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

TL;DR

This paper establishes isomorphisms between three groups related to decorated marked surfaces with vortices, connecting cluster braid groups, braid twist groups, and moduli space fundamental groups, with explicit finite presentations.

Contribution

It proves the isomorphism of three key groups associated with decorated surfaces with vortices and provides their finite presentations, advancing understanding of their algebraic structures.

Findings

Proved the isomorphism between the three groups.

Provided finite presentations of the groups.

Connected cluster braid groups with moduli space fundamental groups.

Abstract

Let $\mathbf{S}$ be a marked surface with vortices (=punctures with extra $\mathbb{Z}_2$ symmetry). We study the decorated version $\mathbf{S}_\bigtriangleup$, 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 $\bigtriangleup$ 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 $\mathbf{S}_\bigtriangleup$ and the fundamental group of Bridgeland-Smith's moduli space of $\mathbf{S}$-framed GMN differentials. Moreover, we give finite presentations of such groups.