Piecewise visual, linearly connected metrics on boundaries of relatively hyperbolic groups

Abstract

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