Quasiconformal automorphism group of domains

Question

My problem refers to the theory of quasiconformal mappings in $\mathbb{C}$: Let $\emptyset \not = D \subseteq \mathbb{C}$ be a domain (i.e. open and connected subset) - for the sake of simplicity, assume that $D$ is simply connected, for example. My question is: Is there a theory on "quasiconformal automorphisms" of $D$ and the corresponding automorphism group? For example, what can be said about the group of quasiconformal automorphisms $\text{Aut}_{QC}(\mathbb{D})$ of the unit disc $\mathbb{D} = \{ z \in \mathbb{C} \, | \, |z| < 1 \}$ with respect to the group structure and the corresponding mappings? And how can such mappings be constructed in an explicit way?

Any answers are highly appreciated! Thank you in advance!

Answer 1

If you mean Kleinian group type of work, yes, quasiconformal automorphisms of the unit disk have been studied in some papers. This is not a very popular topic, because it is quite hard. There are few explicit formulas for quasiconformal mappings, and usually they have to be studied by using indirect methods such as conformal modulus.

There is no explicit representation theorem for the whole class of quasiconformal mappings by using formulas involving rational functions, Taylor series, etc. tools of classical complex analysis. Of course, you can easily construct simple examples of such mappings by taking some basic example of a quasiconformal mapping (for example, a radial stretching type mapping), and then pre/post compose it with Möbius transformation.

Examples can also be constructed by using the complex dilatation (i.e. the measurable Riemann mapping theorem), but that method gives the function as a solution of certain PDE and usually doesn't lead into any nice formulas.

Antti