Geodesic planes in a geometrically finite end and the halo of a measured lamination

TL;DR

This paper classifies the closures of geodesic planes outside the convex core of geometrically finite hyperbolic 3-manifolds, revealing the role of exotic roofs and rays related to bending laminations.

Contribution

It provides a complete classification of geodesic plane closures outside the convex core, introducing the concept of exotic roofs and rays, and establishing their existence in all genera.

Findings

Exotic roofs exist in every genus of geometrically finite ends.

Exotic rays exist if and only if the bending lamination is not a multicurve.

Generic ends in genus 1 contain uncountably many exotic roofs.

Abstract

Recent works [MMO1, arXiv:1802.03853, arXiv:1802.04423, arXiv:2101.08956] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifolds $M$ of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of $M$, giving a complete classification of their closures in $M$. In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end $E$ of $M$, which limits on the convex core boundary $\partial E$, but cannot be separated from the core by a support plane of $\partial E$. A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has finite intersection number with a measured lamination $\mathcal{L}$ but is not asymptotic to any leaf nor eventually disjoint from $\mathcal{L}$. We establish that exotic rays exist if and only if $\mathcal{L}$ is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs. We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased in terms of only the hyperbolic surface $\partial E$ and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus $1$, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.