Local connectedness of boundaries for relatively hyperbolic groups (Q6564523)

A group pair \((\Gamma, \mathbb{P})\) consists of a group \(\Gamma\) together with a finite collection \(\mathbb{P}\) of subgroups of \(\Gamma\). Given a group pair \((\Gamma, \mathbb{P})\), an action of \(\Gamma\) on \(X\) is relative to \(\mathbb{P}\) if each member of \(\mathbb{P}\) has a fixed point in \(X\). A group pair is relatively hyperbolic if the pair admits a relatively hyperbolic action on a graph \(K\). The special case in which \(\mathbb{P} =\emptyset\) is reduces to the notion of a word hyperbolic group. \textit{B. H. Bowditch} [Int. J. Algebra Comput. 22, No. 3, 1250016, 66 p. (2012; Zbl 1259.20052)] (the reviewer points out that article was known since 1997 under the same title as a preprint edited by the University of Southampton) introduces a natural boundary of a relatively hyperbolic pair, the Bowditch boundary, and establishes many of its fundamental properties. \textit{D. V. Osin} [Mem. Am. Math. Soc. 843, 100 p. (2006; Zbl 1093.20025)] extends the notion of relative hyperbolicity to groups that need not be finitely generated.\N\NThe main result of the paper under review is Theorem 1.1: Let \((\Gamma,\mathbb{P})\) be a relatively hyperbolic group pair with connected boundary \(M= \partial(\Gamma, \mathbb{P})\). Then \(M\) is locally connected and every cut point of \(M\) is the fixed point of a parabolic subgroup. The boundary \(M\) has a cut point with stabilizer \(P \in \mathbb{P}\) if and only if \(\Gamma\) splits relative to \(\mathbb{P}\) over a subgroup of \(P\).\N\NThe authors, in Theorem 1.2, deal with the case where the boundary of \((\Gamma, \mathbb{P})\) is connected and contains cut points. They prove that the cut point tree (in the sense of [\textit{P. Papasoglu} and \textit{E. Swenson}, Algebr. Geom. Topol. 6, 1759--1784 (2006; Zbl 1182.54039)]) is simplicial and provides a topological characterization of the boundary as the inverse limit of a tree of compacta (in the sense of [\textit{J. Świątkowski}, Geom. Topol. 24, No. 2, 533--592 (2020; Zbl 1530.20149)]).