On the Sigma invariants of even Artin groups of FC-type

In this paper we study Sigma invariants of even Artin groups of FC-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong homological $n$-link condition for a graph $\Gamma$ and prove that it gives a sufficient condition for a character $\chi:A_\Gamma\to \mathbb{Z}$ to satisfy $[\chi]\in\Sigma^n(A_\Gamma,\mathbb{Z})$. This implies that the kernel $A^\chi_\Gamma=\ker \chi$ is of type $FP_n$. The homotopy counterpart is also proved. Partial results on the converse are discussed. We also provide a general formula for the free part of $H_n(A^\chi_\Gamma;\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $\chi$. This gives a characterization of when $H_n(A^\chi_\Gamma;\mathbb{F})$ is a finite dimensional vector space over $\mathbb{F}$. In the last version we correct a problem in the proof of Lemma 4.3 and also a remark at the end of subsection 3.3.


Introduction
The Sigma-invariants of a group G are certain sets Σ n (G, Z), Σ n (G) of equivalence classes of characters χ : G → R that provide information about the cohomological -in the case of Σ n (G, Z) -and homotopical -for Σ n (G) -finiteness conditions of subgroups lying over the commutator of G. The first version of these invariants was defined by Bieri and Strebel in [10] and the theory was later developed by Bieri-Neumann-Strebel [8], Bieri-Renz [9], and Renz [25]. Usually, it is extremely difficult to compute these invariants explicitly but there are some remarkable cases in which a full computation is available.
One of those cases occurs when G is a right-angled Artin group (RAAG for short). These groups are defined from a given finite graph Γ which will be assumed here to be simple, i.e., without loops or multiple edges between vertices. Associated to Γ one can describe the RAAG, denoted by A Γ , as the group generated by the vertices of Γ with relators of the form [v, w] = 1 for any edge {v, w} of Γ. This is a remarkable family of groups that range between finitely generated free abelian groups (corresponding to complete graphs) and finitely generated free groups (associated with graphs with no edges). Many properties of RAAGs can be determined in terms of the combinatorial properties of the graph. This is precisely the case for their Sigma-invariants, which were computed by Meier-Meinert-VanWyk in [21]. To describe their computation we will need to introduce some terminology.
We recall the concept of link in our context as follows. Fix a simple finite graph Γ as before and denote by V Γ (resp. E Γ ) its set of vertices (resp. edges). If Γ 1 ⊆ Γ is a subgraph and v ∈ V Γ , then the link lk Γ1 (v) of v in Γ 1 is defined as the full subgraph induced by V Γ1 (v) : We extend this definition for subsets ∆ ⊆ Γ by setting lk Γ1 (∆) = ∩ v∈∆ lk Γ1 (v).
We also recall the concept of the flag complex associated with Γ. This the simplicial complex, denoted asΓ, resulting after attaching a (k − 1)-simplex to each k-clique, i.e., to each complete subgraph of k vertices. We use the same notation for arbitrary graphs. Note that, if ∆ ⊆ Γ is a clique and Γ 1 ⊆ Γ a subgraph, thenlk Γ1 (∆) is the intersection withΓ 1 of the ordinary simplicial link of the cell σ associated to ∆, i.e., the subcomplex ofΓ 1 consisting of those simplices τ such that τ ∪ σ is also a simplex ofΓ 1 . Now, let χ : A Γ → R be a character and n ≥ 0 an integer. Consider the full subgraph L χ 0 induced by the vertices v of Γ with χ(v) = 0. Following Meier-Meinert-VanWyk [21], we call L χ 0 the living subgraph of Γ and say that vertices not in L χ 0 are dead. Dead vertices are also called resonant in [11]. We will say that the character χ satisfies the n-link condition if for any clique ∆ ⊆ Γ \ L χ 0 ,l k L χ 0 (∆) is (n − 1 − |∆|)-acyclic. Then Meier-Meinert-VanWyk proved (see Subsection 3.1) that χ ∈ Σ n (G, Z) if and only if χ satisfies the n-link condition. In fact, they also proved the homotopical version of this result that characterizes Σ n (G) in terms of a homotopical n-link condition (with "being (n − 1 − |∆|)connected" instead of "being (n − 1 − |∆|)-acyclic").
