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.