Algebraic Ending Laminations and Quasiconvexity

TL;DR

This paper explores algebraic laminations in hyperbolic groups and uses their relationships to establish quasiconvexity of certain subgroups within these groups.

Contribution

It clarifies various notions of algebraic laminations and applies these concepts to prove quasiconvexity results for subgroups in hyperbolic group extensions.

Findings

Multiple notions of algebraic laminations are explicated.

Relationships between laminations are used to derive new quasiconvexity results.

Quasiconvexity of finitely generated subgroups of the normal subgroup is established.

Abstract

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.