Topological properties of quasiconformal automorphism groups

Abstract

Let G subsetneq mathbb {C} be a bounded, simply connected domain in mathbb {C}, and denote by Q(G):=f:G⟶G|fis a quasiconformal mapping ofGontoG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} Q(G):= \\left\\{ f: G \\longrightarrow G \\ |\\ f \ ext { is a quasiconformal mapping of } G \ ext { onto } G \\ \\right\\} \\end{aligned}$$\\end{document}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 dsup(f,g):=supz∈Gf(z)-g(z),f,g∈Q(G).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} d_{\\sup }(f, g):= \\sup \\limits _{z \\in G} \\left| f(z) - g(z) \\right| , f, g \\in Q(G). \\end{aligned}$$\\end{document}In this paper, we present results concerning topological properties of Q(G) such as completeness, separability, path–connectedness, discreteness and compactness.