Linearity of generalized cactus groups (Q6051015)

For an integer \(n \geq 1\), the cactus group \(J_{n}\) is the group generated by \(s_{p,q}\) for \(1 \leq p < q \leq n\) with relations: \(s_{p,q}^{2}=1\), \([s_{p,q},s_{m,r}]=1\) if \([m,r] \cap [p,q]=\emptyset\), \(s_{p,q}s_{m,r}=s_{p+q-r,p+q-m}s_{p,q}\) if \([m,r] \subset [p,q]\). There is a homomorphism \(g : J_{n} \rightarrow S_{n}\) given by \(s_{p,q} \mapsto \sigma_{p,q}\), where \(\sigma_{p,q}\) reverses the order of elements \(p, \dots, q\) and leaves the rest unchanged. The kernel of \(g\) is the pure cactus group \(\Gamma_{n}\). \textit{J. Mostovoy} in [Arch. Math. 113, No. 3, 229--235 (2019; Zbl 1512.20122)] proved that \(\Gamma_{n}\) embeds into a right-angled Coxeter group (and therefore is residually nilpotent). In the paper under review, the author generalizes Mostovoy's results to cactus groups associated with arbitrary finite Coxeter groups and deduces the linearity of generalized cactus groups.