Cluster braid groups of Coxeter-Dynkin diagrams

Cluster exchange groupoids are introduced by King-Qiu as an enhancement of cluster exchange graphs to study stability conditions and quadratic differentials. In this paper, we introduce the exchange groupoid for any finite Coxeter-Dynkin diagram $\mathrm{\Delta }$ and show that the fundamental group of which is isomorphic to the corresponding braid group associated with $\mathrm{\Delta }$.