Topological properties of quasiconformal automorphism groups

Abstract

AbstractLet $$G \subsetneq \mathbb {C}$$ G ⊊ C be a bounded, simply connected domain in $$\mathbb {C}$$ C , and denote by $$\begin{aligned} Q(G):= \left\{ f: G \longrightarrow G \ |\ f \text { is a quasiconformal mapping of } G \text { onto } G \ \right\} \end{aligned}$$ Q ( G ) : = f : G ⟶ G | f is a quasiconformal mapping of G onto G the quasiconformal automorphism group of G. In a canonical manner, the set Q(G) carries the structure of a (non–abelian) group with respect to composition of mappings. Moreover, we endow the set Q(G) with the topology of uniform convergence by the supremum metric $$\begin{aligned} d_{\sup }(f, g):= \sup \limits _{z \in G} \left| f(z) - g(z) \right| , f, g \in Q(G). \end{aligned}$$ d sup ( f , g ) : = sup z ∈ G f ( z ) - g ( z ) , f , g ∈ Q ( G ) . In this paper, we present results concerning topological properties of Q(G) such as completeness, separability, path–connectedness, discreteness and compactness.