Centers of Artin groups defined on cones

Abstract

We prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point.

1. Introduction

An Artin group A is given by the presentation

\begin{equation*}A = \langle s_1, \dots, s_n| \underbrace{s_is_js_i\cdots}_{m_{ij} \hspace{.5mm} terms} = \underbrace{s_js_is_j\cdots}_{m_{ij} \hspace{.5mm} terms} \rangle\end{equation*}

where $m_{ij} = m_{ji} \geq 2$. The data of an Artin group can be encoded by its defining graph Γ whose vertex set is $V(\Gamma)=\{s_1, \dots, s_n\}$, and each relation involving $s_i, s_j$ with terms of length m_ij corresponds to an edge $(s_i, s_j)$ with label $m_{ij}\leq \infty$. The Artin group with defining graph Γ will be denoted by $A_\Gamma$. Every Artin group has a naturally associated Coxeter group quotient $A_\Gamma \to W_\Gamma$ obtained by adding relations $s_i^2 = 1$ for all $i=1,\dots, n$.

An Artin group A is spherical if the corresponding Coxeter group is finite, and otherwise A is infinite type. A special subgroup of A is a subgroup generated by some subset of $K\subseteq V(\Gamma)$, denoted A_K. Each special subgroup A_K is itself isomorphic to the Artin group with defining graph $\Gamma_K$, where $\Gamma_K$ is the subgraph of Γ induced by K [13].

A 2-labelled join of labelled graphs $\Gamma_1, \Gamma_2$ is a graph join $\Gamma_1*\Gamma_2$ where every edge $(s_1,s_2)$, with $s_1\in V(\Gamma_1)$ and $s_2\in V(\Gamma_2)$, has label 2, and the labels of edges contained in factors $\Gamma_1, \Gamma_2$ remain the same. We denote a 2-labelled join by $\Gamma_1 *_2 \Gamma_2$. If $\Gamma = \Gamma_1 *_2 \Gamma_2$, then $A_\Gamma = A_{\Gamma_1}\times A_{\Gamma_2}$. An Artin group A is irreducible if its defining graph Γ does not split as a 2-labelled join, in which case we also call Γ irreducible.

Each Artin group $A_\Gamma$ admits (a possibly trivial) decomposition into irreducible factors $A_{\Gamma}= A_{\Gamma_1}\times \dots \times A_{\Gamma_n}$, where each $\Gamma_i\subseteq \Gamma$ is irreducible. Equivalently, this corresponds the maximal decomposition of Γ as an n-fold 2-labelled join of subgraphs $\Gamma_1, \dots, \Gamma_n$.

Every irreducible spherical Artin group A has an infinite cyclic centre, and furthermore if $\{s_1, \dots, s_n\}$ is the standard generating set of A then Z(A) is generated by a power of $s_1s_2\cdots s_n$ [3, 6]. If z is a central element in a spherical factor $A_{\Gamma_i}$ of $A_\Gamma$, then $z\in Z(A_\Gamma)$. Conjecturally, all the central elements of $A_\Gamma$ arise from irreducible factors that are spherical.

The Center Conjecture

Let $A_\Gamma$ be an Artin group with irreducible factor decomposition

\begin{equation*}A_\Gamma = A_{\Gamma_1}\times \dots \times A_{\Gamma_n}.\end{equation*}

Then $Z(A_\Gamma) = \mathbb Z^k$ where k is the number of spherical factors $A_{\Gamma_i}$.

When $A = B\times C$ then $Z(A) = Z(B) \times Z(C)$ so, equivalently, the Center Conjecture states that every irreducible Artin group of infinite type has trivial centre.

Apart from the spherical Artin groups, the Center Conjecture also holds for FC-type Artin groups, two-dimensional Artin groups [8] and Euclidean Artin groups [12]. More generally, every Artin group that satisfies the $K(\pi,1)$-conjecture also satisfies the Center Conjecture [10].

A cone point in a graph Γ is a vertex that is adjacent to every other vertex of Γ. Charney and Morris-Wright have shown the Center Conjecture holds for Artin groups whose defining graphs are not cones [4]. In other words, this is the case where the set of cone points of Γ is empty. The proof of Charney and Morris-Wright relies on the geometry of the clique-cube complex, which is a CAT(0) cube complex associated with Γ. Their result can also be deduced from the following proposition, whose proof is brief and follows directly from the presentation of $A_\Gamma$.