Here, we want to extend this result for another remarkable family: even Artin groups of FC-type. Given a finite simple graph Γ as above, one can consider an even labeling on the edges, that is, for any edge e = {u, v}, its label ℓ(e) is an even number. Any such even graph Γ defines an even Artin group A Γ generated by the vertices of Γ and whose relators have the form (uv) k = (vu) k , where ℓ(e) = 2k. These special Artin groups where first considered in detail in [12] and [13]. Note that any subgraph X ⊂ Γ of an even graph Γ generates an even Artin group A X . In addition, an even Artin group is said to have FC-type if A X is of finite type for each clique X ⊂ Γ: this means that the standard parabolic Coxeter group W X , i.e., the quotient of A X by the normal subgroup generated by u 2 ; u ∈ V X is finite. For a character χ on an even Artin group we consider a generalization of the living subgraph as follows (see [22]). Denote m v = χ(v), v ∈ V Γ and m e = m v + m w , e = {v, w} ∈ E Γ . We say that an edge is dead if e has label ℓ(e) > 2 and m e = 0. We will consider the subgraph L χ obtained from Γ after removing all dead vertices and the interior of all dead edges. Note that if all the edges have label precisely 2, i.e., for a RAAG, then L χ 0 = L χ . To state our first main result, we also need to introduce the clique poset, that is the poset of subgroups of A Γ which are generated by cliques of Γ: Note that the poset structure of P is the poset structure of the poset of cliques of Γ. Also, we allow ∆ to be empty, in that case A ∅ = {1}. So the geometric realization of the clique poset is the cone of the barycentric subdivision of the flag complex where the vertex of the cone corresponds to the empty clique.
A special role is played by the subset B χ ⊂ P of those subgroups A ∆ where ∆ ⊆ Γ is a clique such that for each vertex v in ∆ either v is dead or v ∈ e for e a dead edge in ∆. Observe that 1 = A ∅ ∈ B χ . We will see that this is equivalent to asking that the center of A ∆ lies in the kernel of χ, that is, χ(Z(A ∆ )) = 0. Definition 1.1. Let B χ ⊂ P be as above. Assume that for any A ∆ ∈ B χ with |∆| ≤ n the link lk L χ (∆) is (n − 1 − |∆|)-acyclic. Then we say that χ satisfies the strong n-link condition.
Note that the homotopical strong n-link condition implies the strong n-link condition.
Theorem 1.2. Let G = A Γ be an even Artin group of FC-type, and 0 = χ : G → R a character such that the strong n-link condition holds for χ. Then [χ] ∈ Σ n (G, Z).
In the case n = 1, it is known for several types of Artin groups (see Theorem 3.6) that [χ] ∈ Σ n (G, Z) if and only if L χ is connected and dominant. This is equivalent to saying that χ satisfies the strong n-link condition (see Subsection 3.1).
We do not know whether the converse of Theorem 1.2 is true in general, but in Section 5 we prove a partial converse. To do that, we use some of the techniques of [11] to perform computations on the homology groups H n (A χ Γ , F) where F is a field, χ : A Γ → Z is assumed to be discrete and A χ Γ = ker χ. More precisely, we show that these homology groups are finite dimensional as F-vector spaces if and only if certain p-local version of the strong n-link condition holds. Recall that a consequence of the well-known properties of the Sigma-invariants (see Section 2) is that if for a discrete character χ we have [χ] ∈ Σ n (G, Z), then A χ Γ is of type FP n and therefore the homology groups with coefficients over any field must be finite dimensional. As a by-product, an explicit computation of independent interest is provided in Theorem 5.5 for the free part of H n (A χ Γ , F) when seen, via χ, as a module over the principal ideal domain F[t ±1 ]. Moreover, section 6 is devoted to stating and proving a partial homotopic analogue of Theorem 1.2 in Theorem 6.1.

Sigma-invariants
Let G be a finitely generated group. In this section we will consider arbitrary non-trivial characters χ : G → R. We say that two characters χ 1 , χ 2 are equivalent if one is a positive scalar multiple of the other, i.e., if χ 1 = tχ 2 for some t > 0. We denote by [χ] the equivalence class of the character χ and by S(G) the set of equivalence classes of characters. Note that if G/G ′ has finite torsion and free rank r then S(G) can be identified with the sphere S r−1 . The homological Σ-invariants of G are certain subsets which are very useful to understand the cohomological finiteness properties of subgroups of G containing the commutator G ′ .
For a formal definition, consider χ : G → R a character and G χ the monoid G χ = {g ∈ G | χ(g) ≥ 0}. Then There is also a homotopical version We can sketch the definition as follows (see [21]). Let G be a group of type F n . We can choose a CW -model X for the classifying space for G with a single 0-cell and finite n-skeleton. Let Y be the universal cover of X. Then we may identify G with a subset of Y and given a character χ : G → R, we can extend χ to a map χ : Y → R that we denote in the same way. To do that, map the vertex labeled by, say, g to χ(g) and extend linearly to the rest of Y . For and we say that [χ] ∈ Σ n (G) if there is some a such that this map is trivial for all i < n. The reader can find more details about Σ n (G, Z), Σ n (G) in [21]. We recall now only two well-known properties: both Σ n (G, Z), Σ n (G) are open subsets of S(G) that determine the cohomological and homotopical finiteness conditions of subgroups containing the commutator thanks to the following fundamental Theorem: Moreover, if G is of type F n , then N is of type F n if and only if In particular, if χ : G → Z is discrete, we have that ker χ is of type FP n if and only if If R is a commutative ring, one can also define R-Sigma-invariants Σ n (G, R) by substituting the homology groups in the definition above by homology groups with coefficients in R. Theorem 2.1 remains true when FP n is substituted by FP n over R. Moreover we have for any G, R and n ≥ 2 and We will also need the following useful result. Finally, we state here the following result which was proven by Meier-Meinert-VanWyk [21]. This Theorem will be the main tool in the proof of Theorem 1.2 as it was one of the main tools in their description of the Sigma-invariants for RAAGs.

Theorem 2.3. [21, Theorem 3.2]
Let G be a group acting by cell-permuting homeomorphisms on a CW -complex X with finite n-skeleton modulo G. Let χ : G → R be a character such that for any 0 ≤ p ≤ n and any p-cell σ of X the stabilizer G σ is not inside ker χ. Then

Artin groups and their Sigma-invariants
As we have seen in the introduction, Artin groups can be defined in terms of a labeled graph. Using the symmetry of the standard presentation of an Artin group we can show the following.
Proof. Due to the symmetry of the relations in G, there is a well-defined map ϕ : G → G given as ϕ(v) := v −1 for v ∈ V Γ , which defines an automorphism of G. Since χ • ϕ = −χ and both Σ n (G) and Σ n (G, Z) are invariant under automorphisms of G, the result follows.

So we have:
Lemma 3.2. Let G = A Γ be an Artin group and χ : G → Z a discrete character. Then ker χ is of type FP n if and only if [χ] ∈ Σ n (G, Z). In particular, if there is some field F and some Proof. The first statement is a direct consequence of Theorem 2.1 and Proposition 3.1. For the second one, recall that if a group is of type FP n , then after tensoring a finite type resolution of the trivial module by F, one obtains a finite type resolution of projective modules over its group ring and thus it is also FP n over F.

3.1.
Sigma-invariants for RAAGs. The explicit computation of the Sigma-invariants for a particular group is usually very difficult. In [23], Meier and VanWyk computed Σ 1 (A Γ ) for A Γ a RAAG: Recall that L χ 0 is the subgraph obtained from Γ by removing the vertices v with χ(v) = 0. As we are assuming that A Γ is a RAAG, L χ 0 = L χ is the living subgraph defined above. Also, we say that a subgraph ∆ ⊆ Γ is dominating if for any v ∈ Γ \ ∆ there is some w ∈ ∆ linked to v. In other words, the condition of L χ 0 being dominant is equivalent to saying that for every And therefore the Theorem can be reformulated as follows: This can be restated using the 1-link condition defined in the introduction: [χ] ∈ Σ 1 (G) if and only if χ satisfies the homotopical 1-link condition.
Later on, in [21] Meier-Meinert-VanWyk, extending Theorem 3.3, were able to give a full description of the higher Sigma-invariants of a RAAG in terms of the n-link condition.
and only if the n-link condition holds for χ.

3.2.
Some partial results for Artin groups. Not much is known about Σ-invariants of general Artin groups. Let χ : A Γ → R, A Γ an Artin group and Z(S) the center of S ⊂ A Γ . We highlight the following partial result.
We will see below that the hypothesis χ(Z(A ∆ )) = 0 for any ∅ = ∆ ⊆ Γ clique means that B χ consists of the trivial subgroup only. Therefore this result is a particular case of our main Theorem 1.2.
A full characterization is available in few cases only. [3,4], Kochloukova [20]). Assume that one of the following conditions holds: • Γ is a connected tree, • Γ is connected and π 1 (Γ) is free of rank at most 2, • Γ is even and whenever there is a closed reduced path in Γ with all labels bigger than 2, then the length of such path is always odd.
Moreover, the class of Artin groups that satisfy the hypothesis in Theorem 3.6 is known to be closed under graph products and, as a consequence, every FC-type Artin group also does ( [5]).
Other concrete examples of Artin groups satisfying this hypothesis can be found in [4] and [1].
Note that by a similar observation as above, this result can be stated as follows: For Γ connected and with π 1 (Γ) = 1 or free of rank at most 2, then Observe also that here we are not assuming that A Γ is even.

3.
