Intersection of Parabolic Subgroups in Euclidean Braid Groups: a short proof

Abstract

We give a short proof for the fact, already proven by Thomas Haettel, that the arbitrary intersection of parabolic subgroups in Euclidean Braid groups $A[\tilde{A}_n]$ is again a parabolic subgroup. To that end, we use that the spherical-type Artin group $A[B_{n+1}]$ is isomorphic to $A[\tilde{A}_n] \rtimes \mathbb{Z}$.

Figures (2)

• Figure 1: The Coxeter graph $\Tilde{A}_n$.
• Figure 2: The Coxeter graph $B_{n+1}$.

Theorems & Definitions (4)

• Theorem 1: Haettel
• Theorem 2
• Remark 1
• proof : Proof of \ref{['maintheorem']}