Artin HNN-extensions virtually embed in Artin groups

T. Hsu^{∗}  I. J. Leary^{∗}^{†}

February 12, 2008

Abstract

An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.

1 Results

Recall that an Artin system is a finite set S, together with a function m from the 2-element subsets of S to the set {2, 3, 4, …} ∪ {∞}. It may be helpful to consider m as an edge-labelling on the complete graph with vertex set S. The Artin group A = A(S, m) corresponding to an Artin system (S, m) is defined by a presentation. The generators for A are the elements of the set S. There is one relation in the presentation for each unordered pair s, t ∈ S such that m(t, s) = m(s, t) < ∞, of the form

sts ⋯ = tst ⋯,  (1)

where there are m(s, t) symbols on each side of (1). Call these relations the Artin relations associated to (S, m). If S′ is a subset of S, and we write m for the restriction to S′ of m, inclusion induces a group homomorphism from A(S′, m) to A(S, m), and this homomorphism is known to be injective [8, 9]. The image of such a homomorphism is called an Artin subgroup of A(S, m).

An Artin system also gives rise to a Coxeter group W = W (S, m). The generators of the Coxeter group are the elements of S, subject to the Artin relations for (S, m) together with the relations s^{2} = 1 for all s ∈ S. Call these relations the Coxeter relations for W. Note that in the presence of the relations s^{2} = 1 = t^{2}, the Artin relation between s and t is equivalent to the relation (st)^{m(s,t)} = 1. For S′ a subset of S, the natural map W (S′, m) → W (S, m) is injective, and the image of such a homomorphism is called a Coxeter subgroup of W (S, m) [1, 2].

∗This work was done at MSRI, Berkeley, where research is supported in part by NSF grant DMS-0441170

†partially supported by NSF grant DMS-0505471.

The set of labels, L(S, m), for an Artin system (S, m) is the image of m, viewed as a subset of {2, 3, 4, …} ∪ {∞}. An Artin system (S, m) is called right-angled if L(S, m) ⊆ {2, ∞}, and A(S, m) (resp. W (S, m)) is then called a right-angled Artin group (resp. right-angled Coxeter group).

Now suppose ϕ : S′ → S′′ is a bijection between two subsets of S that preserves labels in the sense that

m(s, s′) = m(ϕ(s), ϕ(s′))  for all s, s′ ∈ S′.  (2)

The Artin HNN-extension G(S, m, ϕ) (resp. Coxeter HNN-extension C(S, m, ϕ)) associated to (S, m, ϕ) is the free product of A(S, m) (resp. W (S, m)) and an infinite cyclic group ⟨t⟩, modulo the relations t^{−1}st = ϕ(s) for all s ∈ S′. Thus an Artin HNN-extension (resp. a Coxeter HNN-extension) is a special type of HNN-extension in which the base group is an Artin (resp. a Coxeter) group and the stable letter conjugates an Artin (resp. a Coxeter) subgroup to another Artin (resp. Coxeter) subgroup via the map induced by a bijection of standard generating sets. The set of labels for G(S, m, ϕ) (resp. C(S, m, ϕ)) is the set of labels for the group A(S, m) (resp. W (S, m)).

We require one more definition before stating our main result. For any property P of groups, a group G virtually has property P if there exists a finite-index subgroup H ≤ G such that H has property P.

Theorem 1 Any Artin (resp. Coxeter) HNN-extension virtually embeds in an Artin (resp. Coxeter) group. If L is the set of labels for the HNN-extension, then the Artin (resp. Coxeter) group in which it virtually embeds can be chosen to have its set of labels equal to L ∪ {2, ∞}. In particular right-angled HNNextensions virtually embed in right-angled groups.

Theorem 1 yields the following application, which was our original motivation for considering Artin HNN-extensions.

Corollary 2 For any finite group Q not of prime power order, there exists n and a subgroup G of SL(n, ℤ) such that G virtually has a finite classifying space and G has infinitely many conjugacy classes of subgroups isomorphic to Q.

Proof. Note that by ‘G virtually has a finite classifying space’ we mean that there is a finite index subgroup H ≤ G which has a finite K(H, 1). In [6], a group G is constructed which virtually has a finite classifying space and which contains infinitely many conjugacy classes of subgroup isomorphic to Q. By [6, Thm. 28], we know that G virtually embeds in a right-angled Artin HNNextension. But then, by Theorem 1, G virtually embeds in a right-angled Artin group, and by [3, 5], any right-angled Artin group embeds in SL(m, ℤ) for some m, so the group G embeds in SL(n, ℤ) for some (possibly larger) n.

