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Quasiconformal groups acting on $$B^ 3$$ that are not quasiconformally conjugate to Möbius groups. (English) Zbl 0851.30008
It is shown that there is a quasiconformal group acting on the unit ball of $$\mathbb{R}^3$$ which is not quasiconformally conjugate to any Möbius group. The construction is based on Tukia’s idea on the rigidity of quasiconformal maps in this context [P. Tukia, Ann. Acad. Sci. Fenn., Ser. A I 6, 149-160 (1981; Zbl 0473.30015)]. In the plane the situation is different (Sullivan, Tukia). For $$n\geq 4$$ there exist simpler constructions [G. Martin, Ann. Acad. Sci. Fenn., Ser. A I 11, 179-202 (1986; Zbl 0635.30021)], [O. Martio and J. Väisälä, Math. Ann. 282, No. 3, 423-443 (1988; Zbl 0632.35021)].

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
##### Keywords:
quasiconformal group
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