A major theme in dynamics is that rigidity (of a group action on a space, or of a representation) often comes from some underlying geometric phenomenon. In this talk, I will explain a new result with Maxime Wolff for surface group actions, along these lines. We show that the only source of topological rigidity for surface groups acting on the circle is an underlying geometric structure: if an action is rigid, then it comes from an embedding of the group as a lattice in PSL(2,R) or one of its finite extensions, acting in the standard way. The talk will introduce and motivate the broad theme of our work, and explain the philosophy of how to “reconstruct” a hyperbolic surface out of basic dynamical information.