Remark 3 Corollary 2 was proved in [7] in the special case when Q is not of the form (p-group)-by-cyclic-by-(q-group) for primes p and q.

2 Proofs

To review the standard definitions from [10, I.5], recall that a graph of groups (G_{(−)}, Γ) consists of a directed CW-graph Γ, together with a group G_{v} for each vertex v of Γ (the vertex groups), a group G_{e} for each edge e of Γ (the edge groups), and for each edge e, two injective group homomorphisms ι_{e} : G_{e} → G_{ι(e)} and τ_{e} : G_{e} → G_{τ(e)} (the attaching homomorphisms), where ι(e) denotes the initial vertex of e and τ(e) denotes the terminal vertex of e. We define F (G_{(−)}, Γ) to be the free product of the G_{v} and infinite cyclic groups ⟨t_{e}⟩, where e runs over all edges of Γ, modulo the relations

t_{e}^{−1}ι_{e}(g)t_{e} = τ_{e}(g)  for all e, g ∈ G_{e};  (3)

and for a basepoint P_{0} in Γ, we define π_{1}(G_{(−)}, Γ, P_{0}) to be the subgroup of all words in F (G_{(−)}, Γ) that, after ignoring all syllables from G_{v}’s, define paths in Γ that begin and end at P_{0}. Similarly, for a spanning tree T of Γ, we define π_{1}(G_{(−)}, Γ, T) to be the free product of the G_{v} and infinite cyclic groups ⟨t_{e}⟩, where e runs over all edges of Γ not in T, modulo the relations

ι_{e}(g) = τ_{e}(g)  for all e ∈ T, g ∈ G_{e}, and  (4)
t_{e}^{−1}ι_{e}(g)t_{e} = τ_{e}(g)  for all e ∉ T, g ∈ G_{e}.  (5)

Up to isomorphism, π_{1}(G_{(−)}, Γ, P_{0}) is independent of P_{0} and π_{1}(G_{(−)}, Γ, T) is independent of T, and the two groups are isomorphic, so we call their isomorphism class the fundamental group of (G_{(−)}, Γ), denoted by π_{1}(G_{(−)}, Γ). We also call the elements t_{e} stable letters.

Let k be a positive integer, let V, E, and V_{∗} be groups, and let f : E → V, g : E → V, and h_{i} : V → V_{∗} (i ∈ ℤ/k) be injective group homomorphisms.

1. Let Δ_{1} be a self-loop. Taking vertex group V, edge group E, and attaching maps f and g, we get a graph of groups ({V, E} , Δ_{1}) whose fundamental group is an HNN extension with base group V and associated subgroup E. Let t be the associated stable letter.

2. Let Δ_{k} be a (directed) k-cycle, with vertex set ℤ/k and edges (i, i + 1) (i ∈ ℤ/k). Taking the vertex (resp. edge) groups V_{i} (resp. E_{i}) to be copies of V (resp. E), and the attaching homomorphisms f_{i} : E_{i} → V_{i} and g_{i} : E_{i} → V_{i+1} to be copies of f and g, with all indices running over all i ∈ ℤ/k, we get a graph of groups ({V_{i}, E_{i}} , Δ_{k}). Let {t_{i}} be the set of stable letters in the group F ({V_{i}, E_{i}} , Δ_{k}).

3. Let Γ_{k} be a k-leaved rose, with edges labelled by ℤ/k, and let V_{i}, E_{i}, f_{i}, and g_{i} be defined as in 2. Taking vertex group V_{∗}, edge groups the E_{i}, and attaching homomorphisms h_{i} ∘ f_{i} and h_{i+1} ∘ g_{i}, we get a graph of groups ({V_{∗}, E_{i}} , Γ_{k}).

Lemma 4 Let K be the kernel of the homomorphism ρ : π_{1}({V, E} , Δ_{1}) → ℤ/k defined by ρ(V) = 0 and ρ(t) = 1. Then K = π_{1}({V_{i}, E_{i}} , Δ_{k}), and K embeds in π_{1}({V_{∗}, E_{i}} , Γ_{k}).

