What is the stabilizer of a point \(z_0 \in \mathbb{D}\) in the open unit disk under the group of conformal automorphisms \(\operatorname{Aut}(\mathbb{D})\)? Describe the group structure and express the automorphisms explicitly.
Solution
Let \[ \phi_{z_0}(z)=\frac{z-z_0}{1-\overline{z_0}z}. \] Then \(\phi_{z_0}\in\operatorname{Aut}(\mathbb D)\) and \(\phi_{z_0}(z_0)=0\).
By Schwarz’s lemma, an automorphism of \(\mathbb D\) fixing \(0\) has the form \[ z\longmapsto e^{i\theta}z. \] Hence \[ \operatorname{Stab}_{\operatorname{Aut}(\mathbb D)}(0) =\{z\mapsto e^{i\theta}z\} \cong U(1). \] Conjugating by \(\phi_{z_0}\) gives \[ \operatorname{Stab}(z_0) =\phi_{z_0}^{-1}\,U(1)\,\phi_{z_0}. \] Thus every element of the stabilizer is \[ f_\theta(z) =\phi_{z_0}^{-1}\!\left(e^{i\theta}\phi_{z_0}(z)\right), \] and \[ \boxed{\operatorname{Stab}(z_0)\cong U(1)\cong SO(2)}. \]