Rigidity for planes in hyperbolic $3$-manifolds and slices of Sierpinski carpets

Ratner and Shah showed that every immersed plane in a compact hyperbolic 3-manifold is either closed or dense.  We discuss the extent to which this rigidity persists for hyperbolic 3-manifolds of infinite volume.  The topological type of the 3-manifold plays a decisive role.  Joint work with A. Mohammadi and H. Oh.