Proof. Let θ : π_{1}({V_{i}, E_{i}} , Δ_{k}, 0) → π_{1}({V, E} , Δ_{1}) be the homomorphism defined by θ(v) = v ∈ V for all v ∈ V_{i} and θ(t_{i}) = t. Since θ sends normal forms to normal forms, θ is injective; and since the image of θ is precisely the set of all normal forms t^{±1}v_{1}t^{±1}v_{2} … with t-exponent sum divisible by k, the image of θ is precisely K.

Next, let Δ_{∗} be Δ_{k} with an additional vertex ∗ and k additional edges (i, ∗) (i ∈ ℤ/k). Taking vertex groups to be the V_{i} and V_{∗} (corresponding to ∗), edge groups to be the E_{i} and copies of the V_{i} (corresponding to (i, ∗)), and attaching homomorphisms to be the f_{i} and g_{i}, along with the identity on V_{i} (corresponding to the initial vertex of (i, ∗)) and h_{i} (corresponding to the terminal vertex of (i, ∗)), we get a graph of groups ({V_{∗}, V_{i}, E_{i}} , Δ_{∗}).

Since ({V_{i}, E_{i}} , Δ_{k}) is a sub-(graph of groups) of ({V_{∗}, V_{i}, E_{i}} , Δ_{∗}), by the normal form theorem, its fundamental group K is a subgroup of the fundamental group π_{1}({V_{∗}, V_{i}, E_{i}} , Δ_{∗}). Furthermore, if T_{Δ} is the spanning tree of Δ_{∗} whose edges are the (i, ∗) and T_{Γ} is the (unique) spanning tree of Γ_{k}, by (4)–(5), the group π_{1}({V_{∗}, V_{i}, E_{i}} , Δ_{∗}, T_{Δ}) is isomorphic to π_{1}({V_{∗}, E_{i}} , Γ_{k}, T_{Γ}). The lemma follows.

For the rest of this section, we let S, m, ϕ : S′ → S′′, and G(S, m, ϕ) be as defined in Section 1. Let Λ be the (directed) graph with vertex set S and an edge (s, ϕ(s)) for each s ∈ S′. Since every vertex of Λ has in-degree and out-degree each at most 1, Λ is a disjoint union of loops and (non-closed) paths. Choose an integer k to be a multiple of the length of each loop in Λ and strictly greater than twice the length of each path in Λ.

Let \overline{S} be the quotient of the direct product ℤ/k × S by the equivalence relation generated by (i, s) ∼ (i + 1, ϕ(s)) for all i ∈ ℤ/k and s ∈ S′. More geometrically, let Λ̂ be the (directed) graph with vertex set ℤ/k ×S and an edge ((i, s), (i + 1, ϕ(s))) for each s ∈ S′ and each i ∈ ℤ/k. Then the map (i, s) ↦ s on vertex sets extends to a k-fold covering map π : Λ̂ → Λ, and each element of \overline{S} is the vertex set of a connected component of Λ̂.

For i ∈ ℤ/k, define ψ_{i} : S → \overline{S} by letting ψ_{i}(s) be the equivalence class of (i, s). We would like to define a labelling function \overline{m} on \overline{S} by

\overline{m}(x, y) = \begin{cases}m(s, t) & if x = ψ_{i}(s) and y = ψ_{i}(t) for some i ∈ ℤ/k, \\ ∞ & otherwise. \end{cases} (6)

Lemma 5 The labelling function \overline{m} is well-defined.

The proof of Lemma 5 uses a variation on an idea from [11, Lem. 5.7].

Proof. Let p : (i, s) ↦ i be the natural projection from Λ̂ to the graph Δ_{k} defined above, and let π : Λ̂ → Λ be the covering map defined above. If ℓ̂ is a component of Λ̂ such that ℓ = π(ℓ̂) is a closed loop, then since the length of ℓ divides k, p induces a graph isomorphism from ℓ̂ to Δ_{k}. Similarly, if ℓ̂ is a component of Λ̂ such that ℓ = π(ℓ̂) is a non-closed path in Λ, then p induces a graph isomorphism from ℓ̂ to p(ℓ̂), a path of length strictly less than k/2. It follows that for any two components ℓ̂, ℓ̂′ of Λ̂, the intersection p(ℓ̂) ∩ p(ℓ̂′) is a connected subgraph of Δ_{k}.

