Piecewise Visual, Linearly Connected Metrics on Boundaries of Relatively Hyperbolic Groups

Abstract

Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly connected" if and only if the boundary $\partial (G,\mathcal P)$ has no cut point. Our goal is to produce linearly connected metrics on $\partial (G,\mathcal P)$ that are "piecewise" visual when $\partial (G,\mathcal P)$ contains cut points. %Visual metrics for $\partial (G,\mathcal P)$ are tightly linked to inner products of geodesic rays in "cusped" spaces for $(G,\mathcal P)$. The identity vertex $\ast$ is usually our base point in these cusped spaces and visual metrics depend on this base point. %We say the visual metric $d_p$ on $\partial(G,\mathcal P)$, with base point $p$, is {\it $G$-equivariant} if for points $x_1,x_2\in \partial(G,\mathcal P)$, we have $d_p(x_1,x_2)=d_{gp}(gx_1,gx_2)$ for all $g\in G$. Our main theorem is about graph of groups decompositions of relatively hyperbolic groups $(G,\mathcal P)$, and piecewise visual metrics on their boundaries. We assume that each vertex group of our decomposition has a boundary with linearly connected visual metric or the vertex group is in $\mathcal P$. If a vertex group is not in $\mathcal P$, then it is hyperbolic relative to its adjacent edge groups. Our linearly connected metric on $\partial (G,\mathcal P)$ agrees with the visual metric on limit sets of vertex groups and is in this sense piecewise visual.