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Cores of hyperbolic $$3$$-manifolds and limits of Kleinian groups. II. (English) Zbl 0959.30028
In this paper the authors prove that if $$G$$ is a finitely generated group and if $$(\rho_j)$$ is a sequence of embeddings of $$G$$ into $$\text{PSL}_2(\mathbb{C})$$ as Kleinian groups of a fixed type which converge algebraically to an embedding $$\rho$$ so that $$\Omega(\rho (G))$$ is non-empty then the sequence also converges geometrically to $$\rho(G)$$, i.e. $$\rho(G)$$ is the derived set of $$\cup_j \rho_j(G)$$. Moreover the limit sets $$\Lambda (\rho_j(G))$$ of the $$\rho_j(G)$$ converge to $$\Lambda (\rho(G))$$ in the Hausdorff topology. This is a close approximation to a conjecture of T. Jørgensen. The proof consists of a very delicate and eclectic geometric argument.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57-XX Manifolds and cell complexes
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