Existence of an exotic plane in an acylindrical 3-manifold

TL;DR

This paper constructs an explicit example of a geodesic plane in an acylindrical hyperbolic 3-manifold where the plane's interior intersection is closed but the plane itself is not, showing a failure of a Ratner-type classification.

Contribution

It provides the first explicit example demonstrating that the closure properties of geodesic planes in acylindrical 3-manifolds can differ from previous expectations, challenging the generalization of Ratner's theorem.

Findings

Existence of a geodesic plane with closed interior intersection but non-closed in the manifold.

Counterexample to the conjecture that such planes must be either closed or dense.

Ratner's theorem does not extend straightforwardly to acylindrical 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 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.