  We obtain a number of finiteness results for groups acting on
Gromov-hyperbolic spaces. In particular we show that a torsion-free locally
quasiconvex hyperbolic group has only finitely many conjugacy classes of
$n$-generated one-ended subgroups. We also show that the rank problem is
solvable for the class of torsion-free locally quasiconvex hyperbolic groups
(even though it is unsolvable for the class of all torsion-free hyperbolic
groups). We apply our results to 3-manifold groups. Namely, suppose $G$ is the
fundamental group of a closed hyperbolic 3-manifold fibering over a circle and
suppose that all finitely generated subgroups of $G$ are topologically tame. We
prove that for any $k\ge 2$ the group $G$ has only finitely many conjugacy
classes of non-elementary freely indecomposable $k$-generated subgroups of
infinite index in $G$.
