The concept of “quasiconformal motion” was first introduced by Sullivan and Thurston (in ). Theorem 3 of that paper asserted that any quasiconformal motion of a set in the sphere over an interval can be extended to the sphere. In this paper, we give a counter-example to that assertion. We introduce a new concept called “tame quasiconformal motion” and show that their assertion is true for tame quasiconformal motions. We prove a much more general result that, any tame quasiconformal motion of a closed set in the sphere, over a simply connected Hausdorff space, can be extended as a quasiconformal motion of the sphere. Furthermore, we show that this extension can be done in a conformally natural way. The fundamental idea is to show that the Teichmüller space of a closed set in the sphere is a “universal parameter space” for tame quasiconformal motions of that set over a simply connected Hausdorff space.
"Teichmüller spaces and tame quasiconformal motions." Tohoku Math. J. (2) 70 (4) 607 - 631, 2018. https://doi.org/10.2748/tmj/1546570827