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Cyclic parabolic quasiconformal groups that are not quasiconformal conjugates of Möbius groups. (English) Zbl 0781.30017
P. Tukia published in [Ann. Acad. Sci. Fenn., Ser. A I 6, 89-94 (1981; Zbl 0473.30014)] the first example of a uniformly quasi-isometric and hence quasiconformal group acting on $$R^ n$$, $$n\geq 3$$, which is not a quasiconformal conjugate of any Möbius group. We have analyzed this group and have shown that it contains elements which generate cyclic parabolic uniformly quasi-isometric groups that cannot be conjugated by a quasi-conformal map to a Möbius group. By an argument of Martin these groups can be chosen smooth. Since these groups also act on the upper half-space $$U^ n$$, we can use our result to give a negative answer to a conjecture of Martin and Tukia, where the hope has been that every three- dimensional quasiconformal Fuchsian group, which are groups of quasiconformal homeomorphisms of $$U^ 3$$, was quasiconformally conjugated to a Fuchsian group.
Reviewer: V.Mayer (Bordeaux)

MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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