Canonical JSJ for Relatively Hyperbolic Groups
JSJ decompositions first appeared in the context of 3-manifolds with the work of Jaco-Shalen and Johannson. Generalizations of JSJ theory to finitely generated groups have been studied in different contexts by many authors (Kropholler, Rips-Sela, Papasoglu-Swenson, Bowditch, Guirardel-Levitt, etc.). Guirardel and Levitt have shown that if G is one ended and hyperbolic relative to a finite collection of finitely generated subgroups, then there is a relative JSJ tree for G over the class of elementary subgroups. In a joint work with Chris Hruska we show that Guirardel and Levitt’s result can be obtained from the topology of the Bowditch boundary. This implies that the relative JSJ tree for G is a “relative” quasi-isometry invariant.