×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI