  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.
