Planar boundaries and parabolic subgroups

IF 0.6 3区 数学 Q3 MATHEMATICS Mathematical Research Letters Pub Date : 2024-04-03 DOI:10.4310/mrl.2023.v30.n4.a5
G. Christopher Hruska, Genevieve S. Walsh
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引用次数: 0

Abstract

We study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. We show that if $(G, \mathcal{P})$ is a rigid relatively hyperbolic group pair whose boundary embeds in $S^2$, then the action on the boundary extends to a convergence group action on $S^2$. More generally, if the boundary is connected and planar with no cut points, we show that every element of $\mathcal{P}$ is virtually a surface group. This conclusion is consistent with the conjecture that such a group $G$ is virtually Kleinian. We give numerous examples to show the necessity of our assumptions.
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平面边界和抛物线子群
我们研究了相对双曲群对的鲍迪奇边界,重点是没有切点的情况。我们证明,如果 $(G, \mathcal{P})$ 是刚性相对双曲群对,其边界嵌入 $S^2$,那么边界上的作用会扩展为 $S^2$ 上的收敛群作用。更一般地说,如果边界是连通的、平面的、没有切点,我们就能证明 $\mathcal{P}$ 的每个元素实际上都是一个曲面群。这一结论与这样一个群 $G$ 实际上是克莱因群的猜想是一致的。我们举了许多例子来证明我们假设的必要性。
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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
9
审稿时长
6.0 months
期刊介绍: Dedicated to publication of complete and important papers of original research in all areas of mathematics. Expository papers and research announcements of exceptional interest are also occasionally published. High standards are applied in evaluating submissions; the entire editorial board must approve the acceptance of any paper.
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