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 of the decorated marked surface S, the universal cover of this groupoid can be constructed using decorated triangulations of S. Such a covering graph is a skeleton for a space of suitably framed quadratic differentials on S, which in turn models the space Stab(S) of Bridgeland stability conditions for the 3-Calabi-Yau category associated to S. Finally, we show that Stab(S) is simply connected.