We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi–Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.

中文翻译：

---

集群交换groupoids和框架二次微分

我们引入了与具有潜力的非退化箭袋相关联的集群交换 groupoid，作为集群交换图的增强。在由（未穿孔的）标记表面产生的情况下，其中交换图以标记表面的三角剖分图为模型，我们表明可以使用表面三角剖分的覆盖图来构建该 groupoid 的通用覆盖带有额外的装饰。这个覆盖图是表面上适当框架的二次微分空间的骨架，它反过来模拟了与标记表面相关的 3-Calabi-Yau 类别的 Bridgeland 稳定性条件空间。通过证明覆盖群中的关系在解释为稳定条件空间中的环时是同调微不足道的，