Proposition 1.1. Let $A_\Gamma$ be an Artin group, where $T\subseteq V(\Gamma)$ is the set of cone points of Γ. Then $Z(A_{\Gamma})\subseteq Z(A_T)$.

In particular, if in the above theorem $Z(A_{T}) = \{1\}$, then $Z(A_\Gamma) = \{1\}$, i.e. $A_{\Gamma}$ satisfies the Center Conjecture.

Godelle and Paris showed that if all Artin groups whose defining graph is a clique satisfy the Center Conjecture, then all Artin groups satisfy the Center Conjecture [8]. Our second result gives alternative proofs of this fact and the Center Conjecture for FC-type Artin groups, as well as the Center Conjecture for many new Artin groups (see Figure 1 for some examples).

Theorem 1.2. Let $A_\Gamma$ be an Artin group, where $T\subseteq V(\Gamma)$ is the set of cone points of Γ. If A_T satisfies the Center Conjecture, then $A_\Gamma$ satisfies the Center Conjecture.

Figure 1. New examples of graphs whose associated Artin groups satisfy the Center Conjecture. The red subgraphs are spanned by the set of cone points. The middle graph can have any labels. The diagram on the right represents an infinite class of such graphs; the double edges represent joins, with all labels equal to 3.

To prove Proposition 1.1, we consider splittings of non-clique Artin groups. This generalizes the result of [4] using a different method. Then to deduce Theorem 1.2 from Proposition 1.1 we use a retraction map (first used in [5] and recently discussed more explicitly in [1]) to prove some combinatorial restrictions on central elements of irreducible Artin groups; namely that any spelling of a central element which does not use every generator cannot be strictly positive or strictly negative.

While preparing this manuscript, Kato and Oguni announced a proof that all Artin groups whose defining graphs are not cliques are acylindrically hyperbolic and obtained our results as corollaries [11]. Their methods are a generalization of the techniques used in [4] and are therefore distinct from the techniques used here.

2. Central elements are generated by cone points

Let Γ be a graph and let $x, y \in V(\Gamma)$ so that $m_{x,y} = \infty$. A direct consequence of the presentation of $A_\Gamma$ is that it splits as the following amalgamated product:

\begin{equation*} A_\Gamma = A_{\Gamma - \{x\}} *_{A_{\Gamma - \{x, y\}}} A_{\Gamma - \{y\}}. \end{equation*}

The following lemma can be proven by considering normal forms in amalgamated free products and was utilized in [8].

Lemma 2.1. Let $G = H_1 *_K H_2$ be the free product of $H_1, H_2$ amalgamated over K. Then $Z(G) = Z(H_1) \cap Z(H_2) \subseteq K$.

Using these two results, we can prove our first theorem.

Proposition 1.1. Let $A_\Gamma$ be an Artin group, where $T\subseteq V(\Gamma)$ is the set of cone points of Γ. Then $Z(A_{\Gamma})\subseteq Z(A_T)$.

Proof. Let $R = V(\Gamma) - T$ and $r \in R$. There exists some $q_r \in V(\Gamma) - T$ so that $m_{r, q_r} = \infty$. Thus

\begin{equation*} A_\Gamma = A_{\Gamma-\{r\}} *_{A_{\Gamma - \{r, q_r\}}} A_{\Gamma - \{q_r\}}. \end{equation*}

By Lemma 2.1, this implies that for every $r \in R$ we have $Z(A_\Gamma) \subseteq {A_{\Gamma - \{r, q_r\}}} \subseteq A_{\Gamma - \{r\}}.$ In particular, we have

\begin{equation*} Z(A_\Gamma) \subseteq \bigcap_{r \in R} A_{\Gamma - \{r\}}. \end{equation*}

In [13], van der Lek showed that for any $S, Q \subseteq V(\Gamma)$ we have $A_S \cap A_Q = A_{S\cap Q}$. In particular, $\bigcap_{r \in R} A_{\Gamma - \{r\}} = A_{\Gamma - R} = A_T$. Since $z \in Z(A_\Gamma)$, z is also central in A_T.

We obtain the following corollary immediately.

Corollary 2.2. Suppose that Γ is a graph with cones points T, and suppose A_T has trivial centre. Then $A_\Gamma$ has trivial centre.

In particular, if $T = \emptyset$ then $Z(A_\Gamma) = \{1\}$ and we recover [4, Theorem 3.3]. The condition that $Z(A_T) = \{1\}$ is not a necessary condition: if $T = \{t\}$ then $A_T \cong \mathbb{Z}$, so A_T does not have trivial centre. On the other hand, if there is any vertex of Γ which does not commute with t then $Z(A_\Gamma)$ is trivial.

