Narrow Results

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.