Planar boundaries and parabolic subgroups (Q6131209)

In the paper under review the authors study the Bowditch boundaries of relatively hyperbolic group pairs, focusing on the case where there are no cut points. The main result is Theorem 8.7: Suppose \(G\) is one ended and \((G,\mathcal{P})\) is relatively hyperbolic. If \((G,\mathcal{P})\) is rigid and \(M=\partial(G,\mathcal{P})\) topologically embeds in \(S^{2}\), then the action of \(G\) on \(M\) extends to an action on \(S^{2}\) by homeomorphisms (see Definition 2.10 for a precise explanation of the terminology used). Using the previous result the authors characterize the peripheral subgroups in the one ended case, even though the action may not extend to \(S^{2}\).