We consider two families of subgroups of a group. Assume that each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.
"On Relative Hyperbolicity for a Group and Relative Quasiconvexity for a Subgroup." Tokyo J. Math. 42 (1) 83 - 112, June 2019. https://doi.org/10.3836/tjm/1502179263