Corollary 2.3. Suppose Γ is a graph with a single cone point t. Then $A(\Gamma)$ satisfies the Center Conjecture, i.e. $Z(A_\Gamma) = \{1\}$ if and only if there exists $s \in S$ so that $m_{st} \neq 2$.

Proof. Assume there exists $s \in S$ with $m_{s,t}\neq 2$. By Proposition 1.1, if $z \in Z(A_{\Gamma})$ then $z \in A_{\{t\}} = \langle t \rangle$. Let $z = t^i$. Since $A_{\{t, s\}}$ is spherical, the centre of $A_{\{t, s\}}$ is generated by a power of ts. Hence z is central in $A_{\{t, s\}}$ only if i = 0.

If $m_{st} = 2$ for all $s \in S - \{t\}$, then $A_\Gamma \cong \langle t \rangle \times A_S$, so $\langle t \rangle \leq Z(A_\Gamma)$.

In the following section, we generalize this argument to larger sets of cone points.

3. Cone sets satisfying the Center Conjecture

3.1. Retraction map

The main goal of this subsection is Theorem 3.4. Its proof relies on the retraction map $\pi_X :A_\Gamma \to A_X$ described in [1, 5]. The definition of the map π_X relies on passing to the Coxeter group $W_\Gamma$. We re-establish notation here so that we can differentiate easily between elements in the Coxeter group and elements in the Artin group.

Fix a graph Γ. Let $\Sigma = \{\sigma_v | v \in V(\Gamma)\}$ and $S = \{s_v | v \in V(\Gamma)\}$ be generating sets for the Artin and Coxeter groups $A_\Gamma$ and $ W_\Gamma$, respectively. Let $\theta: A_\Gamma \to W_\Gamma$ the natural surjection sending $\sigma_v \mapsto s_v$. The kernel of θ is the pure Artin group $P\!A_\Gamma$. Let $X \subseteq V(\Gamma)$. We denote the subgroup of $W_\Gamma$ generated by X by W_X and similarly for the corresponding subgroup of $P\!A_\Gamma$ and subsets of $\Sigma, S$.

Let $X, Y$ be two subsets of S. An element $w \in W_\Gamma$ is (X, Y)-reduced if it is of minimal length amongst the elements of the double coset $W_X w W_Y$. For any element $w \in W_\Gamma$, the length of w, denoted $|w|$, is the shortest length of a word in S needed to express w.

Lemma 3.1. (Lemma 2.3 [9])

Let $X, Y \subseteq V(\Gamma)$ and let $w \in W_\Gamma$. There exists a unique (X, Y)-reduced element in $W_X w W_Y.$ Furthermore, the following are equivalent:

• • an element $w \in W$ is $(X, \emptyset)$-reduced,

• • $|sw| \gt |w|$ for all $s \in X$ and

• • $|v\cdot w| = |v|+|w|$ for all $v\in W_X$.

The map is defined as follows.

Definition 3.2. ([1])

Let $(\Sigma\cup \Sigma^{-1})^*$ denote the free monoid on $\Sigma\,\cup\, \Sigma^{-1}$. Let $X \subseteq V(\Gamma)$, and let $\widehat{\alpha} = \sigma_{v_1}^{\varepsilon_1}\sigma_{v_2}^{\varepsilon_2} \cdots \sigma_{v_p}^{\varepsilon_p} \in (\Sigma\cup \Sigma^{-1})^*.$

Set $u_0 = 1 \in W_\Gamma$, and for $i \in \{1, \dots, p\}$ set $u_i = s_{v_1}s_{v_2}\cdots s_{v_i} \in W_\Gamma$. We can write each u_i as $u_i = v_iw_i$ where $v_i \in W_X$ and w_i is $(X, \emptyset)$-reduced.

Now define

\begin{equation*} t_i = \begin{cases} w_{i-1}s_{v_i}w_{i-1}^{-1} &\mbox{if } \varepsilon = 1, \\ w_i s_{v_i}w_i^{-1} &\mbox{if } \varepsilon = -1. \end{cases} \end{equation*}

If $t_i \notin S_X$ we set $\tau_i = 1$. Otherwise, we have $t_i \in S_X$. Thus $t_i = s_{x_i}$ for some $x_i \in X$, and we set $\tau_i = \sigma_{x_i}^{\varepsilon_i}\in \Sigma_X\cup \Sigma_X^{-1}$.

