On canonical splittings of relatively hyperbolic groups

A JSJ decomposition of a group is splitting that allows one to classify all possible splittings the over certain family edge groups. Although decompositions are not unique in general, Guirardel–Levitt have constructed canonical decomposition, tree cylinders, which classifies relatively hyperbolic groups elementary subgroups. In this paper, we give new topological construction and show depends only on homeomorphism type Bowditch boundary. Furthermore, cylinders admits natural action by homeomorphisms particular, quasi-isometry (G, ?) acts naturally cylinders.