An action of a finitely generated group G on a manifold M is called "geometric" if it comes from an embedding of G as a lattice in a Lie group acting transitively on M. In this talk, I will explain new joint work with Maxime Wolff that characterizes geometric actions of surface groups on the circle by (a strong form of) topological rigidity. Part of our motivation comes from the notion of nonlinear analogs of character varieties for spaces of group actions, and I will describe some of this perspective in the talk.
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