  Let $1 \to K \longrightarrow G \stackrel{\pi}\longrightarrow Q$ be an exact
sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let
$G_1=\pi^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed
surface group or a free group and $Q$ is convex cocompact), we show that
infinite index quasiconvex subgroups of $G_1$ are quasiconvex in $G$. Related
results are proven for metric bundles, developable complexes of groups, and
graphs of groups.
