Intersection of parabolic subgroups in Euclidean braid groups: a short proof (Q6639471)

In the paper under review the authors give a short proof for the fact, already proven by \textit{T. Haettel} in [``Lattices, injective metrics and the \(K (\pi,1)\) conjecture'', Preprint, \url{arXiv:2109.07891}], that the arbitrary intersection of parabolic subgroups in Euclidean braid groups \(A[\widetilde{A}_{n}]\) is again a parabolic subgroup. To do this, they use that the spherical-type Artin group \(A[B_{n+1}]\) is isomorphic to \(A[\widetilde{A}_{n}]\rtimes \mathbb{Z}\).