Labeled four cycles and the \(K(\pi,1)\)-conjecture for Artin groups (Q6647737)

Let \(W_{S}\) be a Coxeter group with generating set \(S\) and \(R \subseteq W_{S}\) the set of all reflections. Let \(\rho : W_{S} \rightarrow GL(V)\) be the canonical representation of \(W_{S}\), then each element in \(\rho(R)\) acts as linear reflection on \(V\), and the action of \(W_{S}\) on \(V\) stabilizes the interior \(I\) of a convex cone in \(V\), called the Tits cone. For each \(r \in R\), let \(H_{r}=\{ v^{\rho(r)}=v \mid v \in I \}\) and let \(M(W_{S})= \big (I \times I \big ) \setminus \bigcup_{r \in R} \big (H_{r} \times H_{r} \big)\). The \(K(\pi,1)\)-conjecture for reflection arrangement complements, predicts that the space \(M(W_{S})\) is aspherical for any Coxeter group.\N\NGiven an Artin group with presentation graph \(\Gamma\), its Dynkin diagram \(\Lambda\) is obtained by removing all open edges of \(\Gamma\) labeled by 2, and add extra edges labeled by \(\infty\) between vertices of \(\Gamma\) which are not adjacent. A Dynkin diagram is spherical or affine, if the associated Coxeter group is spherical or affine. If \(A_{\Lambda}\) is an Artin group with Dynkin diagram \(\Lambda\), then the vertices in the presentation graph are called nodes.\N\NIn this interesting and well-written paper the author proves the following results. Let \(\Lambda\) be a tree Dynkin diagram and suppose there exists a collection \(E\) of open edges with labels \(\geq 6\). Then\N\NTheorem 1.1: If each component of \(\Lambda \setminus E\) is either spherical, or locally reducible, then \(A_{\Lambda}\) satisfies the \(K(\pi,1)\)-conjecture.\N\NTheorem 1.2: If each component of \(\Lambda \setminus E\) is either spherical, or affine, or locally reducible and if some conditions (stated in Corollary 8.6) for affine Artin groups of type \(\widetilde{B}\), \(\widetilde{D}\), \(\widetilde{E}\) and \(\widetilde{F}\), then \(A_{\Lambda}\) satisfies the \(K(\pi,1)\)-conjecture.\N\NThe reviewer also reports the first part of Theorem 1.3. Suppose \(\Lambda\) is a connected Dynkin diagram with an induced cyclic subgraph \(C\). Suppose that for each node \(s\in C\), the component \(\Lambda_{s}\) of \(\Lambda \setminus \{s \}\) that contains remaining nodes of \(C\) is either spherical or a locally reducible tree. Then \(A_{\Lambda}\) satisfies the \(K(\pi,1)\)-conjecture.\N\NAn Artin group is of hyperbolic cyclic type if its Dynkin diagram is a circle, and the associated reflection group acts cocompactly on \(\mathbb{H}^{n}\). Their Dynkin diagrams are either \((m,n,r)\)-triangles (whose \(K(\pi,1)\)-conjecture were proved in [\textit{R. Charney} and \textit{M. W. Davis}, J. Am. Math. Soc. 8, No. 3, 597-627 (1995; Zbl 0833.51006)]) or belong to the 6 examples with edge labeling \((3,4,3,3)\), \((3,4,3,4)\), \((3,4,3,5)\), \((3,5,3,3)\), \((3,5,3,5)\), \((3,3,3,3,4)\). The results of this paper imply that all hyperbolic cyclic type Artin groups satisfy the \(K(\pi,1)\)-conjecture. This article also proves several group theoretic properties for hyperbolic cyclic type Artin groups, including the existence of parabolic closure.