On canonical splittings of relatively hyperbolic groups (Q6584677)

A word hyperbolic group \(G\) has a natural Gromov boundary at infinity \(\partial G\), whose topological properties are often closely related to properties of \(G\). A theorem of this type states that \(\partial G\) is homeomorphic to a circle if and only if \(G\) is a cocompact Fuchsian group (see [\textit{P. Tukia}, J. Reine Angew. Math. 391, 1--54 (1988; Zbl 0644.30027); \textit{D. Gabai}, Ann. Math. (2) 136, No. 3, 447--510 (1992; Zbl 0785.57004); \textit{A. Casson} and \textit{D. Jungreis}, Invent. Math. 118, No. 3, 441--456 (1994; Zbl 0840.57005)]). \textit{B. Bowditch} [Acta Math. 180, No. 2, 145--186 (1998; Zbl 0911.57001)] used this result to prove that each hyperbolic group with connected boundary has a canonical JSJ tree for splittings over \(2\)-ended subgroups. A JSJ decomposition of a group is a splitting that allows one to classify all possible splittings of the group over a certain family of edge groups. Although JSJ decompositions are not unique in general, \textit{V. Guirardel} and \textit{G. Levitt} [Geom. Topol. 15, No. 2, 977--1012 (2011; Zbl 1272.20026)] constructed a canonical JSJ decomposition, the tree of cylinders, which classifies splittings of relatively hyperbolic groups over elementary subgroups.\N\NIn the paper under review, the authors give a new topological construction of the Guirardel-Levitt tree of cylinders, and they show that this tree depends only on the homeomorphism type of the Bowditch boundary. Furthermore, the tree of cylinders admits a natural action by the group of homeomorphisms of the boundary. In particular, the quasi-isometry group of \((G, \mathbb{P})\) acts naturally on the tree of cylinders.