obtained from Γ bydeleting all vertices w of Γ and the edges attached to them such that w = t and χ ( w ) = 0. Thus the set of vertices of S is V ( L ( χ )) ∪ { t } . We write V for V ( L ( χ )).Furthermore in S all vertices from V are linked by edges labelled by 2. We define Q as the free abelian group with a basis V , and consider the epimorphism ofgroups π : G = G Γ → G S = Q ∗ h t i that is the identity on the vertices of S and sends all vertices w to 1, where w = t and χ ( w ) = 0. Here h t i is infinite cyclic. We set Q = G S /G ′ S ≃ Q ⊕ Z . Since Ker( π ) ⊆ Ker( χ ) we deduce that χ induces a character of G S that we denoteby χ S . Finally it remains to show that [ χ S ] ∈ Σ ( G S /G ′′ S ) c and applying againLemma 2.3 for the epimorphism G/G ′′ → G S /G ′′ S induced by π we can deduce that[ χ ] ∈ Σ ( G/G ′′ ) c .We will show that Σ ( G S /G ′′ S ) c = S ( G S /G ′′ S ). By [11] [ µ ] ∈ Σ ( G S /G ′′ S ) if andonly if for every prime ideal P of Z Q minimal subject to I ⊆ P we have that Z Q/P is finitely generated as Z Q µ -module, where µ is the character of G S /G ′ S inducesby µ , I = ann Z Q ( G ′ S /G ′′ S ) and we view G ′ S /G ′′ S as Z Q -module with Q acting onthe left by conjugation. Thus if we show that I = 0 we are done, since in this case P = 0 and Z Q/P ≃ Z Q is not finitely generated as Z Q µ -module for any non-zeroreal character µ .Note that G ′ S /G ′′ S is generated as Z Q -module by w ( q i ) := t − q − i tq i for 1 ≤ i ≤ m , where V = { q , . . . , q m } and has relations that come from the Hall-Wittidentity (where all commutators are left normed)1 = [ q i , t − , q j ] t . [ t, q − j , q i ] q j . [ q j , q − i , t ] q i = [ q i , t − , q j ] t . [ t, q − j , q i ] q j Since [ q i , t − ] = q − i tq i t − = tw ( q i ) t − we have[ q i , t − , q j ] t = ( tw ( q i ) − t − q − j tw ( q i ) t − q j ) t = w ( q i ) − t − q − j tw ( q i ) t − q j t = w ( q i ) − w ( q j ) q − j w ( q i ) q j w ( q j ) − = w ( q i ) − w ( q j )( w ( q i ) q j ) w ( q j ) − . Since [ t, q − j ] = t − q j tq − j = q j w ( q j ) − q − j we have[ t, q − j , q i ] q j = ( q j w ( q j ) q − j q − i q j w ( q j ) − q − j q i ) q j =( w ( q j ) q − j ( w ( q j ) q − j q i ) − ) q j = w ( q j )( w ( q j ) q − j q i q j ) − . Then in G S /G ′′ S since [ w ( q k ) , w ( q s ) q p ] = 1 and [ q k , q s ] = 1 for 1 ≤ k, s, p ≤ m wehave1 = [ q i , t − , q j ] t . [ t, q − j , q i ] q j = w ( q i ) − w ( q j )( w ( q i ) q j ) w ( q j ) − w ( q j )( w ( q j ) q − j q i q j ) − =(6.1) w ( q i ) q j .w ( q i ) − w ( q j )( w ( q j ) q i ) − = ( q − j w ( q i )) .w ( q i ) − .w ( q j ) . ( q − i w ( q j )) − . Recall that we view G ′ S /G ′′ S as a left Z Q -module via conjugation and it is generatedas Z Q -module by w ( q ) , . . . , w ( q m ) subject only to the relations (6.1) since G S isthe free product of Q and h t i . Then (6.1) written additively as an element of P i Z Qw ( q i ) becomes 0 = ( q − j − w ( q i ) − ( q − i − w ( q j ) . Then