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Register a new userMobius Transformations of the Circle Form a Maximal Convergence Group
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We investigate the relationship between quasisymmetric and convergence groups acting on the circle. We show that the Mobius transformations of the circle form a maximal convergence group. This completes the characterization of the Mobius group as a maximal convergence group acting on the sphere. Previously, Gehring and Martin had shown the maximality of the Mobius group on spheres of dimension greater than one. Maximality of the isometry (conformal) group of the hyperbolic plane as a uniform quasi-isometry group, uniformly quasiconformal group, and as a convergence group in which each element is topologically conjugate to an isometry may be viewed as consequences.
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We compare a Gromov hyperbolic metric with the hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between the Gromov hyperbolic metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for the Gromov hyperbolic metric under some families of M{o}bius transformations.
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