Teichmüller spaces and tame quasiconformal motions
Let $E$ be a subset of the Riemann sphere $\widehat{\mathbb{C}}$ and $X$ a connected Hausdorff space with base point $x_0$; a \textit{motion of $E$ over $X$} is a map $\phi: X \times E \to \widehat{\mathbb{C}}$ such that $\phi$ is the identity map evaluated in $x_0$, and injective for all $x \in X$ (so $X$ is the parameter space of the motion $\phi$). In the paper [Acta Math. 157, 243--286 (1986; Zbl 0619.30026)], \textit{D. P. Sullivan} and \textit{W. P. Thurston} introduced the concept of \textit{quasiconformal motion} and claimed that every quasiconformal motion of a subset of $\widehat{\mathbb{C}}$ over the interval $I = [0,1]$ extends to a quasiconformal motion of $\widehat{\mathbb{C}}$; the first main result of the present paper is a counterexample to that claim. Then the authors introduce the concept of a \textit{tame quasiconformal motion} and show that the claim remains true for tame quasiconformal motions; more generally, they show that every tame quasiconformal motion of a subset of $\widehat{\mathbb{C}}$ over a simpy connected Hausdorff space extends to a quasiconformal motions of $\mathbb{C}$ (in a ``conformally natural way''). ``The fundamental idea is to show that the Teichmüller space of a closed set in the sphere is a univeral parameter space for tame quasiconformal motions of that set over a simply connected Hausdorff space.''
scientific article; zbMATH DE number 7040978