in the localisation Z Q [ q − , . . . , q m − ] ⊗ Z Q ( G ′ S /G ′′ S ) we have (after omiting ⊗ ) w ( q i ) = 1 q − j − q − i − w ( q j ) , hence Z Q [ q − , . . . , q m − ] ⊗ Z Q ( G ′ S /G ′′ S ) is a cyclic Z Q [ q − , . . . , q m − ]-module and Z Q [ 1 q − , . . . , q m − ⊗ Z Q ( G ′ S /G ′′ S ) ≃ Z Q [ 1 q − , . . . , q m − . Finally since the ring homomorphism Z Q → Z Q [ q − , . . . , q m − ] induced by theidentity on Q and Z is injective we deduce that I = 0.2) Suppose now that L ( χ ) is disconnected and χ is a discrete character. Thenin (5.4) from the proof of Theorem A we showed that [ χ ] / ∈ Σ ( G/G ′′ ). N THE Σ -INVARIANT OF EVEN ARTIN GROUPS 13 3) Finally suppose that L ( χ ) is disconnected without any restrictions on χ .By Theorem A and Proposition B Σ ( G ) c is a rationally defined polyhedron. Inparticular since [ χ ] ∈ Σ ( G ) c there is a sequence of discrete characters χ i of G such that [ χ ] is the limit of the sequence [ χ i ] in S ( G ) and [ χ i ] ∈ Σ ( G ) c . Thensince the Σ - Conjecture for Artin groups holds for G we have that either L ( χ i )is disconnected or L ( χ i ) is not dominant in Γ. Let χ i be the character of G/G ′′ induced by χ i . Then by part 1) and 2) [ χ i ] ∈ Σ ( G/G ′′ ) c . Since Σ ( G/G ′′ ) c is aclosed subset of S ( G/G ′′ ) and [ χ ] is the limit of the sequence [ χ i ] in S ( G/G ′′ ) wededuce that [ χ ] ∈ Σ ( G/G ′′ ) c . 7. Some examples Finally we remark why the method used in this paper generally does no applyfor arbitrary Artin groups or even Artin groups that do not satisfy the assumptionsof Theorem A. It boils down to the fact that in general the maximal metabelianquotient G/G ′′ does not contain sufficient information to calculate Σ ( G ).1) Let Γ be a graph with vertices V ∪ W , where V = { v, s } , W = { u, w } , allvertices are linked by an edge and 2 m xy denotes the label of the edge that links x and y . We set m v,s = m u,w = 1 , m u,v = m v,w = m w,s = 2 , m s,u = 3 . Consider the character χ : G = G Γ → R that sends V to 1 and W to -1. Let χ be the character of G/G ′′ induced by χ and χ be the character of G/G ′ inducedby χ . We will show that Ker( χ ) is finitely generated though the Σ -Conjecture forArtin groups predicts that [ χ ] , [ − χ ] / ∈ Σ ( G ), hence if this prediction holds Ker( χ )is not finitely generated but we do not know whether this is the case. Justifyingthis by concrete calculation is surprisingly hard.To calculate G/G ′′ we use the previous calculations from the proof of TheoremA and we deduce that G ′ /G ′′ is generated as Z Q -module by e v ∧ e u , e v ∧ e w , e s ∧ e u , e s ∧ e w subject to the relations( v − e s ∧ e u − ( s − e v ∧ e u = 0 = ( v − e s ∧ e w − ( s − e v ∧ e w , ( w − e s ∧ e u − ( u − e s ∧ e w = 0 = ( w − e v ∧ e u − ( u − e v ∧ e w and(1 + uv ) e v ∧ e u = 0 = (1 + vw ) e v ∧ e w , (1 + us + ( us ) ) e s ∧ e u = 0 = (1 + sw ) e s ∧ e w . By [11] the Σ ( G/G ′′ ) depends on the minimal prime ideals P above I = ann Z