3. An easier particular case: direct products of Artin dihedral groups. It will be important below to understand the Sigma-invariants of the finite type Artin subgroups A ∆ of a given even Artin group of FC-type A Γ . In general, finite type Artin groups are direct products of finite type irreducible Artin groups and the only possible irreducible finite type Artin groups are those of dihedral type, which are the Artin groups associated to a single edge. In the even case the edge is labeled by an even integer, say 2ℓ and the associated group is The (homotopical) Sigma-invariants for irreducible Artin groups of finite type have been described in [22, Section 2]. In the particular case of a dihedral Artin group we have the following result.
Proof. For the homotopical result, see [22, p. 76]. For the homological one note that Σ 1 (G) = Σ 1 (G, R) and If A Γ is an even Artin group of FC-type and A ∆ is a finite type subgroup with ∆ ⊆ Γ, then ∆ must be a clique and A ∆ is a direct product of even Artin dihedral groups and possibly a factor which is free abelian of finite rank. In this subsection we will give a full description of the Sigma-invariants of such an A ∆ . But we will consider the slightly more general case of a product of arbitrary Artin dihedral groups and possibly a free abelian groups of finite rank. Assume Using [7, Theorem 1.4] and the fact that according to Lemma 3.7 the R-Sigma-invariants for Artin dihedral groups for R = Z and R = Q coincide, we deduce (the upper script c means the complementary of the corresponding subset) where ⋆ is the join product in the corresponding spheres (see [7]). We have where χ i maps the standard generators of G i to 1 and −1 resp. As a consequence, if we order the factors so that G 1 , . . . , G t are precisely those which are dihedral of even type, and denote by v t+1 , . . . v s the vertices not involved in dihedral type edges, we have Note that χ(Z(G)) = 0 is equivalent to m e = 0 for e ∈ E Γ if ℓ(e) > 2 and χ(v i ) = 0 for t + 1 ≤ i ≤ s. For m < t, Σ m (G, Z) c consists of characters that vanish in all the vertices v i for t+1 ≤ i ≤ s and in at least t − m of the factors G i for 1 ≤ i ≤ t, having zero m e -value in the rest.
3.4. Coset posets. In this section we prove some results on coset posets that will be used in the main proofs later on.
Definition 3.8. Let G be a group and P a poset (ordered by inclusion) of subgroups of G. The coset complex C G (P) (or simply C(P) if the group G is clear by the context) is the geometric realization of the poset GP of cosets gS where g ∈ G and S ∈ P. In other words, it is the geometric realization of the simplicial complex whose k-simplices are the chains where g 0 , . . . , g k ∈ G and S 0 , . . . , S k ∈ P. Let P χ be the subposet of P consisting of those subgroups S ∈ P such that χ| S = 0. It yields a subcomplex C(P) χ of C(P). We identity C(P) with its geometric realization.
We will consider posets of subgroups P having a height function h : P → Z + ∪ {0} such that whenever S T both sit in P, we have h(S) < h(T ). We also assume that there is a bound for the height of the elements of P. We denote that bound by h(P). Now, assume we have a subposet H ⊆ P. Then C(H) is a subcomplex of C(P). We want to compare the homology of C(P) with the homology of C(H). Let be a k-simplex in C(P). Using the height function h we set . This implies that we can use this function to define a subcomplex For s ≥ 0, simplices in D s but not in D s−1 are of the form such that there is some 0 ≤ i ≤ k with S i ∈ P \ H of height precisely s and S j ∈ H for j > i.
Fix an S ∈ P \ H of height precisely s and consider the set of all simplices of the form with S i = S and S j ∈ H for each j > i. Those simplices lie in D s . The boundary ∂σ of such a k-simplex σ consists of k − 1-simplices τ which are of the same form except of the case when and then τ ∈ D s−1 . Consider now the complex Z S which is the geometric realization of C S (P S )\ {S} for the poset P S = {S ∩ T | T ∈ P} (note that Z S could be empty if S is empty). And let J S be the poset The join Z S ⋆ |J S | is in a natural way an S-space and we may form the induced G-space . Its chain complex is a G-complex that consists of induced modules of the form C • (Z S ⋆ |J S |) ↑ G S . We claim that the quotient complex D s+1 /D s can be decomposed as whereC •+1 is the augmented chain complex shifted by one. To see it, consider a simplex σ as in (2) and put g = xy for y ∈ S so that σ : x(yS 0 ⊆ · · · ⊆ yS i−1 ⊆ S ⊆ S i+1 ⊆ · · · ⊆ S k ).
