Algebraic Ending Laminations and Quasiconvexity

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.