Can I find automorphism group of the domain $\mathbb{D} - a $ ? where $\mathbb{D}$ is the unit disc with center 0 in the complex plane and $a$ is a point in $\mathbb{D}$
2026-03-29 01:35:46.1774748146
finding an automorphism group of the unit disc
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A holomorphic map of a punctured disk to itself is bounded (because the disk is bounded), so the isolated singularity at the puncture is removable. Thus, an automorphism of a punctured disk extends to an automorphism of the disk...
EDIT: with a single point missing from the disk, the extended automorphism must fix that point. The group of holomorphic automorphisms of the open disk is transitive, so without loss of generality the missing/fixed point is $0$. Then Schwarz inequality proves that the automorphisms of the disk fixing $0$ consist of rotations.
With two (or more) points missing, again send one to $0$, and then ask how many rotations fix a non-zero point. No non-trivial ones. So the group would be trivial...