Proof. Using Mayer-Vietoris one can determine the homology groups H r (Z S ⋆ |J S |) in terms of H i (Z S ) and H j (|J S |) for i + j < r so the hypothesis imply that the complex Z S ⋆ |J S | is By equation (3), for 0 ≤ j ≤ n. And this implies the result: to see it assume by induction that D s+1 is R-(n − 1)acyclic (the beginning of the induction is D h(P) = C(P) which is (n − 1)-acyclic) and consider the long exact homology sequence of the short exact sequence of complexes 0 Thus also D s is (n − 1)-acyclic. Eventually, C(H) = D 0 is (n − 1)-acyclic.
Remark 3.10. As noted by an anonymous referee of this paper, Lemma 3.9 can also be proven using Morse theory, having h H c as the Morse function. The complex Z S ⋆ |J S | is the link of S in D s+1 so the hypothesis of Lemma 3.9 is in fact a condition on the acyclicity of the link. The proof presented in this paper exhibits the fact from Morse theory that acyclicity of the links yields isomorphisms between the homology groups of the involved subcomplexes.
Remark 3.11. In Lemma 3.9 we can substitute the instances of "being acyclic" by "being R acyclic" for any unital commutative ring R.

proof of Theorem 1.2
As stated above, our proof of Theorem 1.2 is based in Theorem 2.3. To do that we need a suitable complex X.
Let A Γ be an Artin group and consider the clique poset (recall that a clique is a complete subgraph). If the Artin group A Γ is of FC-type, then any clique of Γ generates a finite type Artin subgroup so the coset complex C G (P) = C(P) of P is the modified Deligne complex for A Γ considered by Charney and Davis in [16]. In [19] the modified Deligne complex is used to construct what is called the clique cube-complex which is a CAT-(0) cube complex.
For Artin groups of FC-type, the modified Deligne complex was shown to be contractible in [16] but for completeness, we give a direct easy proof of the fact that the coset complex C G (P) is contractible in general, i.e., for arbitrary Artin groups A Γ possibly without the FC condition. Proof. We argue by induction on the number of vertices of Γ. Observe first that the result is obvious if Γ is complete, because then G itself lies in P. If Γ is not complete we may decompose This decomposition induces a decomposition of G as the free product with amalgamation Let C(P 1 ), C(P 2 ) and C(P 0 ) be the corresponding coset complexes for the clique posets of A Γ1 , A Γ2 and A Γ0 . By induction we may assume that they are all contractible. Consider the poset The geometric realization of this poset is precisely the barycentric subdivision of the Bass-Serre tree associated to the free amalgamated product above. Consider the map Observe that each clique of Γ is a subgraph either of Γ 1 or of Γ 2 . Moreover, if gS ⊆ gT with S and T both in P, then f (gS) ⊆ f (sT ) so it is a well-defined poset map and for any gA Γi ∈ BS, i.e., it is the coset poset of the clique poset of A Γi shifted by g. By induction, the posets C AΓ i (P i ) have contractible geometric realizations for i = 0, 1, 2. So we may apply Quillen poset map Lemma ( [6]) and deduce that f induces a homotopy equivalence between the geometric realizations. As the geometric realization |BS| is contractible, we deduce the same for the geometric realization |C(P)|.
Note that the Artin group acts on the clique poset so we have a nice action on the geometric realization. However, this is not what we need to apply Theorem 2.3 because the stabilizers of this action are not nice enough. In order to construct our suitable X, we will also need an auxiliary Lemma.
Lemma 4.2. Let ∆ be a (non empty) complete graph with s vertices and with S := A ∆ even of FC-type and consider the simplicial complex Z S with simplices hS 0 ⊆ hS 1 ⊆ · · · ⊆ hS r for h ∈ A ∆ and each S j a special proper subgroup of A ∆ . Then Z S is (s − 2)-acyclic.
Proof. We will denote ∆ v = ∆ \ {v}. For each g ∈ S, v ∈ ∆ let gX v be the subcomplex of Z S induced by {gA Ω | Ω ⊆ ∆ v }. Obviously, the set {gX v | g ∈ S, v ∈ ∆} is a covering of Z S as each special proper subgroup of A ∆ is inside one of the form gA ∆v .
