Acylindrical hyperbolicity of Artin groups associated with graphs that are not cones (Q6619329)

Let \(\Gamma\) be a finite simple graph and \(A_{\Gamma}\) the Artin group associated with \(\Gamma\). If the Coxeter group \(W_{\Gamma}\) associated with \(A_{\Gamma}\) is finite, then \(A_{\Gamma}\) is said to be of finite type, otherwise, it is of infinite type.\N\N\textit{R. Charney} and \textit{R. Morris-Wright} [Proc. Am. Math. Soc. 147, No. 9, 3675--3689 (2019; Zbl 1483.20068)] showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and the actions on them.\N\NIn the paper under review, the authors generalize this result by developing their study and formulating some additional discussion. The main result is Theorem 1.4: Let \(A_{\Gamma}\) be an Artin group associated with a graph \(\Gamma\) where \(\Gamma\) has at least three vertices. Suppose that \(\Gamma\) is not a cone. Then, the following are equivalent: \N\begin{itemize}\N\item[(1)] \(A_{\Gamma}\) is irreducible, that is, it cannot be decomposed as a join of two subgraphs such that all edges between them are labeled by \(2\); \N\item[(2)] \(A_{\Gamma}\) has a WPD contracting element with respect to the isometric action on the clique-cube complex; \N\item[(3)] \(A_{\Gamma}\) is acylindrically hyperbolic; \N\item[(4)] \(A_{\Gamma}\) is directly indecomposable, that is, it can not be decomposed as a direct product of two non-trivial subgroups.\N\end{itemize}