Finally, we define

\begin{equation*} \widehat\pi_X(\widehat{\alpha}) = \tau_1 \dots \tau_p \in (\Sigma_X\cup \Sigma_X^{-1})^*. \end{equation*}

We collect in the following proposition several properties of the map $\widehat{\pi}_X$ and the induced map π_X (see Proposition 3.3(1)) that will be of use in later results.

Proposition 3.3. Let $X, Y \subseteq V(\Gamma)$, and $\alpha \in (\Sigma\cup\Sigma^{-1})^*$.

1. (i) [1, Prop 2.3(1)] Themap $\widehat\pi_X: (\Sigma\cup \Sigma^{-1})^* \to (\Sigma_X\cup \Sigma^{-1}_X)^*$ induces a set-map $\pi_X: A_\Gamma \to A_X$.

2. (ii) [1, Prop 2.3(2)] For all $\alpha \in A_X$ we have $\pi_X(\alpha) = \alpha$.

3. (iii) [1, Prop 2.3(3)] The restriction of π_X to $P\!A_\Gamma $ is a homomorphism $\pi_X:P\!A_\Gamma \to P\!A_X$.

4. (iv) [7, Prop 0.2(vi)] Let $A_\Gamma ^+, A_X^+$ denote the Artin monoids on $V(\Gamma), X$, respectively. Then $\pi_X: A_\Gamma ^+ \to A_X^+$.

5. (v) [7, Prop 0.2(ix)] If $\omega \in A_Y$ then $\pi_X(\omega) \in A_{Y\cap X}$.

6. (vi) [7, Prop 0.3(ii)] For any $\omega \in A_X$ we have $\pi_X(\omega\alpha) = \omega \pi_X(\alpha).$

Theorem 3.4. Let $A_\Gamma$ be an irreducible Artin group and let z be either a central element of $A_\Gamma$ of infinite order, or a non-trivial central element of $P\!A_\Gamma$, such that z admits a positive spelling in $V(\Gamma)$. There does not exist any $K \subsetneq V(\Gamma)$ so that $z \in A_K$.

Proof. Suppose z is central in $A_\Gamma$ and has infinite order, and suppose that $z \in A_K$ for some $K \subsetneq V(\Gamma)$. Suppose further that z is a positive word on K; that is, there exists an $\widehat\alpha \in K^*$ so that $z = \widehat{\alpha}$. Since z is central, $\theta(z)$ lies in the centre of $W_\Gamma$. In particular, it has finite order [2]. By possibly passing to a finite positive power of z, we can assume that $z \in Z(P\!A_\Gamma)$.

Without loss of generality, by possibly passing to a smaller subset of K we can assume that all the letters of K appear in $\widehat\alpha$. We claim that there exists $t \in K$ and $s \in V(\Gamma) - K$ so that $m_{st} \neq 2$. Indeed, if there is no such $t \in K$, then $\Gamma = K *_2 (\Gamma - K)$, which contradicts the assumption that $A_\Gamma$ is irreducible. Let $\widehat\alpha'$ denote a cyclic permutation of $\widehat\alpha$ beginning with t. Note that every cyclic permutation of $\widehat\alpha$ is a conjugate of $\widehat\alpha$, and therefore $\widehat\alpha$ represents z, since z is central. Consider the map $\pi_{\{s, t\}}: A_\Gamma \to A_{\{s, t\}}$. By Proposition 3.3(6), we know that $\widehat\pi_{\{s, t\}}(\widehat\alpha')$ starts with the letter t, and by Proposition 3.3(4) we know that $\widehat\pi_{\{s, t\}}(\widehat\alpha')$ is a positive word. In particular, $\pi_{\{s, t\}}(z)$ is non-trivial. By Proposition 3.3(5), $\pi_{\{s, t\}}(z)\in A_{K\cap \{s, t\}} = A_{\{t\}}$, and therefore $\pi_{\{s, t\}}(z)$ is a positive power of t.

By Proposition 3.3(3), the map $\pi_{\{s, t\}}:P\!A_\Gamma \to P\!A_{\{s, t\}}$ is a surjective homomorphism, so $\pi_{\{s, t\}}(z)$ is central in $P\!A_{\{s, t\}}$. But since $A_{\{s, t\}}$ is spherical, its centre is generated by a power of st. In particular, no non-trivial power of t commutes with any non-trivial power of s in $A_{\{s,t\}}$. Therefore z = 1.

