Decorated Marked Surfaces with vortices: Cluster braid group vs. braid twist group

Let $\mathbf{S}$ be a marked surface with vortices (=punctures with extra ${\mathbb{Z}}_2$ symmetry). We study the decorated version ${\mathbf{S}}_{△}$, where the ${\mathbb{Z}}_2$ symmetry lifts to the relation that the fourth power of the braid twist of any collision path (connecting a decoration in $△$ and a vortex) is identity. We prove the following three groups are isomorphic: King-Qiu’s cluster braid group associated to $\mathbf{S}$, the braid twist group of ${\mathbf{S}}_{△}$ and the fundamental group of Bridgeland-Smith’s moduli space of $\mathbf{S}$-framed GMN differentials. Moreover, we give finite presentations of such groups.