Observe that for any ∆ 1 ⊂ ∆, if g 1 , g 2 ∈ S are such that And if we have ∆ 1 , ∆ 2 and g 1 , g 2 ∈ S so that g 1 A ∆1 ∩ g 2 A ∆2 = ∅, then for some x ∈ A ∆1 , y ∈ A ∆2 , g 1 x = g 2 y so [18]). This implies that any set of pairwise distinct g 0 X v0 , . . . , g k X v k has empty intersection if two of the v i 's are equal and also that whenever such as set has non empty intersection, then the intersection is the subcomplex induced by {gA Ω | Ω ⊆ ∆ v1 ∩ . . . ∩ ∆ v k } which is contractible (it is the geometric realization of a poset with a maximal element). The last property implies that Z S is homotopy equivalent to the nerve of the covering (see [14, Chap. VII Theorem 4.4]) which we denote N ∆ . The nerve has k-simplices the sets {g 0 X v0 , . . . , g k X v k } with non empty intersection and the discussion above implies that the cardinality of such as set can be at most s so this nerve is a complex of dimension s − 1. To deduce that this complex is (s − 2)-connected one can use [17, (4.18), Theorem (2.15)], but it is also possible to give a direct argument as follows. As we are assuming that S is even of FC-type, we have a decomposition where each S i = A ∆i is either infinite cyclic or Artin dihedral (of even type). This means that for each g ∈ S and v ∈ ∆, gA ∆v = g i A ∆v for some g i ∈ S i where i is the index such that v ∈ ∆ i . From this one deduces that N ∆ is the join where each N ∆i is the nerve associated to the same construction but for S i . Moreover, if S i is infinite cyclic, obviously N Si is a disjoint union of points, thus non empty and if S i is Artin dihedral, then it is easy to see that N Si is 0-connected. As s = a + 2b where a is the number of S i 's which are infinite cyclic, we get the result.
. At this point, we are able to construct the desired X. Note that Theorem 2.3 together with the next result imply Theorem 1.2.
In the introduction we have defined the subposet B χ of P as those A ∆ for ∆ ⊆ Γ clique such that for each vertex v in ∆ either v is dead or v ∈ e for e a dead edge in ∆, this includes the case when ∆ = ∅. The hypothesis that A Γ is even of FC-type implies that A ∆ is a direct product of dihedral groups (corresponding to edges with label bigger than 2) and of infinite cyclic groups (generated by dead vertices). Taking into account that the center of the dihedral group generated by, say, x and y is the infinite cyclic group generated by xy we see that for such a ∆ we have χ(Z(A ∆ )) = 0. The converse is also obvious. So we have Let A σ ∈ L S . Then for T = A (σ⋆∆) , we have S ≤ T and T ∈ P. We claim that T ∈ J S . To see it, note that as σ = ∅ and σ ⊆ L χ , Z(A σ ) = 0. As Γ is even of FC-type, this implies that either there is some v ∈ σ, v ∈ Z(A σ ) with χ(v) = 0 or there are v, w ∈ σ, (vw) k ∈ Z(A σ ) for some k with χ(v) + χ(w) = 0. Again, the condition that Γ is of FC-type implies that in the second case, (vw) k ∈ Z(T ) so χ(Z(T )) = 0. In the first case, either v ∈ Z(T ) so again χ(Z(T )) = 0 or there is some w ∈ ∆ with (vw) k ∈ Z(T ) for some k. In this case moreover w ∈ Z(S) thus χ(w) = 0. Therefore χ((vw) k ) = 0 and again χ(Z(T )) = 0. The claim follows and therefore we have a well defined poset map f : L S → J S A σ → A (σ⋆∆) . We claim that this map induces a homotopy equivalence between the corresponding geometric realizations. To see it, let T ∈ J S and consider By Quillen's poset map Theorem (see [6]) it suffices to check that the poset f −1 ≤T has contractible geometric realization. Put T = A ν . Then ν is a clique of Γ such that ν = ∆ ⋆ σ for some ∅ = σ clique in lk Γ (∆). We can describe σ as a join σ = e 1 ⋆ · · · ⋆ e t ⋆ p 1 ⋆ . . . p s where each e i is a single edge with label > 2 and each p i is a single point and all the edges not in some e i are labeled by 2. We may order them so that χ(e 1 ), . . . , χ(e l ) = 0, χ(e l+1 ), . . . , χ(e t ) = 0, χ(p 1 ), . . . , χ(p r ) = 0, χ ( p r+1 ), . . . , χ(p s ) = 0. Then σ ∩ L χ = w 1 ⋆ · · · ⋆ w l ⋆ e l+1 ⋆ · · · ⋆ e t ⋆ p r+1 ⋆ · · · ⋆ p s where each w i is the disconnected set consisting of the two vertices of each e i has as barycentric subdivision precisely the geometric realization of the poset f −1 ≤T . As e l+1 ⋆ · · · ⋆ e t ⋆ p r+1 ⋆ · · · ⋆ p s is either contractible or empty, so show that f −1 ≤T is contractible we only have to show that e l+1 ⋆ · · · ⋆ e t ⋆ p r+1 ⋆ · · · ⋆ p s is not empty. As T ∈ J S , χ(Z(T )) = 0. If there is some v vertex of ν with v ∈ Z(T ) and χ(v) = 0 then the fact that χ(Z(S)) = 0 implies v ∈ σ so v belongs to {p r+1 , . . . , p s }. So we are left with the case when χ(vw) = 0 for v, w vertices of an edge of ν with label > 2. If, say, v lies in ∆, then v ∈ Z(S) so χ(v) = 0 and we deduce χ(w) = 0. Moreover, in this case the FC-condition implies that w is in the center of A σ , i.e., w belongs to {p r+1 , . . . , p s }. So we may assume that both v, w lie in σ so the edge joining them lies in the set {e l+1 , . . . , e t }.