Q ( A ),where A = G ′ /G ′′ and Q = G/G ′ . More precisely [ χ ] ∈ Σ ( G/G ′′ ) if and only if forevery prime ideal P of Z Q minimal subject to I ⊆ P we have that Z Q/P is finitelygenerated as Z Q χ -module.If I = ann Z Q ( e v ∧ e u ) , I = ann Z Q ( e v ∧ e w ) , I = ann Z Q ( e s ∧ e u ) , I = ann Z Q ( e s ∧ e w ) then I I I I ⊆ I ∩ I ∩ I ∩ I = I ⊆ P hence for some i we have I i ⊆ P . Here we consider the case i = 1. The other casesare similar. Note that1 + uv, (1 + su + ( su ) )( s − , (1 + vw )( w − , (1 + sw )( s − w − ∈ I ⊆ P. If s − , w − / ∈ P , since P is a prime ideal we get1 + uv, su + ( su ) , vw, sw ∈ P. But the ideal in Z Q -generated by 1 + uv, su + ( su ) , vw, sw is the wholering Z Q , a contradiction. Thus we can assume that s − w − P . Since χ ( s ) = 0 , χ ( w ) = 0 we deduce that Z Q/P is finitely generated as Z Q χ -module and as Z Q − χ -module. This holds for any proper prime ideal P above I and similarly for any proper prime ideal P above I i , hence for any proper primeideal P above I . Then [ χ ] , [ − χ ] ∈ Σ ( G/G ′′ ), so Ker( χ ) is finitely generated.2) Consider the graph Γ that is triangle with labels 3,4 and 6. By [2] the Σ -Conjecture for Artin groups holds for G = G Γ , hence Σ ( G ) c = ∅ , hence G ′ is notfinitely generated. We can do calculations with Fox derivatives similar to the onesfrom Section 3 and show that in this case G ′ /G ′′ is a finitely generated abeliangroup though G ′ is not finitely generated. References [1] Almeida, K.; The BNS-invariant for Artin groups of circuit rank 2 , Journal of Group Theory,v. 21, p. 189 - 228, 2018.[2] Almeida, K.; Kochloukova, D.; The Σ -invariant for Artin groups of circuit rank 1 , ForumMathematicum, v. 27, p. 2901 - 2925, 2015.[3] Almeida, K.; Lima, F. ; Finite Graph Product Closeness for a Conjecture on the BNS-invariant of Artin Groups , preprint[4] Bestvina, M.; Brady, N.; Morse theory and finiteness properties of groups , Invent. Math. 129,no. 3 (1997), 445 - 470.[5] Bieri, R.; Homological Dimension of Discrete Groups, The sigma invariants of Thompson’s group F ,Groups Geom. Dyn. 4 (2010), no. 2, 263 - 273.[7] Bieri R.; Groves, J.; The geometry of the set of characters induced by valuations, J. reineangew. Math., Band 347, 1984, S. 168 - 195.[8] Bieri, R.; Neumann, W. D.; Strebel, R.; A geometric invariant of discrete groups , Invent.Math. 90 (1987), no. 3, 451 - 477.[9] Bieri, R.; Renz B.; Valuations on free resolutions and higher geometric invariants of groups ,Comment. Math. Helv. 63 (1988), 464-497.[10] Bieri, R.; Strebel, R.; Valuations and finitely presented metabelian groups , Proc. LondonMath. Soc. (3)41 (1980), 439 - 464.[11] Bieri, R.; Strebel, R.; A geometric invariant for modules over an abelian group , J. ReineAngew. Math. 322 (1981), 170 - 189.[12] Brown, K. S.; Cohomology of Groups, Springer, 1982.[13] Blasco-Garcia, R.; Cogolludo-San Agustn, J. I.; Martinez-Perez, C.; On the Sigma invariantsof even Artin groups of FC type, in preparation.