The universal relatively hyperbolic structure on a group and relative   quasiconvexity for subgroups
  We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a group and relative quasiconvexity for subgroups of the group. 
Yoshifumi Matsuda, Shin-ichi Oguni, Saeko Yamagata
math.GR