We finish this section with a couple of example to illustrate how to apply Theorem 1.2.  Incidentally, in this case, the hypothesis of Corollary 5.7 below is satisfied for p = 2, hence the converse of Theorem 1.2 also holds true so

The free part of the homology groups of Artin kernels
Let A Γ be an even Artin group of FC-type and χ : A Γ → Z a discrete character. In this section, we are interested in the homology groups H n (A χ Γ , F) where F is a field of characteristic p (either zero or a prime). More precisely, we want to characterize when they are finite F dimensional and, more generally, to compute their free part when seen as F[t ±1 ]-modules via χ.
To do that we first develop a p-local version of some of the notions that we used above. We say that an edge e of Γ with label 2l e is p-dead if m e = χ(v) + χ(w) = 0 and p |l. In [11], p-dead edges were called F-resonant (recall that F is a field of characteristic p).
We set L χ p for the subgraph of Γ that we get when we remove dead vertices and open p-dead edges. This notation is consistent with L χ 0 because no edge can be 0-dead. Note that the set of dead edges is the union of the sets of p-dead edges where p runs over all prime numbers and therefore L χ = p prime L χ p .
Let B χ p be the set of those A ∆ ∈ P for a clique ∆ ⊆ Γ such that for each vertex v in ∆ either v is dead or v ∈ e for e a dead edge in ∆ (this includes the case when ∆ = ∅). Note that as edges which are p-dead for some p are dead, this condition implies that χ(Z(A ∆ )) = 0.
Definition 5.1. Assume that for any A ∆ ∈ B χ p with |∆| ≤ n the linklk L χ p (∆) is p−(n−1−|∆|)acyclic, meaning that its homology up to degree (n − 1 − |∆|) with coefficients in a field of characteristic p vanishes. Then we say that χ satisfies the strong p-n-link condition.
The homology groups H n (A χ Γ , F) are precisely the homology groups of the F-chain complex C n (Sal χ Γ ) described in [11,Section 2]. This complex was obtained using the χ-cyclic cover of the Salvetti complex of A Γ (see [26,15,24]) and has C n (Sal whereC(Γ) n−1 is the augmented chain complex of the flag complexΓ shifted by one. The differential of C n (Sal χ Γ ) can be described as follows (see [11], after Remark 2.3). For each edge e ∈ E Γ let 2l e be its label in Γ and denote q k (x) = (x k − 1)/(x − 1). Then for any X ⊆ Γ complete we have where we are denoting X v the clique obtained from X by removing v and In particular, ifl e = 1, then ql e (t me ) = 1 and if m e = 0, ql e (t me ) =l e . Recall that an edge e ∈ E Γ is called p-dead if m e = 0 and p |l e ; otherwise will be called p-living. So we see that in (4), the coefficient b v,X vanishes if either v is dead or belongs to a p-dead edge in X.
Let I be the augmentation ideal of the ring R = F[t ±1 ], i.e., the kernel of the augmentation map R → F with t → 1, R is a principal ideal domain and I is the ideal generated by t − 1.
Since I is a prime ideal, we can localize and get a new ring R I . We can also localize the complex C n (Sal χ Γ ) and get a new complex C n (Sal χ Γ ) I with n-term C n (Sal χ Γ ) I = R I ⊗ FC (Γ) n−1 whose differential we also denote by ∂ χ n . Since localizing is flat, the R-free part of the homology of (R⊗ FC (Γ) n−1 , ∂ χ n ) has the same rank as the R I -free part of the homology of (R I ⊗ FC (Γ) n−1 , ∂ χ n ). But in this complex we can normalize over the living vertices and p-living edges in the following way.
