Piecewise Visual, Linearly Connected Metrics on Boundaries of Relatively Hyperbolic Groups
Matthew Haulmark, Michael L. Mihalik

TL;DR
This paper develops piecewise visual, linearly connected metrics on the boundaries of relatively hyperbolic groups with cut points, extending the understanding of boundary metrics beyond the no cut point case.
Contribution
It introduces a method to construct piecewise visual, linearly connected metrics on boundaries of relatively hyperbolic groups with cut points, generalizing previous results.
Findings
Constructed piecewise visual metrics on boundaries with cut points.
Established conditions for linearly connected metrics based on group decompositions.
Extended the class of boundaries where visual metrics can be applied.
Abstract
Suppose a finitely generated group is hyperbolic relative to a set of proper finitely generated subgroups of . Established results in the literature imply that a "visual" metric on is "linearly connected" if and only if the boundary has no cut point. Our goal is to produce linearly connected metrics on that are "piecewise" visual when contains cut points. %Visual metrics for are tightly linked to inner products of geodesic rays in "cusped" spaces for . The identity vertex is usually our base point in these cusped spaces and visual metrics depend on this base point. %We say the visual metric on , with base point , is {\it -equivariant} if for points $x_1,x_2\in \partial(G,\mathcal…
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