Homeomorphism types of the Bowditch boundaries of infinite-ended relatively hyperbolic groups (Q6661334)

The notion of a hyperbolic group was introduced by \textit{M. Gromov} in the seminal paper [Publ., Math. Sci. Res. Inst. 8, 75--263 (1987; Zbl 0634.20015)]. \textit{B. H. Bowditch} [Int. J. Algebra Comput. 22, No. 3, 1250016, 66 p. (2012; Zbl 1259.20052)], introduced a natural boundary for such groups, the Bowditch boundary, and established many of its fundamental properties. \textit{A. Martin} and \textit{J. Swiatkowski}, in [J. Group Theory 18, No. 2, 273--289 (2015; Zbl 1329.20054)], proved that the topology of the Gromov boundary of \(A\ast B\), where \(A\) and \(B\) are hyperbolic groups, is uniquely determined by the topology of the Gromov boundary of \(A\) and \(B\).\N\NIn this paper, the author proves Theorem 1.1: For \(n \geq 2\), suppose \(G_{1}=A_{1} \ast \ldots \ast A_{n}\) and \(G_{2}=B_{1} \ast \ldots \ast B_{n}\) are two free products of non-elementary relatively hyperbolic groups. Suppose that, for \(1 \leq i \leq n\), the Bowditch boundary of \(A_{i}\) is homeomorphic to the Bowditch boundary of \(B_{i}\). Then the Bowditch boundary of \(G_{1}\) is homeomorphic to the Bowditch boundary of \(G_{2}\).\N\NFor a graph of groups \(\mathcal{G}\) whose vertex groups are relatively hyperbolic, let \(h(\mathcal{G})\) be the set of homeomorphism types (without multiplicity) of the Bowditch boundaries of non-elementary relatively hyperbolic vertex groups of \(\mathcal{G}\). In Theorem 1.2 the author generalizes Theorem 1.1 to graphs of relatively hyperbolic groups by showing that if \(\mathcal{G}\) and \(\mathcal{H}\) are two finite graphs of groups, and some particular conditions are satisfied, then the equality \(h(\mathcal{G}) =h(\mathcal{H})\) implies that the Bowditch boundary of the fundamental group of \(\mathcal{G}\) is homeomorphic to the Bowditch boundary of the fundamental group of \(\mathcal{H}\).