Let G be a countable group. We show there is a topological relationship
between the space CO(G) of circular orders on G and the moduli space of actions
of G on the circle; as well as an analogous relationship for spaces of left
orders and actions on the line. In particular, we give a complete
characterization of isolated left and circular orders in terms of strong
rigidity of their induced actions of G on S1 and R.
As an application of our techniques, we give an explicit construction of
infinitely many nonconjugate isolated points in the spaces CO(F_{2n}) of
circular orders on free groups disproving a conjecture from Baik--Samperton,
and infinitely many nonconjugate isolated points in the space of left orders on
the pure braid group P_3, answering a question of Navas. We also give a
detailed analysis of circular orders on free groups, characterizing isolated
orders