[14] Blasco-Garcia, R.; Martinez-Perez, C.; Paris, L.; Poly-freeness of even Artin groups of FCtype ,Groups Geometry and Dynamics, (2019), 13 (1), 309 - 325.[15] Delzant, T.; L’invariant de BieriNeumannStrebel des groupes fondamentaux desvari´et´esk¨ahl´eriennes, Math. Ann. 348, 119 - 125 (2010).[16] Funke, F.; Kielak, D.; Alexander and Thurston norms, and the Bieri-Neumann-Strebelinvariants for free-by-cyclic groups , Geom. Topol., 22 (2018), 2647 - 2696.[17] Kielak, D.; The Bieri-Neumann-Strebel invariants via Newton polytopes , Inventiones Math.,219 (2019), 1009 - 1068.[18] Kochloukova, D. H.; On the Σ -invariants of the generalised R. Thompson groups of type F ,Journal of Algebra 371, 430 - 456.[19] Kochloukova, D. H., On subdirect products of type F P m of limit groups, J. Group Theory13,(2010), 1 - 19.[20] Kochloukova, D.; Lima, F.; On the Bieri-Neumann-Strebel-Renz invariants of residually freegroups , to appear in Proc. of Edinburgh Math. Soc.[21] Lodha, Y.; Zaremsky, M.; The BNSR-invariants of the LodhaMoore groups, and an exotic-simple group of type F ∞ , arXiv:2007.12518 N THE Σ -INVARIANT OF EVEN ARTIN GROUPS 15 [22] Meier, J.; Geometric invariants for Artin groups , Proc. London Math. Soc. (3) 74 (1997)151-173.[23] Meier, J.; Meinert, H.; van Wyk, L.; On the Σ -invariants of Artin Groups , Topology and itsApplications, 110 (2001), 71-81.[24] Meier, J.; Meinert, H.; vanWyk, L.; Higher generation subgroup sets and the Σ -invariants ofgraph groups , Comment. Math. Helv. 73 (1998) 22-44.[25] Meinert, H.; The homological invariants for metabelian groups of finite Pr¨uer rank: a proofof the Σ m -conjecture , Proc. London Math. Soc. (3) 72 (1996), no. 2, 385 - 424.[26] Mendon¸ca, L.; On the Sigma-invariants of wreath products , Pacific Journal of Mathematics;v. 298, ( 2019) n. 1, p. 113-139.[27] Orlandi-Korner, L.A.; The BieriNeumannStrebel invariant for basis-conjugating automor-phisms of free groups, Proc. Am. Math. Soc. 128, (2000), 1257 - 1262.[28] Renz, B.; Geometrische Invarianten und Endlichkeitseigenschaften von Gruppen . Disserta-tion, Frankfurt/M. 1988.[29] Thurston, W. P.; A norm for the homology of 3-manifolds , Mem. Amer. Math. Soc.59(1986),99130.[30] Wall, C. T. C. ; Finiteness Conditions for CW-Complexes , Annals of Mathematics, 81(1965), 1, 56 - 69.[31] Witzel, S.; Zaremsky, M.; The Σ -invariants of Thompsons group F, via Morse theory , Topo-logical Methods in Group Theory, London Math. Soc. Lecture Note Ser. Vol. 251,173194,Cambridge University Press, 2018.[32] Zaremsky, M.; On the Σ -invariants of generalized Thompson groups and Houghton groups ,Int. Math. Res. Not. IMRN. Vol. 2017, Issue 19, 58615896.[33] Zaremsky, M.; On normal subgroups of the braided Thompson groups, Groups Geom. Dyn.12, (2018), 6592.[34] Zaremsky, Matthew C. B.; Symmetric automorphisms of free groups, BNSR-invariants, andfiniteness properties , Michigan Math. J. 67 (2018), no. 1, 133158., Michigan Math. J. 67 (2018), no. 1, 133158.