The topic of my talk is what the possibilities are for the closure of a totally geodesic immersion of a hyperbolic plane into a complete hyperbolic 3 manifold.  For finite volume hyperbolic 3 manifolds, work of Shah and Ratner shows that strong rigidity properties hold: any such immersed plane is either closed or dense.  I will describe recent joint work with Curt McMullen and Amir Mohammadi  which shows some of this rigidity persists in some cases of infinite volume hyperbolic 3-manifolds

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