Automorphisms action on $\mathbb C$ If we know that the $$Aut \ \mathbb C=\{z\mapsto az+b:a\in \mathbb C^*,b\in \mathbb C\}$$ What are the subgroups of $Aut \ \mathbb C$ that act withiut fixed points and properly discontinuously on $\mathbb C$? How to do that in steps?

Ok, so lets start by identifying what group we're actually working with. We're acting with the action of a fractional linear transformation right? So we are looking at subgroups of $PSL(2, \mathbb{C})$. Ok, so with the restrictions given we know that we have to be in the matrix group generated by
$$
\begin{bmatrix} a & b \\
0 & 1
\end{bmatrix}
$$
where $a \in \mathbb{C}^*, b \in \mathbb{C}$. 
Lets look at when we have a fixed point. A fixed point is when $f(z) = z$ so $az + b = z$. Thus we have a fixed point at $z = \frac{-b}{a-1}$ and thus we must have $z$ be the point at infinity. This also means that $a$ must be 1. Now we've reduced our group to unitriangular matrices whose upper right entry is in $\mathbb{C}$. But with only this condition we may have matrices $A, B$ where the upper right element is different and not an integer multiple of eachother. 
Now we need to verify that this action is properly discontinuous, that is that if for all $z \in \mathbb{C}$ there is an open neighborhood $U_z$ of $z$ such that $g \in G |U_z \cap g*U_z = \emptyset$ is finite. Clearly we can't take all $b \in \mathbb{C}$ then. So far our group is the action of translation by some amount.  How can we possibly restrict out group? Well for any $b$ we pick, we will get $b\mathbb{Z}$ as the cyclic subgroup, suppose we pick $b, b' \in \mathbb{C}$, $b \neq b', b,b' \neq 0$ where $b, b'$ are not rational multples, then we can multiply elements to get infinitely many elements who only translate by too small an amount. So we must fix some $b \in \mathbb{C}$ to generate our group. 
Putting this all together, we have that for any fixed $b$ in $\mathbb{C}$ we get a matrix group of unitriangular 2x2 matrices with integer multiples of $b$ in the top right, this group is canonically isomorphic to unitriangular matrices with entries in $\mathbb{Z}$.
