Topological properties of quasiconformal automorphism groups

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