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

TL;DR

This paper proves that in certain infinite-volume hyperbolic 3-manifolds, geodesic planes are either closed or dense, with only countably many being closed, establishing new topological rigidity results.

Contribution

It introduces the first topology-based rigidity theorems for geodesic planes in convex cocompact acylindrical hyperbolic 3-manifolds of infinite volume.

Findings

Any geodesic plane in the convex core is either closed or dense.

Only countably many geodesic planes are closed.

The results depend solely on the topology of the manifold.

Abstract

Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$. In this paper we show that any geodesic plane in $M^*$ 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.