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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
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