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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.

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 57M50 General geometric structures on low-dimensional manifolds
##### Keywords:
Kleinian group; quasi-conformal conjugacy
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##### References:
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