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

IF 1.5 1区 数学 Q1 MATHEMATICS
Tina Torkaman , Yongquan Zhang
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引用次数: 0

Abstract

Recent works [22], [23], [3], [33] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifold 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 ∂E, but cannot be separated from the core by a support plane of ∂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 a finite intersection number with a measured lamination L but is not asymptotic to any leaf nor eventually disjoint from L. We establish that exotic rays exist if and only if 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 only in terms of the hyperbolic surface ∂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.

几何有限端中的大地平面和测量层叠的光环
最近的研究 [22]、[23]、[3]、[33] 揭示了无限体积几何有限双曲三芒星 M 凸核中大地平面的拓扑行为。在本文中,我们将重点研究 M 的凸核之外的其余测地平面,给出它们在 M 中的闭包的完整分类。这里,奇异屋顶是包含在 M 的端 E 中的大地平面,它限制在凸核边界 ∂E上,但不能通过 ∂E 的支撑平面与凸核分离。这里的奇异射线是指与测量层理 L 有有限交点数,但不渐近于任何叶片也不最终与 L 不相交的大地射线。我们还证明,满足比奇异射线更强条件的大地射线的存在,即满足双曲面 ∂E 和弯曲层理的条件,足以证明奇异屋顶的存在。因此,我们证明了具有奇异屋顶的几何有限端在每一属中都存在。此外,在第 1 属中,当末端与穿刺环同构时,一般的末端(在贝雷范畴的意义上)包含不可计数的奇异屋顶。
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来源期刊
Advances in Mathematics
Advances in Mathematics 数学-数学
CiteScore
2.80
自引率
5.90%
发文量
497
审稿时长
7.5 months
期刊介绍: Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.
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