Let X ⊆ Γ be a clique and put Let µ X be the multiplicity of t − 1 as a factor of a X . Then where h X is a unit in our ring R I . Observe that for any v ∈ X we have We can choose an integer k such that k|X| ≥ µ X for any X ⊆ Γ clique. Let X ⊆ Γ be a clique and setσ Recall that the summand associated to each v ∈ X vanishes if either v is dead or it belongs to a p-dead edge in X. Otherwise, using (6) we see that that summand is, up to a sign, Hence, if we denote by F χ p (X) the subgraph that we get from X when we remove dead vertices and closed p-dead edges we have Observe that F χ p (X) ⊆ X ∩ L χ p where L χ p is the p-living subgraph defined in the introduction, i.e., the subgraph of Γ that we get when we remove dead vertices and open p-dead edges. Now, for each n letC n (Sal χ Γ ) I be the sub R I -module of C n (Sal χ Γ ) I generated by theσ X , |X| = n. The computations above imply that (C n (Sal χ Γ ) I , ∂ χ n ) is a subcomplex of (C n (Sal χ Γ ) I , ∂ χ n ) and by definition each quotient C n (Sal χ Γ ) I /C n (Sal χ Γ ) I is a R I -torsion module. Using the long exact homology sequence we see that the R I -free part of the homology of (C n (Sal χ Γ ) I , ∂ χ n ) equals the R I -free part of the homology of (C n (Sal χ Γ ) I , ∂ χ n ). So from now on we consider this last complex.
We define a new map d χ n :C n (Sal Lemma 5.2. With the notation above we have Proof. i) is obvious. For ii), take a ∈ im ∂ χ n . Then a = (t − 1)a 1 and a = ∂ χ n (b) so a 1 = d χ n (b) ∈ im d χ n so a = (t − 1)a 1 ∈ im d χ n . For iii), the fact that im ∂ χ n ⊆ im d χ n ∩ I kC n (Sal χ Γ ) I is obvious because of ii). Conversely, take a ∈ im d χ n ∩ I kC n (Sal χ Γ ) I . Then a = d χ n (b) and the fact that a ∈ I kC n (Sal χ Γ ) I together with the definition of d χ n implies that also b ∈ I kC n (Sal χ Γ ) I so b = (t − 1) k b 1 and ∂ χ n (b 1 ) = 1 (t−1) k ∂ χ n (b) = d χ n (b) = a thus a ∈ im ∂ χ n . Finally, iv) follows from iii). Example 5.9. Let G = DA 4 × DA 6 where DA 4 (resp. DA 6 ) is the dihedral Artin group associated to the edge with label 4 (resp. 6). Then G = A Γ , where Γ is a full graph with 4 vertices and two disjoint edges labeled with 4 and 6. Denote the standard generators of the factor DA 4 by v, w and the standard generators of the factor DA 6 by x, y and consider the character χ : G → Z induced by χ(v) = χ(x) = 1, χ(w) = χ(y) = −1. Taking into account the computation of the Sigma-invariants for this type of groups that we performed in Subsection 3.3, we see that [χ] ∈ Σ 2 (G, Z) so its kernel A χ Γ is not of type FP 2 . In fact G does not satisfy the strong 2-link condition. To see it, note that B χ = {∅, e 1 , e 2 } where e 1 = (v, w) and e 2 = (x, y) and L χ is a square with vertices v, x, w, y (that we get when we remove the interior of e 1 and e 2 from Γ). For X = ∅ ∈ B χ we have lk L χ (∅) = L χ . Then H 2−1−|X| (lk L χ (X)) = H 1 (L χ ) = Z = 0.
It is easy to see that also the strong 3-link condition fails: to check it consider for example X = e 1 , its link in L χ consists of the isolated vertices x and y.
We claim however that this χ does satisfy the strong p-n-link condition for each p (zero or a prime). As a consequence, for any field F, Assume first that p = 2. Then B χ 2 = {∅, e 1 } and L χ 2 is the graph obtained from Γ when we remove the open edge e 1 . Then lk L χ 2 (∅) = L χ 2 and lk L χ 2 (e 1 ) = e 2 and both associated flag complexes are contractible.
Example 5.10. Things are very different if we consider for example G 1 = DA 4 × DA 4 and χ as before. Then one easily checks that the strong 2-2-link condition fails so dim F H 2 (A χ Γ , F) is infinite.

The homotopic invariants
In this section we explain how to modify the statement of Theorem 1.2 to obtain the analogous homotopic result.
Basically, we have to change the hypothesis to the homotopic version. As we have said in Definition 1.1, we define the strong homotopic n-link condition as follows: Consider again the set B χ ⊂ P of those A ∆ in the clique poset such that χ(Z(A ∆ )) = 0. Assume that for any A ∆ ∈ B χ with |∆| ≤ n the link lk L χ (σ) is (n − 1 − |σ|)-connected. Then we say that χ satisfies the strong homotopic n-link condition.
Theorem 6.1. Let G = A Γ be an even Artin group of FC-type, and 0 = χ : G → R a character such that the strong homotopic n-link condition holds for χ. Then [χ] ∈ Σ n (G).
Proof. The proof follows that of Theorem 1.2 in its homotopic version. The homotopic analogue of Lemma 3.9 can be proved using relative homotopy groups, and the (n − 1)-connectedness of  MR 80918