Note on the residual finiteness of Artin groups

Theorem 1.1.

Let Γ be a Coxeter graph, let A=AΓ, and let P be an admissible partition of S such that

1. the group AX is residually finite for all X∈𝒫,

2. the Coxeter graph Γ/𝒫 is either even and triangle free, or a forest.

Then A is residually finite.

Corollary 1.2.

The following statements hold.

1. If Γ is even and triangle free, then A is residually finite.

2. If Γ is a forest, then A is residually finite.

Remark.

When Γ is a forest, the Artin group A is the fundamental group of a graph manifold with boundary by Gordon [10], and is thus virtually special by Przytycki and Wise [13], hence is linear and residually finite.

Remark.

Recall that the Coxeter groupW of Γ is the quotient of A by the relations s2=1, s∈S. We say that Γ is of spherical type if W is finite. We say that a subset X of S is free of infinity if ms,t≠∞ for all s,t∈X. We say that Γ is of FC type if for each free of infinity subset X of S the Coxeter graph ΓX is of spherical type. Artin groups of FC type were introduced by Charney and Davis [4] in their study of the K⁢(π,1) problem for Artin groups and there is an extensive literature on them. It is easily checked that any triangle free Coxeter graph (in particular any forest) is of FC type. So, all the groups that appear in Corollary 1.2 are of FC type. On the other hand, we know by Blasco-Garcia, Martinez-Perez and Paris [1] that all even Artin groups of FC type are residually finite (which gives an alternative proof to Corollary 1.2 (1)). The next challenge would be to prove that all Artin groups of FC type are residually finite. Another interesting challenge would be to prove that all three generators Artin groups are residually finite.

Theorem 2.1 (Boler and Evans [2]).

Let G1 and G2 be two residually finite groups and let L be a common subgroup of G1 and G2. Assume that both G1 and G2 split as semi-direct products G1=H1⋊L and G2=H2⋊L. Then G=G1∗LG2 is residually finite.

Lemma 2.2.

If a Coxeter graph Γ has one or two vertices, then A=AΓ is residually finite.

Proof.

If Γ has only one vertex, then A≃ℤ which is residually finite. Suppose that Γ has two vertices s,t. If ms,t=∞, then A is a free group of rank 2 which is residually finite. If ms,t≠∞, then Γ is of spherical type, hence, by Digne [8] and Cohen and Wales [6], A is linear, and therefore A is residually finite. ∎

Lemma 2.3.

Let Γ be a Coxeter graph and let A=AΓ. Let s∈S. Set Y=S∖{s}, denote by Γ1,…,Γℓ the connected components of ΓY and, for i∈{1,…,ℓ}, denote by Yi the set of vertices of Γi. If AYi∪{s} is residually finite for all i∈{1,…,ℓ}, then A is residually finite.

Proof.

We argue by induction on ℓ. If ℓ=1, then Y1∪{s}=S and AY1∪{s}=A, hence A is obviously residually finite. Suppose that ℓ≥2 and that the inductive hypothesis holds. We set X1=Y1∪⋯∪Yℓ-1∪{s}, X2=Yℓ∪{s} and X0={s}. Let G1=AX1, G2=AX2, and L=AX0≃ℤ. The groups G1 and G2 are residually finite by induction. It is easily seen in the presentation of A that A=G1∗LG2. Furthermore, the homomorphism ρ1:G1→L which sends t to s for all t∈X1 is a retraction of the inclusion map L↪G1, hence G1 splits as a semi-direct product G1=H1⋊L. Similarly, G2 splits as a semi-direct product G2=H2⋊L. We conclude by Theorem 2.1 that A is residually finite. ∎

Lemma 2.4.

Let Γ be a Coxeter graph, let A=AΓ, and let P be an admissible partition of S such that

1. the group AX is residually finite for all X∈𝒫,

2. the Coxeter graph Γ/𝒫 has at most two vertices.

Then A is residually finite.

Proof.

If |𝒫|=1, there is nothing to prove. Suppose that |𝒫|=2 and one of the elements of 𝒫 is a singleton. We set 𝒫={X,Y} where X=S∖{t} and Y={t} for some t∈S. If there is no edge in Γ connecting t to an element of X, then A=AX*AY, hence A is residually finite. So, we can assume that there is an edge connecting t to an element s∈X. Note that this edge is unique by the definition of admissibility. We denote by Γ1,…,Γℓ the connected components of ΓX∖{s} and, for i∈{1,…,ℓ}, we denote by Xi the set of vertices of Γi. For all i∈{1,…,ℓ} the group AXi∪{s} is residually finite since AXi∪{s}⊂AX. On the other hand, A{s,t} is residually finite by Lemma 2.2. Noticing that the connected components of ΓS∖{s} are precisely Γ1,…,Γℓ and {t}, we deduce from Lemma 2.3 that A is residually finite.