So now, suppose we have i, j ∈ ℤ/k and s, s′, t, t′ ∈ S such that x = ψ_{i}(s) = ψ_{j}(s′) and y = ψ_{i}(t) = ψ_{j}(t′). Equivalently, suppose the vertices (i, s) and (j, s′) lie in the same component x of Λ̂ and the vertices (i, t) and (j, t′) lie in the same component y of Λ̂. Since p induces graph isomorphisms from x and y to their images p(x) and p(y), respectively, and p(x) ∩ p(y) is connected, for some n ≥ 0, either both (j, s′) = (i + n, ϕ^{n}(s)) and (j, t′) = (i + n, ϕ^{n}(t)), or both (i, s) = (j + n, ϕ^{n}(s′)) and (i, t) = (j + n, ϕ^{n}(t′)). In either case, by (2), we see that m(s, t) = m(s′, t′).

Proofof Theorem 1. Taking the Artin case first, let V = A(S, m), E = A(S′, m), and V_{∗} = A(\overline{S}, \overline{m}) (which is well-defined by Lemma 5). Also, let f : E → V be induced by the inclusion of S′ in S and let g : E → V be induced by ϕ : S′ → S′′. Finally, since k is a multiple of the length of each loop in Λ and strictly longer than each path in Λ, every lift of a loop or path in Λ intersects each image ψ_{i}(S) at most once, which means that each of the maps ψ_{i} : S → \overline{S} is injective and each ψ_{i} induces an injection h_{i} : V → V_{∗}. Applying Lemma 4 with this choice of V, E, V_{∗}, f, g, and h_{i}, we see that G(S, m, ϕ) has a subgroup of index k that embeds in G_{∗} = π_{1}({V_{∗}, E_{i}} , Γ_{k}).

It remains to show that G_{∗} is an Artin group. By (5), we see that G_{∗} has a presentation with generators \overline{S} ∪ \{t_{i}\} (i ∈ ℤ/k), subject to the Artin relations for (\overline{S}, \overline{m}) together with the relations

t_{i}^{−1}ψ_{i}(s)t_{i} = ψ_{i+1}(ϕ(s))  (7)

for all i ∈ ℤ/k and s ∈ S′. However, since (i, s) = (i+1, ϕ(s)) in \overline{S}, (7) becomes the Artin relation ψ_{i}(s)t_{i} = t_{i}ψ_{i}(s). Therefore, if

m^{+}(x, y) = \begin{cases}\overline{m}(x, y) & if x, y ∈ \overline{S}, \\ 2 & if \{x, y\} = \{t_{i}, ψ_{i}(s)\} for some i ∈ ℤ/k, \\ ∞ & otherwise, \end{cases}  (8)

we see that G_{∗} = A(\overline{S} ∪ \{t_{i}\} , m^{+}).

Now consider the Coxeter case. Replacing V = A(S, m), E = A(S′, m), and V_{∗} = A(\overline{S}, \overline{m}) with V = W (S, m), E = W (S′, m), and V_{∗} = W (\overline{S}, \overline{m}), the previous proof carries through, except that G_{∗} is not a Coxeter group. However, consider W^{+} = W (\overline{S} ∪ \{u_{i}, u′_{i}\} , m^{+}), where

m^{+}(x, y) = \begin{cases}\overline{m}(x, y) & if x, y ∈ \overline{S}, \\ 2 & if \{x, y\} = \{u_{i}, ψ_{i}(s)\} for some i ∈ ℤ/k, \\ 2 & if \{x, y\} = \{u′_{i}, ψ_{i}(s)\} for some i ∈ ℤ/k, \\ ∞ & otherwise. \end{cases}  (9)

It follows from Tits’ solution to the word problem for Coxeter groups that the homomorphism θ : G_{∗} → W^{+} defined by θ(s) = s for s ∈ \overline{S} and θ(t_{i}) = u_{i}u′_{i} is injective, as reduced (graph of groups) words in G_{∗} are sent to reduced (Coxeter) words in W^{+}. The theorem follows.

3 An open question

Can Theorem 1 be extended to “Artin graphs of Artin groups” (defined similarly to Artin HNN-extensions)? Note that since an “Artin amalgamated free product” is itself an Artin group, it is enough to consider multiple-Artin-HNN-extensions. In that case, the key point is to generalize Lemma 5, but it is not clear to us how (or if) that can be done.

One may of course also consider the analogous Coxeter question(s).

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T. Hsu Department of Mathematics San José State University San José CA 95192-0103 USA

hsu@math.sjsu.edu

I. J. Leary Department of Mathematics The Ohio State University 231 W. 18th Avenue Columbus OH 43210 USA

leary@math.ohio-state.edu