We prove that a relatively hyperbolic pair $(G,P)$ has Bowditch boundary a 2-sphere if and only if it is a 3-dimensional Poincare duality pair. We prove this by studying the relationship between the Bowditch and Dahmani boundaries of relatively hyperbolic groups.

中文翻译：

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在具有$ S ^ 2 $ Bowditch边界的组上

我们证明了一个相对双曲对$（G，P）$当且仅当它是3维庞加莱对偶对时才具有2个球面的Bowditch边界。我们通过研究相对双曲线群的Bowditch和Dahmani边界之间的关系来证明这一点。