  We introduce the notions of geometric height and graded (geometric) relative
hyperbolicity in this paper. We use these to characterize quasiconvexity in
hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and
convex cocompactness in mapping class groups and $Out(F_n)$.
  Corrigendum: there is an unfortunate mistake in the statement and the proof
of Proposition 5.1. This affects one direction of the implications of the main
theorem. A correction is given, that states that given a quasi-convex subgroup
of a hyperbolic (or relatively hyperbolic) group, the graded relative
hyperbolic structure holds with respect to saturations of I-fold intersections,
that are stabilizers of limit sets of I-fold intersections.
