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I've been reading Greene and Krantz's Functions Theory of One Complex Variable, and have been getting stuck on some of the problems what he calls Aut($\Omega$), I.e., the automorphism group of $\Omega$ where Aut is a collection of conformal mappings. He then goes on to Topologize the group by saying a sequence $\{f_j\} \subseteq $ Aut($\Omega$) if it does so uniformly on compact sets. I thus have the general following questions related to these problems (6.3-6.12):

What sorts of domains yield different properties of Aut($\Omega$)?

For example, I know naturally that if $\Omega$ is conformally equivalent to the unit disk, $\Delta$, then $f$ must take the form: $f(z)=e^{i \theta}\bigg( \frac{z+a}{1+\overline{a}z}\bigg)$. Also, if $\Omega$ as defined in the problem is conformally equivalent to $\mathbb{C}$, then $f$ must take the form: $f(z)=a+bz, a \neq 0$.

So, would either $\Delta$ or $\mathbb{C}$ be examples of a bounded domain when Aut$(\Omega)$ is not compact. Or would they or some other domain be an example of when it is compact but not finite; finite but not trivial; finite and trivial; etc.

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    $\Omega = \mathbb C\setminus\{0\}$ also has noncompact $\text{Aut}(\Omega)$: it contains $\Omega$ (as multiplication). – 2017-02-17
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    So would any annulus be a bounded domain when Aut($\Omega$) is not compact? - and do u mind just clarifying what you mean by "it contains"? – 2017-02-17
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    For the nondegenerate annulus $Aut$ is $\cong U(1)$ and hence compact. This is the only case (up to conformal equivalence) when $Aut$ is compact but infinite. An example of a finite but nontrivial would be a disk with two points removed. An example of trivial would be a disk with three generic points removed. For "most" domains $Aut$ is trivial. – 2017-02-17

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