Existence of an exotic plane in an acylindrical 3-manifold

Let P be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold M . Assume that P ^∗ = M ^∗ ∩ 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 [9] it is shown that P ^∗ is either closed or dense in M ^∗. Moreover, there are at most countably many planes with P ^∗ closed, and in all previously known examples, P was also closed in M . In this note we show more exotic behavior can occur: namely, we give an explicit example of a pair (M, P) such that P ^∗ is closed in M ^∗ but P is not closed in M . In particular, the answer to the question above is no. Thus Ratner’s theorem fails to generalize to planes in acylindrical 3-manifolds, without additional restrictions.