Limits of geometrically tame Kleinian groups (Q582417)

Let \(\Gamma\) be a finitely generated Kleinian group with the associated hyperbolic manifold \(M={\mathbb{H}}^ 3/\text{Russian{E}}\). Suppose that \(\Gamma\) is geometrically tame, i.e. the minimal convex hyperbolic 3- manifold in \({\mathbb{H}}^ 3/\Gamma\) which is a deformation retract of \({\mathbb{H}}^ 3/\Gamma\) has finite volume. Consider the deformation space of the hyperbolic structure of M, and let AH(M) denote the space of faithful representations with discrete image from the fundamental group of M to \(PSL_ 2(C)\) modulo conjugacy with the topology of algebraic convergence. In his main theorem, the author unifies and extends work of Thurston, Morgan-Shalen and himself to determine when a sequence in AH(M) has a converging subsequence. When \(\Gamma\) is a Fuchsian group of first kind, he proves as another main result that every geometrically finite group of first kind, he proves as another main result that every geometrically finite group in AH(M) is a boundary group for the subspace of quasi- Fuchsian groups. This result can be regarded as a generalization of a theorem of Abikov on regular b-groups.