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Topologically conjugate Kleinian groups. (English) Zbl 0845.30032
Let \(\Gamma_1\) and \(\Gamma_2\) be finitely generated, torsion free Kleinian groups (acting on the 3-dimensional hyperbolic space and on its boundary \(S^2\)). We suppose that both \(\Gamma_1\) and \(\Gamma_2\) have positive injectivity radius. We also suppose that neither \(\Gamma_1\) and \(\Gamma_2\) can be expressed as a free product of two of their subgroups. Then the author poves that if there is a homeomorphism conjugating the action of \(\Gamma_1\) on \(S^2\) to that of \(\Gamma_2\) there is a quasi-conformal homeomorphism with the same property. The theorem is deduced from a theorem of Minsky by techniques of Bonahon and Waldhausen.

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M50 General geometric structures on low-dimensional manifolds
Full Text: DOI
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