  We explicate a number of notions of algebraic laminations existing in the
literature, particularly in the context of an exact sequence $$1\to H\to G \to
Q \to 1 $$ of hyperbolic groups. These laminations arise in different contexts:
existence of Cannon-Thurston maps; closed geodesics exiting ends of manifolds;
dual to actions on $\R-$trees.
  We use the relationship between these laminations to prove quasiconvexity
results for finitely generated infinite index subgroups of $H$, the normal
subgroup in the exact sequence above.
