Geodesic planes in the convex core of an acylindrical 3-manifold

Abstract

Let M be a convex cocompact, acylindrical hyperbolic 3-manifold of infinite volume, and let $M^{\ast }$ denote the interior of the convex core of M. In this paper we show that any geodesic plane in $M^{\ast }$ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theorems for planes in convex cocompact 3-manifolds of infinite volume that depend only on the topology of M.