Existence of an exotic plane in an acylindrical 3-manifold

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 arXiv:1802.03853 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.