In the case that Γ is not a clique, together with Proposition 1.1 this implies that no central element of $A_\Gamma$ admits a positive spelling.

3.2. Proof of Theorem 1.2

We are now ready to prove the main theorem of this note.

Theorem 1.2. Let $A_\Gamma$ be an Artin group, where $T\subseteq V(\Gamma)$ is the set of cone points of Γ. If A_T satisfies the Center Conjecture, then $A_\Gamma$ satisfies the Center Conjecture.

Proof. If Γ is a clique, then $V(\Gamma) = T$, so by assumption $A_\Gamma$ satisfies the Center Conjecture. Suppose instead that Γ is not a clique.

It suffices to prove the theorem for irreducible Γ. Indeed, if $\Gamma = \Gamma_1*_2 \dots *_2\Gamma_n$ then $Z(A_\Gamma) = Z(A_{\Gamma_1}) \times \dots \times Z(A_{\Gamma_n})$ so the Center Conjecture holds for Γ if and only if it holds for each irreducible factor $\Gamma_i$. The set of cone-points T of Γ is the union of the sets $T_i = T\cap V(\Gamma_i)$ of cone points in each $\Gamma_i$. Furthermore, if T splits as $T = {T'_1}*_2 \cdots *_2 {T'_n}$ then each T_i is a union of sets $T'_{i_1}, \dots, T'_{i_j}.$ Hence $A_{T_i}$ satisfies the Center Conjecture for each i because A_T satisfies the Center Conjecture.

Suppose that Γ is irreducible. By Proposition 1.1, $Z(A_\Gamma) \subseteq Z(A_T)$. Let $A_T = A_{T_1}\times \cdots \times A_{T_m}\times \cdots \times A_{T_n}$ be the irreducible factor decomposition of A_T, where the factor $A_{T_i}$ is spherical if and only if $i \leq m$. Since A_T satisfies the Center Conjecture, $Z(A_T) = \langle z_1, \dots, z_m\rangle\simeq \mathbb Z^m$ where z_i is a positive element of $A(T_i)$ generating $Z(A_{T_i})$ such that any spelling of z_i uses every letter of T_i at least once. Let z be an arbitrary non-trivial element $Z(A_\Gamma)$. Then z can be expressed as $z = z_1^{k_1} \cdots z_m^{k_m}$ for some $k_i \in \mathbb{Z}$ for $i=1,\dots, m$. Since z is non-trivial, we know that at least one of the k_i is non-zero. Up to renumbering the factors and replacing z with z ⁻¹, we may assume that $k_1 \gt 0$. There is some k > 0 so that $z^k \in P\!A_\Gamma.$ Let $x = z_1^{kk_1}$ and $y = z_{2}^{kk_{j+1}}\cdots z_m^{kk_m}$ so that $z^k = xy$. Let $\widehat\alpha_x, \widehat\alpha_y$ denote minimal spellings of $x, y$, respectively. Note that $\widehat\alpha_x$ is positive.

There must exist some $r \in V(\Gamma)-T, t \in T_1$ so that $m_{rt}\neq 2$. Indeed, if this is not the case then $\Gamma = (\Gamma-T_1) *_2 {T_1}$, contradicting the irreducibility of Γ. Since all cyclic permutations of $\widehat\alpha_x$ represent x, we may take t to be the initial letter of $\widehat\alpha_x\widehat\alpha_y$. Now consider the map $\pi_{T_1 \cup \{r\}}: A_\Gamma \to A_{T_1 \cup \{r\}}$. We have $\pi_{T_1 \cup \{r\}}(z^k) \in Z(P\!A_{T_1 \cup \{r\}}).$

By Proposition 3.3(6), $\pi_{T_1 \cup \{r\}}(xy) = x\pi_{T_1 \cup \{r\}}(y)$, since $x\in A_{T_1\cup \{r\}}$. Since $y \in A_{T - \{T_1\cup\{r\}\}}$, by Proposition 3.3(5), we have $\pi_{T_1 \cup \{r\}}(y) = 1$. So $\pi_{T_1 \cup \{r\}}(z^k) = x$. Thus, x is a positive central element of $P\!A_{T_1 \cup \{r\}}$ which does not use every letter of $T_1 \cup \{r\}$. By Theorem 3.4, x = 1, contradicting the choice of k ₁.