On parabolic subgroups of Artin groups (Q6594748)

Let \(A_{\Gamma}\) be the Artin group associated to a finite simplicial graph \(\Gamma=(V(\Gamma), E(\Gamma))\), if \(X \subseteq V(\Gamma)\), then let \(A_{X}\) be the subgroup of \(A_{\Gamma}\) generated by \(X\). A group of the form \(A_{X}\) is called a standard parabolic subgroup of \(A_{\Gamma}\) and a subgroup conjugate to a standard parabolic subgroup is simply called a parabolic subgroup.\N\NIt was proven by \textit{H. Van der Lek} [The homotopy type of complex hyperplane complements. Nijmegen: University of Nijmegen (PhD Thesis) (1983)] that \(A_{X}\) is canonically isomorphic to the Artin group \(A_{\langle X \rangle}\) (where \(\langle X \rangle\) is the induced subgraph of \(X\) in \(\Gamma\)) and that the class of standard parabolic subgroups is closed under intersection. It is conjectured that the same result holds for the class consisting of all parabolic subgroups.\N\NA subset \(X\subseteq V(\Gamma)\) is free of infinity if \(\{v, w \} \in E(\Gamma)\) for all \(v, w \in Y\), \(v \not = w\). In this case, the subgroup \(A_{X}\) is called a complete standard parabolic subgroup. A subgroup conjugate to a complete standard parabolic subgroup is called a complete parabolic subgroup.\N\NThe main result is (see Theorem 1.1): Let \(A_{\Gamma}\) be an Artin group, then the following two properties are equivalent:\N\begin{itemize}\N\item[(\(\mathsf{Int}\))] For each free of infinity subset \(Y \subseteq V(\Gamma)\) and for all parabolic subgroups \(P_{1}\), \(P_{2}\) of \(A_{Y}\), the intersection \(P_{1}\cap P_{2}\) is a parabolic subgroup.\N\item[(\(\mathsf{Int+-}\))] For each complete parabolic subgroup \(P_{1}\) and for each parabolic subgroup \(P_{2}\) of \(A_{\Gamma}\), the intersection \(P_{1}\cap P_{2}\) is a parabolic subgroup.\N\end{itemize}\NAn Artin group is of spherical type if the associated Coxeter group is finite and an Artin group is of FC-type if all complete standard parabolic subgroups are of spherical type. It was proven by \textit{M. Cumplido} et al. [Adv. Math. 352, 572--610 (2019; Zbl 1498.20088)], that intersections of parabolic subgroups in a finite type Artin group are parabolic. Hence, an immediate consequence of Theorem 1.1 is \N\NCorollary 1.2: Let \(A_{\Gamma}\) be an Artin group of FC-type and \(P_{1}\), \(P_{2}\) be two parabolic subgroups. If \(P_{1}\) is complete, then \(P_{1} \cap P_{2}\) is parabolic.\N\NFurther, the authors connect the intersection behavior of complete parabolic subgroups of \(A_{\Gamma}\) to fixed point properties and to automatic continuity of \(A_{\Gamma}\) using Bass-Serre theory and a generalization of the Deligne complex.