Geodesic planes in a geometrically finite end and the halo of a measured lamination
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| Format: | Preprint |
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2022
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| _version_ | 1866913468787982336 |
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| author | Torkaman, Tina Zhang, Yongquan |
| author_facet | Torkaman, Tina Zhang, Yongquan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_03937 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Geodesic planes in a geometrically finite end and the halo of a measured lamination Torkaman, Tina Zhang, Yongquan Geometric Topology Dynamical Systems 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. |
| title | Geodesic planes in a geometrically finite end and the halo of a measured lamination |
| topic | Geometric Topology Dynamical Systems |
| url | https://arxiv.org/abs/2210.03937 |