Suppose a finitely generated group $G$ is hyperbolic relative to ${𝒫}$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a “visual” metric on $\partial (G,{𝒫})$ is “linearly connected” if and only if the boundary $\partial (G,{𝒫})$ has no cut point. Our goal is to produce linearly connected metrics on $\partial (G,{𝒫})$ that are “piecewise” visual when $\partial (G,{𝒫})$ contains cut points. Our main theorem is connected to graph of groups decompositions of relatively hyperbolic groups $(G,{𝒫})$ by work of B. Bowditch. We describe piecewise visual linearly connected metrics on connected boundaries of relatively hyperbolic groups. Our metric on $\partial (G,{𝒫})$ agrees with the visual metric on limit sets of vertex groups and is in this sense piecewise visual.

假设一个有限生成群$G$是双曲线的${𝒫}$的一组适当的有限生成子群$G$. 文献中的既定结果意味着“视觉”指标$\partial (G,{𝒫})$是“线性连接的”当且仅当边界$\partial (G,{𝒫})$没有切点。我们的目标是在$\partial (G,{𝒫})$这是“分段”视觉的$\partial (G,{𝒫})$包含切割点。我们的主要定理与相对双曲群的群分解图有关$(G,{𝒫})$B. Bowditch 的作品。我们描述了相对双曲线组的连接边界上的分段视觉线性连接度量。我们的指标$\partial (G,{𝒫})$同意顶点组极限集的视觉度量，并且在这个意义上是分段视觉的。