For a long time many questions have been answered for hyperbolic 3 manifolds of finite volume, but they are wide open for those of infinite volume. One such question is classification of possible closures of a geodesic plane in a hyperbolic 3-manifold. The main obstacle in the infinite volume case is scarcity of recurrence of unipotent flows, without which we cannot use unipotent dynamics.

In this course, we will discuss recent joint work with McMullen, Mohammadi, and in part with Benoist which establishes a closed or dense dichotomy for a geodesic plane in the interior of the convex core of M for any geometrically finite acylindrical hyperbolic 3-manifold. This discussion boils down to classifying the closures of SL(2, R) orbits in the quotient space Gamma\SL(2,C) where Gamma is a geometrically finite Zariski dense subgroup. When Gamma is not a lattice, no SL(2,R) orbit is dense in Gamma\SL(2,C). Description of possible SL(2,R)-orbit closures involves the canonical fractal set associated to Gamma, called the limit set of Gamma.

We will explain how some of the main techniques in unipotent dynamics can be adapted to the setting of infinite volume, and how the topological structure of the manifold Gamma\H^3 affects the recurrence property of unipotent flows.