Geodesic planes in hyperbolic 3-manifolds

Abstract

Let P be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold M. Assume that P^=M^\cap P is nonempty, where M^* is the interior of the convex core of M. Does this condition imply that P is either closed or dense in M? A positive answer would furnish an analogue of Ratner's theorem in the infinite volume setting. In the first part of the thesis, we show that Ratner's theorem fails to generalize to planes in convex cocompact acylindrical 3-manifolds, giving a negative answer to the question above. In contrast, it is shown in [MMO2] that the dichotomy does hold if we restrict to M^: P^ is either closed or dense in M^* for any geodesic plane P intersecting M^. Our result is achieved by exhibiting an exotic plane, i.e. an explicit example of a pair (M,P) such that P^ is closed in M^* but P is not closed in M. Recently, the results of [MMO2] have been generalized to certain, but not all, geometrically finite acylindrical hyperbolic 3-manifolds in [BO]. Notably, the Apollonian manifold M_A, whose limit set is the classical Apollonian gasket, is excluded. In the second part of the thesis, we study the geometry and topology of elementary planes in M_A. The existence of these planes leads to the following failure of isolation, in contrast to the convex cocompact case: there exist sequences of closed geodesic planes limiting only on elementary planes. On the other hand, we show that certain rigidity still holds: the area of an elementary plane in M_A is uniformly bounded above, and the union of all elementary planes is closed.