# Finite presentations for spherical/braid twist groups from decorated marked surfaces

## TL;DR

This paper provides finite presentations for braid and spherical twist groups associated with decorated surfaces, facilitating the study of stability conditions in related 3-Calabi-Yau categories.

## Contribution

It introduces explicit finite presentations for these groups derived from decorated surfaces, linking geometric and categorical structures.

## Key findings

- Finite presentation for the braid twist group of decorated surfaces.
- Finite presentation for the spherical twist group of 3-Calabi-Yau categories.
- Application to the simply connectedness of stability condition spaces.

## Abstract

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability conditions on the 3-Calabi-Yau category is simply connected in the sequel.