Geodesic planes in geometrically finite acylindrical 3-manifolds

TL;DR

This paper investigates the behavior of geodesic planes in geometrically finite acylindrical hyperbolic 3-manifolds, showing they are either closed or dense, with only countably many closed planes, extending previous results to a broader class.

Contribution

It extends the classification of geodesic planes from convex cocompact to geometrically finite acylindrical hyperbolic 3-manifolds, and explores the influence of arithmeticity on their topological behavior.

Findings

Geodesic planes in M* are either closed or dense.

There are only countably many closed geodesic planes.

Topological behavior in covers by arithmetic manifolds is governed by the base manifold.

Abstract

Let $M$ be a geometrically finite acylindrical hyperbolic 3-manifold and let $M^*$ denote the interior of the convex core of M. We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$. These results were obtained earlier by McMullen, Mohammadi, and the second named author when M is convex cocompact. As a corollary we obtain that when $M$ covers an arithmetic hyperbolic 3-manifold $M_0$, the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$. We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.