M\"obius Transformations of the Circle Form a Maximal Convergence Group

We investigate the relationship between quasisymmetric and convergence groups acting on the circle. We show that the M\"obius transformations of the circle form a maximal convergence group. This completes the characterization of the M\"obius group as a maximal convergence group acting on the sphere. Previously, Gehring and Martin had shown the maximality of the M\"obius 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.