Conformal automorphisms of countably connected regions (Q2841083)

A circular region is an open connected set \(D\) such that each component of the complement of \(D\) is either a single point or a closed spherical disk. The conformal automorphism group of \(D\) is the group of all conformal maps of \(D\) to itself. Every region of connectivity less than three is conformally equivalent to either \(\mathbb C_{\infty}\), \(\mathbb C\), \(\mathbb D\), \(\mathbb C\setminus\{0\}\), \(\mathbb D\setminus\{0\}\), or an annulus. The main result of the article is the following theorem.NEWLINENEWLINETheorem 1.1. The conformal automorphism group of a countably connected circular region of connectivity at least three is either a Fuchsian group or a discrete elementary group of Möbius transformations. Furthermore, each Fuchsian group and discrete elementary group arises as the conformal automorphism group of a countably connected circular region.NEWLINENEWLINECorollary 1.2. Each countably connected region of connectivity at least three is conformally equivalent to a region whose conformal automorphism group is either a Fuchsian group or a discrete elementary group of Möbius transformations.NEWLINENEWLINEAccording to Theorem 1.1, the conformal automorphism group of a countably connected circular region of connectivity at least three is conjugate by a Möbius transformation to a discrete subgroup of conformal isometries of either \(\mathbb D\), \(\mathbb C\), \(\mathbb C_{\infty}\) or \(\mathbb C\setminus\{0\}\). For punctured spheres, the author proved a stronger version of Theorem 1.1: The conformal automorphism group of a countably connected punctured sphere of connectivity at least three is a discrete elementary group of Möbius transformations.