Quasi-Isometries, Boundaries and JSJ-Decompositions of Relatively   Hyperbolic Groups
  We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometries and outer automorphisms, as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups. 
Bradley Groff
math.GR