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. The famous rigidity results of Mostow and Margulis say that many lattices in semi-simple Lie groups have a very rigid structure — what can we say about rigidity of geometric actions?

In this talk, I'll introduce a circle of ideas giving the basic framework and motivation for these and related questions, and then explain new joint work with Maxime Wolff in the low dimensional setting where we have a complete "best possible" answer.