  Given a finitely generated group $G$ that is relatively finitely presented
with respect to a collection of peripheral subgroups, we prove that every
infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of
$G$ is conjugate into a peripheral subgroup. As an application, we obtain a
trichotomy for subgroups of relatively hyperbolic groups. Moreover we prove the
existence of the relative exponential growth rate for all subgroups of limit
groups.