Now assume that |𝒫|=2 and both elements of 𝒫 are of cardinality ≥2. Set 𝒫={X,Y}. If there is no edge in Γ connecting an element of X with an element of Y, then A=AX*AY, hence A is residually finite. So, we can assume that there is an edge connecting an element s∈X to an element t∈Y. Again, this edge is unique. Let Ω1,…,Ωp be the connected components of ΓX∖{s} and let Γ1,…,Γq be the connected components of ΓY. We denote by Xi the set of vertices of Ωi for all i∈{1,…,p} and by Yj the set of vertices of Γj for all j∈{1,…,q}. The group AXi∪{s} is residually finite since Xi∪{s}⊂X for all i∈{1,…,p}, and, by the above, the group AYj∪{s} is residually finite for all j∈{1,…,q}. It follows by Lemma 2.3 that A is residually finite. ∎

Remark.

Alternative arguments from Pride [12] and/or from Burillo and Martino [3] can be used to prove partially or completely Lemma 2.4.

Lemma 2.5.

Let Γ be a Coxeter graph, let A=AΓ, and let P be an admissible partition of S such that

1. the group AX is residually finite for all X∈𝒫,

2. the Coxeter graph Γ/𝒫 is even and triangle free.

Then A is residually finite.

Proof.

We argue by induction on the cardinality |𝒫| of 𝒫. The case |𝒫|≤2 is covered by Lemma 2.4. So, we can suppose that |𝒫|≥3 and that the inductive hypothesis holds. Since Γ/𝒫 is triangle free, there exist X,Y∈𝒫 such that none of the elements of X is connected to an element of Y. We set U1=S∖X, U2=S∖Y, and U0=S∖(X∪Y). We have A=AU1*AU0AU2 and, by the inductive hypothesis, AU1 and AU2 are residually finite. Since Γ/𝒫 is even, the inclusion map AU0↪AU1 admits a retraction ρ1:AU1→AU0 which sends t to 1 if t∈Y and sends t to t if t∈U0. Similarly, the inclusion map AU0↪AU2 admits a retraction ρ2:AU2→AU0. By Theorem 2.1 it follows that A is residually finite. ∎

Lemma 2.6.

Let Γ be a Coxeter graph, let A=AΓ, and let P be an admissible partition of S such that

1. the group AX is residually finite for all X∈𝒫,

2. the Coxeter graph Γ/𝒫 is a forest.

Then A is residually finite.

Proof.

We argue by induction on |𝒫|. The case |𝒫|≤2 being proved in Lemma 2.4, we can assume that |𝒫|≥3 plus the inductive hypothesis. Set Ω=Γ/𝒫. Let Ω1,…,Ωℓ be the connected components of Ω. For i∈{1,…,ℓ} we denote by 𝒫i the set of vertices of Ωi and we set Yi=⋃X∈𝒫iX and Γi=ΓYi. The set 𝒫i is an admissible partition of Yi and Γi/𝒫i=Ωi is a tree for all i∈{1,…,ℓ}. Moreover, we have A=AY1*⋯*AYℓ, hence A is residually finite if and only if AYi is residually finite for all i∈{1,…,ℓ}. So, we can assume that Ω=Γ/𝒫 is a tree.

Since |𝒫|≥3, Ω has a vertex X of valence ≥2. Choose Y∈𝒫 connected to X by an edge of Ω. Let s∈X and t∈Y such that s and t are connected by an edge of Γ. Recall that by definition s and t are unique. Let Q′ be the connected component of Ω𝒫∖{X} containing Y, let 𝒫Q′ be the set of vertices of Q′, let U′=⋃Z∈𝒫Q′Z, let U=U′∪{s}, and let 𝒫Q=𝒫Q′∪{{s}}. Observe that 𝒫Q is an admissible partition of U, that AZ is residually finite for all Z∈𝒫Q, that ΓU/𝒫Q is a tree, and that |𝒫Q|<|𝒫|. By the inductive hypothesis it follows that AU is residually finite. Let R be the connected component of Ω𝒫∖{Y} containing X, let 𝒫R be the set of vertices of R, and let V=⋃Z∈𝒫RZ. Observe that 𝒫R is an admissible partition of V, that AZ is residually finite for all Z∈𝒫R, that ΓV/𝒫R is a tree, and that |𝒫R|<|𝒫|. By the inductive hypothesis it follows that AV is residually finite. Let Δ1,…,Δq be the connected components of ΓS∖{s}. Let i∈{1,…,q}. Let Zi be the set of vertices of Δi. It is easily seen that either Zi∪{s}⊂U, or Zi∪{s}⊂V, hence, by the above, AZi∪{s} is residually finite. We conclude by Lemma 2.3 that A is residually finite. ∎