Note on the residual finiteness of Artin groups (Q1748045)

Let \(S\) be a finite set. With \(S\) a symmetric square matrices \(M=(m_{s t})_{s,t\in S}\) (indexed by the elements of \(S\)) with \(m_{s t}\in \mathbb{N}\cup \{\infty \}\) and \(m_{s s}=1\) and \(m_{s t}\geq 2\) for \(s\neq t\) can be associated. These kind of matrices are called Coxeter matrices. For each Coxeter matrix, a graph (Coxeter graph) \(\Gamma \) is associated. The set of vertices \(V(\Gamma )\) consists of the elements of \(S\), two vertices \(s\neq t\) are connected by an edge if \(m_{s t} \neq \infty \). Note that two Coxeter matrices \(M=(m_{s t})_{s,t\in S}\) and \(N=(n_{s t})_{s,t\in S}\) define the same Coxeter graph over the same finite set \(S\) if \(m_{s t}=\infty \) whenever \(n_{s t}=\infty \). The graph \(\Gamma \) is said to be even if \(m_{s t}\) is either even or \(\infty \) for all \(s, t\in S\), \(s\neq t\). The graph \(\Gamma \) is said to be triangle free if it has no full subgraph which is a triangle. Let \(a\) and \(b\) be two letters and \(m\) an integer greater than 1 then \(\prod (a,b : m)\) denotes the word \(aba\dots \) of length \(m\) (note that the word ends on \(a\) or \(b\) depending on \(m\) odd or even). The Artin group corresponding to a Coxeter graph \(\Gamma \) (which is associated with the Coxeter matrix \(M=(m_{s t})_{s,t\in S}\)) is the group with the presentation \(A=A_{\Gamma }=\langle S \mid \prod (s,t : m_{s t})=\prod (t,s : m_{s t})\) for all \(s,t\in S, s\neq t, m_{s t}\neq \infty \rangle \). Although in some cases Artin groups are residually finite (for example, Artin groups of spherical type, right-angled Artin groups and even Artin groups of FC type), the problem to decide for an Artin group to be residually finite remains open in general. This paper contributes to this direction. Let \(X \subset S\), \(\Gamma _{X}\) be the full subgraph of \(\Gamma \) spanned by \(X\) and \(A_{X}\) be the subgroup of \(A\) generated by \(X\). In [\textit{H. Van der Lek}, The homotopy type of complex hyperplane complements. Nijmegen: Radboud-University (Ph.D. thesis) (1983)], it is proved that \(A_{X}\) is the Artin group of \(X\). A partition \(\mathcal{P}\) of \(S\) is admissible if, for all \(X,Y \in \mathcal{P}\), \(X\neq Y\), there is at most one edge in \(\Gamma\) connecting an element of \(X\) with an element of \(Y\). An admissible partition \(\mathcal{P}\) of \(S\) determines a new Coxeter graph \(\Gamma /\mathcal{P}\) defined as follows. The set of vertices of \(\Gamma /\mathcal{P}\) is \(\mathcal{P}\). Two distinct elements \(X,Y\in \mathcal{P}\) are connected by an edge labelled with \(m\) if there exist \(s \in X\) and \(t \in Y\) such that \(m_{s t}=m\) (\(m\neq \infty \)). The main result of this paper is the following. Theorem. Let \(\Gamma \) be a Coxeter graph, let \(A=A_{\Gamma }\), and let \(\mathcal{P}\) be an admissible partition of \(S\) such that \begin{itemize} \item[(a)] the group \(A_{X}\) is residually finite for all \(X\in \mathcal{P}\), \item[(b)] the Coxeter graph \(\Gamma /\mathcal{P}\) is either even and triangle free, or a forest. \end{itemize} Then \(A\) is residually finite. Corollary. The following statements hold. \begin{itemize} \item[(1)] If \(\Gamma\) is even and triangle free, then \(A\) is residually finite. \item[(2)] If \(\Gamma\) is a forest, then \(A\) is residually finite. \end{itemize} As it is pointed out in the paper, Corollary (2) follows from results in [\textit{C. M. Gordon}, Bol. Soc. Mat. Mex., III. Ser. 10, 193--198 (2004; Zbl 1100.57001)] and [\textit{P. Przytycki} and \textit{D. T. Wise}, J. Topol. 7, No. 2, 419--435 (2014; Zbl 1331.57023)]. Also, an alternative proof of Corollary (1) follows from results in [\textit{R. Blasco-Garcia} et al., Groups Geom. Dyn. 13, No. 1, 309--325 (2019; Zbl 1498.20086)].