A characterization of simple and double connectivity (Q1128353)

Let \(\Omega\) be a region in the complex plane, and let \(\Aut(\Omega)\) denote the group of all conformal automorphisms of \(\Omega\). \(\mathbb{T}\) denotes the unit circle, regarded as a multiplicative group. Consider the condition (c) There is an isomorphism \(\Phi\) of \(\mathbb{T}\) onto a subgroup \({\mathcal G}\) of \(\Aut(\Omega)\) such that the map \((e^{i\theta}, z)\to f_\theta (z)\) is continuous from \(\mathbb{T}\times \Omega \to\Omega\), where \(f_\theta:= \Phi(e^{i\theta})\). The main result of the paper is that (c) is a necessary and sufficient condition for \(\Omega\) to be either simply or doubly connected. The necessity is not difficult to prove. One takes \(\Phi(e^{i\theta}): =\psi^{-1} \circ(e^{i\theta} \psi)\) or \(\Phi(e^{i\theta}): =\psi^{-1} \circ(e^{-i\theta} \psi)\) where \(\psi\) is a conformal map of \(\Omega\) onto the canonical region equivalent to \(\Omega\). The proof of sufficiency is more delicate, though still elementary. One studies the orbits \(O_z: =\{f_\theta(z): e^{i\theta} \in\mathbb{T}\}\), and shows that they provide a foliation of \(\Omega\). A corollary of the proof yields that there is at most one point of \(\Omega\) fixed by every conformal automorphism in \({\mathcal G}\); if there is such a fixed-point, then \(\Omega\) is simply connected, and otherwise \(\Omega\) is doubly connected. This easily implies a 1934 result of G. Aumann and C. Carathéodory: Suppose that \(\Omega\) is a bounded region of the complex plane, that \(a\) is a point of \(\Omega\), and that the group \(\Aut (\Omega,a)\) of conformal automorphisms of \(\Omega\) that fix \(a\) is not finite. Then \(\Omega\) is simply connected. Finally, let \(\Omega\) be bounded and simply connected, and let \(a\) be the point fixed by all the automorphisms of \({\mathcal G}\). Then a technique of H. Cartan can be used to exhibit a conformal map \(F\) of \(\Omega\) onto a disk with \(F(a)\) its center.