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.