Cyclic Parabolic Quasiconformal Groups That Are Not Quasiconformal Conjugates of M Obius Groups

P. Tukia published in T2] the rst example of a uniformly quasi-isometric and hence quasiconformal group acting on R n , n 3, which is not a quasiconformal conjugate of any MM obius 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 MM obius 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.