Characterising large-type Artin groups (Q6647809)

An Artin group is of large-type if all the labels of the presentation graph are at least \(3\). It is well known that the class of large-type Coxeter groups is not invariant under isomorphism: the dihedral group \(D_{12}\) of order \(12\) (with presentation graph an edge with label \(6\)) is isomorphic to the direct product \(D_{6} \times C_{2}\) (with presentation graph a triangle with labels \(2, 2\) and \(3\)). By contrast, the main result in the paper under review is:\N\NTheorem A. The class of large-type Artin groups is invariant under isomorphism.\N\NUnderstanding whether classes of Artin groups are invariant under isomorphism, has wider implications for the Isomorphism Problem for Artin groups, that is, the problem of determining which presentation graphs yield isomorphic Artin groups. Thanks to Theorem A, the authors characterize Artin groups that are rigid, that is, that have only one presentation graph up to isomorphism of presentation graphs, using the notion of twist-equivalent presentation graphs in the sense of [\textit{N. Brady} et al., Geom. Dedicata 94, 91--109 (2002; Zbl 1031.20035)]. Corollary B: Let \(A_{\Gamma}\) be a large-type Artin group. If \(A_{\Gamma'}\) is any Artin group isomorphic to \(A_{\Gamma}\), then \(\Gamma'\) is twist-equivalent to \(\Gamma\). In particular, \(A_{\Gamma}\) is rigid if and only if any presentation graph twist-equivalent to \(\Gamma\) is isomorphic to \(\Gamma\).