  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.
