1 Background The theory of holomorphic motions, introduced by Mané-Sad-Sullivan [1] and advanced by Slodkowski [2], has had a significant impact on the theory of quasiconformal mappings. A reasonably thorough account of this is given in our book [3]. In [4, 5] we established some classical distortion theorems for quasiconformal mappings and used the theory to develop connections between Schottk...